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lands that are 12 miles apart (see figure). For a trip between the islands, there is enough fuel for a 20-mile trip. 6 7 0 0 f t 1.10 radians 3250 ft 0.84 radian (a) Find the angle between the two lines of sight to the peaks. (b) Approximate the amount of vertical climb that is necessary to reach the summit of each peak. 2. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long. Island 1 Island 2 12 mi Not drawn to scale (a) Explain why the region in which the boat can travel is bounded by an ellipse. (b) Let 0, 0 represent the center of the ellipse. Find the coordinates of each island. (c) The boat travels from one island, straight past the other island to the vertex of the ellipse, and back to the second island. How many miles does the boat travel? Use your answer to find the coordinates of the vertex. (d) Use the results from parts (b) and (c) to write an equation for the ellipse that bounds the region in which the boat can travel. 6. Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points 10, 2 2, 2 and is 6. 7. Prove that the graph of the equation (a) Find an equation that models the shape of the room. Ax2 Cy2 Dx Ey F 0 (b) How far apart are the two foci? (c) What is the area of the floor of the room? (The area of an ellipse is A ab. ) 3. Find the equation(s) of all parabolas that have the -axis as the axis of symmetry and focus at the origin. x 4. Find the area of the square inscribed in the ellipse below. y x 2 a 2 + y 2 b2 = 1 x is one of the following (except in degenerate cases). Conic (a) Circle (b) Parabola (c) Ellipse Condition A C A 0 or C 0 AC > 0 (but not both) (d) Hyperbola AC < 0 8. The following sets of parametric equations model projectile motion. x v0 cos t x v0 cos t y v0 sin t (a) Under what circumstances would you use each model? y h v0 sin t 16t2 (b) Eliminate the parameter for each set of equations. (c) In which case is the path of the moving object not affected by a change in the velocity ? Explain. v 809 333202_100R.qxd 12/8/05 9:12 AM Page 810 9. As t equations increases, the ellipse given by the parametric 4 x cos t and y 2 sin t is traced out counterclockwise. Find a parametric representation for which the same ellipse is traced out clockwise. 10. A hypocycloid has the parametric equations x a b cos t b cosa b b t and y a b sin t b sina b b t. (a) Use a graphing utility to graph the hypocycloid for each value of and Describe each graph. a 3, b 1 a 10, b 1 a 4, b 3 a b. a 2, b 1 a 4, b 1 a 3, b 2 (b) (d) (c) (e) (f) 11. The curve given by the parametric equations x 1 t2 1 t2 and y t1 t2 1 t2 is called a strophoid. (a) Find a rectangular equation of the strophoid. (b) Find a polar equation of the strophoid. (c) Use a graphing utility to graph the strophoid. 12. The rose curves described in this chapter are of the form r a cos n or r a sin n n is a positive integer that is greater than or equal to and for some noninteger values of Describe the where 2. Use a graphing utility to graph r a sin n graphs. r a cos n n. −3 4 −4 12( r = e cos − 2 cos 4 + sin 5 θ θ θ ( FIGURE FOR 14 (a) The graph above was produced using 0 ≤ ≤ 2. Does this show the entire graph? Explain your reasoning. (b) Approximate the maximum -value of the graph. Does instead of r this value change if you use 0 ≤ ≤ 2? 0 ≤ ≤ 4 Explain. 15. Use a graphing utility to graph the polar equation r cos 5 n cos 0 ≤ ≤ n 5 to n 5. for the integers for As you graph these equations, you should see the graph change shape from a heart to a bell. Write a short paragraph n explaining what values of produce the heart portion of the n curve and what values of produce the bell portion. 16. The planets travel in elliptical orbits with the sun at one focus. The polar equation of the orbit of a planet with one focus at the pole and major axis of length 2a is r 1 e 2a 1 e cos e where is the eccentricity. The minimum distance (perihelion) from the sun to a planet is and The the maximum distance (aphelion) is length of the major axis for the planet Neptune is a 9.000 109 kilometers and the eccentricity is e 0.0086. The length of the major axis for the planet Pluto is kilometers and the eccentricity is r a1 e r a1 e. a 10.813 109 e 0.2488. 13. What conic section is represented by the polar equation (a) Find the polar equation of the orbit of each planet. r a sin b cos ? 14. The graph of the polar equation r ecos 2 cos 4 sin5 12 is called the butterfly curve, as shown in the figure. 810 (b) Find the perihelion and aphelion distances for each planet. (c) Use a graphing utility to graph the polar equation of each planet’s orbit in the same viewing window. (d) Do the orbits of the two planets intersect? Will the two planets ever collide? Why or why not? (e) Is Pluto ever closer to the sun than Neptune? Why is Pluto called the ninth planet and Neptune the eighth planet? 333202_0A01.qxd 12/6/05 2:09 PM Page A1 Appendix A Review of Fundamental Concepts of Algebra A.1 Real Numbers and Their Properties What you should learn • Represent and classify real numbers. • Order real numbers and use inequalities. • Find the absolute values of real numbers and find the distance between two real numbers. • Evaluate algebraic expressions. • Use the basic rules and properties of algebra. Why you should learn it Real numbers are used to represent many real-life quantities. For example, in Exercise 65 on page A9, you will use real numbers to represent the federal deficit. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. Real numbers Irrational numbers Rational numbers Integers Noninteger fractions (positive and negative) Negative integers Whole numbers Natural numbers Zero FIGURE A.1 Subsets of real numbers Real Numbers Real numbers are used in everyday life to describe quantities such as age, miles per gallon, and population. Real numbers are represented by symbols such as 5, 9, 0, 4 3 , 0.666 . . . , 28.21, ,2, and 332. Here are some important subsets (each member of subset B is also a member of set A) of the real numbers. The three dots, called ellipsis points, indicate that the pattern continues indefinitely. 1, 2, 3, 4, . . . 0, 1, 2, 3, 4, . . . . . . , 3, 2, 1, 0, 1, 2, 3, . . . Set of natural numbers Set of whole numbers Set of integers A real number is rational if it can be written as the ratio where For instance, the numbers q 0. pq of two integers, 1 3 0.3333 . . . 0.3, 1 8 0.125, and 125 111 1.126126 . . . 1.126 3.145 are rational. The decimal representation of a rational number either repeats as in . 173 A real number that cannot be written 55 as the ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers or terminates as in 0.5 1 2 2 1.4142135 . . . 1.41 and 3.1415926 . . . 3.14 are irrational. (The symbol means “is approximately equal to.”) Figure A.1 shows subsets of real numbers and their relationships to each other. Real numbers are represented graphically by a real number line. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure A.2. The term nonnegative describes a number that is either positive or zero. Negative direction −4 −3 −2 −1 0 1 2 3 4 Positive direction Origin FIGURE A.2 The real number line As illustrated in Figure A.3, there is a one-to-one correspondence between real numbers and points on the real number line. − 5 3 0.75 π −3 −2 −1 0 1 2 3 Every real number corresponds to exactly one point on the real number line. FIGURE A.3 One-to-one −2.4 2 −3 −2 −1 0 1 2 3 Every point on the real number line corresponds to exactly one real number. A1 333202_0A01.qxd 12/6/05 2:09 PM Page A2 A2 Appendix A Review of Fundamental Concepts of Algebra Ordering Real Numbers One important property of real numbers is that they are ordered. is less than is denoted by the inequality Definition of Order on the Real Number Line a b a If and are real numbers, a b of and is greater than b be described by saying that a ≤ b is less than or equal to a means that inequality b ≥ a b means that are inequality symbols. is greater than or equal to b if a < b. a a. b, b a is positive. The order This relationship can also and writing b > a. The and the inequality The symbols <, >, ≤, and ≥ a −1 0 1 b 2 FIGURE A.4 the left of b. a < b if and only if a lies to x ≤ 2 0 1 2 3 4 FIGURE A.5 −2 −1 FIGURE A. Geometrically, this definition implies that a < b if and only if a lies to the left of on the real number line, as shown in Figure A.4. b Example 1 Interpreting Inequalities Describe the subset of real numbers represented by each inequality. a. x ≤ 2 b. 2 ≤ x < 3 Solution a. The inequality x ≤ 2 shown in Figure A.5. denotes all real numbers less than or equal to 2, as b. The inequality 2 ≤ x < 3 inequality” denotes all real numbers between including 3, as shown in Figure A.6. means that x ≥ 2 2 and x < 3. and 3, including This “double but not 2 Now try Exercise 19. Inequalities can be used to describe subsets of real numbers called intervals. are the endpoints of In the bounded intervals below, the real numbers each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open interval are not included in the interval. and a b Bounded Intervals on the Real Number Line Notation a, b Interval Type Closed Inequality a ≤ x ≤ b The reason that the four types of intervals at the right are called bounded is that each has a finite
length. An interval that does not have a finite length is unbounded (see page A3). a, b a, b a, b Open Graph 333202_0A01.qxd 12/6/05 2:09 PM Page A3 Appendix A.1 Real Numbers and Their Properties A3 or , Note that whenever you write intervals containing you always use a parenthesis and never a bracket. This is because these symbols are never an endpoint of an interval and therefore not included in the interval. The symbols negative infinity, do not represent real numbers. They are simply convenient symbols used to describe the 1, unboundedness of an interval such as positive infinity, and , 3. or , , Unbounded Intervals on the Real Number Line Notation a, a, , b , b Interval Type Open Open Inequality , Entire real line < x < Graph a a b b x x x x x Example 2 Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. c. All x in the interval b. m is at least 3, 5 3. Solution a. The statement “c is at most 2” can be represented by b. The statement “m is at least 3, 5 c. “All x in the interval 3 ” can be represented by ” can be represented by c ≤ 2. m ≥ 3. 3 < x ≤ 5. Now try Exercise 31. Example 3 Interpreting Intervals Give a verbal description of each interval. a. 1, 0 b. 2, c. , 0 Solution a. This interval consists of all real numbers that are greater than 1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 29. The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a b, a < b, or a > b. Law of Trichotomy 333202_0A01.qxd 12/6/05 2:09 PM Page A4 A4 Appendix A Review of Fundamental Concepts of Algebra Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Definition of Absolute Value If is a real number, then the absolute value of a a is a a, if a ≥ 0 a, if a < 0 . Notice in this definition that the absolute value of a real number is never negative. For instance, if The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute value is 0. So, 5 5 5. a 5, 0 0. then Evaluating the Absolute Value of a Number Example 4 x x Evaluate for (a) x > 0 and (b) x < 0. Solution a. If x > 0, then x x and b. If x < 0, then x x and x x x x x x 1. x x 1. Now try Exercise 47. Properties of Absolute Values a ≥ 0 ab ab 1. 3. 2. 4. a a b a a b, b 0 Absolute value can be used to define the distance between two points on the 3 and 4 is 7 real number line. For instance, the distance between −3 −2 −1 0 1 2 3 4 FIGURE A.7 and 4 is 7. The distance between 3 3 4 7 7 as shown in Figure A.7. Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is da, b b a a b. 333202_0A01.qxd 12/6/05 2:09 PM Page A5 Appendix A.1 Real Numbers and Their Properties A5 Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers. The letters are variables, and combinations of letters and numbers are algebraic expressions. Here are a few examples of algebraic expressions. 5x, 2x 3, 4 x 2 2 , 7x y Definition of an Algebraic Expression An algebraic expression is a collection of letters (variables) and real numbers (constants) combined using the operations of addition, subtraction, multiplication, division, and exponentiation. The terms of an algebraic expression are those parts that are separated by addition. For example, x 2 5x 8 x 2 5x 8 x 2 are the variable terms and 8 is the constant term. has three terms: The numerical factor of a variable term is the coefficient of the variable term. For instance, the coefficient of and the coefficient of 5x 5, and is 1. x 2 is 5x To evaluate an algebraic expression, substitute numerical values for each of the variables in the expression. Here are two examples. Expression 3x 5 3x 2 2x 1 Value of Variable x 3 x 1 Substitute 33 5 312 21 1 Value of Expression 9 5 4 3 2 1 0 When an algebraic expression is evaluated, the Substitution Principle is then a can be replaced by b in any expression used. It states that “If involving a.” In the first evaluation shown above, for instance, 3 is substituted for x in the expression 3x 5. a b, Basic Rules of Algebra There are four arithmetic operations with real numbers: addition, multiplication, subtraction, and division, denoted by the symbols or /. Of these, addition and multiplication are the two primary operations. Subtraction and division are the inverse operations of addition and multiplication, respectively. , and , , or Definitions of Subtraction and Division Subtraction: Add the opposite. Division: Multiply by the reciprocal. a b a b If b 0, then ab a1 b is the additive inverse (or opposite) of b, and In these definitions, is the multiplicative inverse (or reciprocal) of b. In the fractional form a is the numerator of the fraction and b is the denominator. b . a b 1b ab, 333202_0A01.qxd 12/6/05 2:09 PM Page A6 A6 Appendix A Review of Fundamental Concepts of Algebra Because the properties of real numbers below are true for variables and algebraic expressions as well as for real numbers, they are often called the Basic Rules of Algebra. Try to formulate a verbal description of each property. For instance, the first property states that the order in which two real numbers are added does not affect their sum. Basic Rules of Algebra c Let b, a, and be real numbers, variables, or algebraic expressions. Property Commutative Property of Addition: Commutative Property of Multiplication: Associative Property of Addition: Associative Property of Multiplication: Distributive Properties: Additive Identity Property: Multiplicative Identity Property: Additive Inverse Property: Multiplicative Inverse Property: a b b a ab ba a b c a b c abc abc ab c ab ac a bc ac bc , Example 4x x 2 x 2 4x 4 xx 2 x 24 x x 5 x 2 x 5 x 2 2x 3y8 2x3y 8 3x5 2x 3x 5 3x 2x y 8y y y 8 y 5y 2 0 5y2 4x 21 4x 2 5x3 5x3 0 x 2 4 1 1 x 2 4 Because subtraction is defined as “adding the opposite,” the Distributive Properties are also true for subtraction. For instance, the “subtraction form” of ab c ab ac is ab c ab ac. Properties of Negation and Equality Let a and b be real numbers, variables, or algebraic expressions. Notice the difference between the opposite of a number and a negative number. If is already a, negative, then its opposite, is positive. For instance, if a 5, then a a (5) 5. Property 1. 2. 3. 4. 5. 1a a a a ab ab ab ab ab a b a b 6. If 7. If 8. If 9. If then then a b, a b, a ± c b ± c, ac bc and a ± c b ± c. ac bc. then a b. c 0, then a b. Example 17 7 6 6 53 5 3 53 2x 2x x 8 x 8 x 8 1 3 0.5 3 2 42 2 16 2 1.4 1 7 5 3x 3 4 ⇒ x 4 1 ⇒ 1.4 7 5 333202_0A01.qxd 12/6/05 2:09 PM Page A7 Appendix A.1 Real Numbers and Their Properties A7 The “or” in the Zero-Factor Property includes the possibility that either or both factors may be zero. This is an inclusive or, and it is the way the word “or” is generally used in mathematics. Properties of Zero Let a and b be real numbers, variables, or algebraic expressions. 1. a 0 a and a 0 a 2. a 0 0 3. 0 a 0, a 0 4. a 0 is undefined. 5. Zero-Factor Property: If ab 0, then a 0 or b 0. Properties and Operations of Fractions Let a, b, c, and d be real numbers, variables, or algebraic expressions such that d 0. b 0 and 1. Equivalent Fractions: 2. Rules of Signs and a b a b if and only if ad bc. 3. Generate Equivalent Fractions: a b ac bc , c 0 4. Add or Subtract with Like Denominators. Add or Subtract with Unlike Denominators: a b ± c d ad ± bc bd 6. Multiply Fractions: a b c d ac bd 7. Divide Fractions: a b c d a b d c ad bc , c 0 Example 5 Properties and Operations of Fractions In Property 1 of fractions, the phrase “if and only if” implies two statements. One statement cd, ad bc. is: If The other statement is: If ad bc, where d 0, b 0 then ab cd. ab then and a. Equivalent fractions: 3 x 3 5 x 5 3x 15 b. Divide fractions: c. Add fractions with unlike denominators: x 3 2x 5 Now try Exercise 103. 3 2 7 x 5 x 3 2x 3 5 2 7 x 3 11x 15 14 3x ab c, If a, b, and c are integers such that then a and b are factors or divisors of c. A prime number is an integer that has exactly two positive factors — itself and 1—such as 2, 3, 5, 7, and 11. The numbers 4, 6, 8, 9, and 10 are composite because each can be written as the product of two or more prime numbers. The number 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be written as the product of prime numbers in precisely one way (disregarding order). For instance, the prime factorization of 24 is 24 2 2 2 3. 333202_0A01.qxd 12/6/05 2:09 PM Page A8 A8 Appendix A Review of Fundamental Concepts of Algebra A.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. A real number is ________ if it can be written as the ratio of two integers, where p q q 0. 2. ________ numbers have infinite nonrepeating decimal representations. 3. The distance between a point on the real number line and the origin is the ________ ________ of the real number. 4. A number that can be written as the product of two or more prime numbers is called a ________ number. 5. An integer that has exactly two positive factors, the integer itself and 1, is called a ________ number. 6. An algebraic expression is a collection of letters called ________ and real numbers called ________. 7. The ________ of an algebraic expression are those parts separated by addition. 8. The numerical factor of a variable term is the ________ of the variable term. 9. The ________ ________ states that if ab 0, then a 0 or b 0. In Exercises 1– 6, determine w
hich numbers in the set are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and (e) irrational numbers. 1. 2. 3. 4. 5. 6. 3, 2, 0, 1, 4, 2, 11 3, 0, 3.12, 5 4 , 3, 12, 5 2, 5, 2 9, 7 5, 7, 7 2.01, 0.666 . . . , 13, 0.010110111 . . . , 1, 6 2.3030030003 . . . , 0.7575, 4.63, 10, 475, , 1 3, 1 2 25, 17, 12 2, 7.5, 1, 8, 22 5 , 9, 3.12, 1 11.1, 3, 6 , 13 7, 2 In Exercises 19–30, (a) give a verbal description of the subset of real numbers represented by the inequality or the interval, (b) sketch the subset on the real number line, and (c) state whether the interval is bounded or unbounded. 19. 21. 23. 25. 27. 29. x ≤ 5 x < 0 4, 5 20. 22. 24. 26. 28. 30, 2 In Exercises 7–10, use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. In Exercises 31–38, use inequality notation to describe the set. 7. 9. 5 8 41 333 8. 10. 1 3 6 11 31. All x in the interval 32. All y in the interval 2, 4 6, 0 33. y is nonnegative. 34. y is no more than 25. In Exercises 11 and 12, approximate the numbers and place the correct symbol (< or >) between them. 35. 36. t k is at least 10 and at most 22. is less than 5 but no less than 11. 12. −2 −1 0 1 2 3 −7 −6 −5 −4 −3 −2 −1 4 0 In Exercises 13–18, plot the two real numbers on the real number line. Then place the appropriate inequality symbol (< or >) between them. 13. 15. 17. 4, 8 3 2, 7 6, 2 5 3 14. 16. 18. 3.5, 1 1, 16 3 8 7, 3 7 37. The dog’s weight W 3. is more than 65 pounds. r 38. The annual rate of inflation but no more than 5%. is expected to be at least 2.5% In Exercises 39–48, evaluate the expression. 39. 41. 43. 45. 47. 10 40. 42. 44. 46. 48. 0 4 1 3 3 33 x 1 x 1 , x > 1 333202_0A01.qxd 12/6/05 2:09 PM Page A9 Appendix A.1 Real Numbers and Their Properties A9 In Exercises 49–54, place the correct symbol (<, >, or =) between the pair of real numbers. (a) Complete the Expenditures . table. Hint: Find Receipts – 49. 50. 51. 52. 53. 54. 33 44 55 66 22 (2)2 In Exercises 55–60, find the distance between a and b. 55. 56. 57. 58. 59. 60. a 126, b 75 a 126, b 75 a 5 2, b 0 4, b 11 a 1 5 , b 112 a 16 a 9.34, b 5.65 75 4 Budget Variance In Exercises 61–64, the accounting department of a sports drink bottling company is checking to see whether the actual expenses of a department differ from the budgeted expenses by more than $500 or by more than 5%. Fill in the missing parts of the table, and determine whether each actual expense passes the “budget variance test.” Budgeted b Expense, Actual Expense, a 61. Wages $112,700 $113,356 62. Utilities $9,400 $9,772 63. Taxes $37,640 $37,335 64. Insurance $2,575 $2,613 a b 0.05b 65. Federal Deficit The bar graph shows the federal government receipts (in billions of dollars) for selected years from 1960 through 2000. (Source: U.S. Office of Management and Budget ( 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 2025.2 1032.0 192.8 92.5 1960 1970 517.1 1980 Year 1990 2000 Year 1960 1970 1980 1990 2000 Expenditures (in billions) $92.2 $195.6 $590.9 $1253.2 $1788.8 Surplus or deficit (in billions) (b) Use the table in part (a) to construct a bar graph showing the magnitude of the surplus or deficit for each year. 66. Veterans The table shows the number of living veterans (in thousands) in the United States in 2002 by age group. Construct a circle graph showing the percent of living veterans by age group as a fraction of the total number of (Source: Department of Veteran Affairs) living veterans. Age group Number of veterans Under 35 35– 44 45–54 55– 64 65 and older 2213 3290 4666 5665 9784 In Exercises 67–72, use absolute value notation to describe the situation. x 67. The distance between and 5 is no more than 3. 68. The distance between and x 10 is at least 6. 69. 70. y y is at least six units from 0. is at most two units from a. 71. While traveling on the Pennsylvania Turnpike, you pass milepost 326 near Valley Forge, then milepost 351 near Philadelphia. How many miles do you travel during that time period? 72. The temperature in Chicago, Illinois was 48 last night at midnight, then 82 at noon today. What was the change in temperature over the 12-hour period? 333202_0A01.qxd 12/6/05 2:09 PM Page A10 A10 Appendix A Review of Fundamental Concepts of Algebra In Exercises 73–78, identify the terms. Then identify the coefficients of the variable terms of the expression. (b) Use the result from part (a) to make a conjecture about the value of 5n as approaches 0. n 73. 75. 77. 7x 4 3x2 8x 11 4x3 x 2 5 74. 76. 78. 6x3 5x 33x2 1 3x4 x2 4 106. (a) Use a calculator to complete the table. 1 10 100 10,000 100,000 n 5n In Exercises 79–84, evaluate the expression for each value of x. (If not possible, state the reason.) Expression 4x 6 9 7x x 2 3x 4 x 2 5x 4 x 1 x 1 x x 2 79. 80. 81. 82. 83. 84. Values (a) (a) (a) (ab) (b) (b) (ba) x 1 (b) x 1 (a) x 2 (b) x 2 In Exercises 85–96, identify the rule(s) of algebra illustrated by the statement. 85. x 9 9 x 86. 2 1 2 1 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. h 6 h 6 1, 1 h 6 x 3 x 3 0 2x 5x z x 5 x x y 10 x y 10 x3y x 3y 3xy 3t 4 3 t 3 4 7 12 1 1 7 7 712 1 12 12 In Exercises 97–104, perform the operation(s). (Write fractional answers in simplest form.) 97. 99. 101. 103. 1 6 5 3 16 16 5 5 12 8 12 1 4 2x x 3 4 98. 100. 102. 104. 6 7 10 11 4 7 6 33 6 4 5x 2 6 9 8 13 66 105. (a) Use a calculator to complete the table. 1 0.5 0.01 0.0001 0.000001 n 5n (b) Use the result from part (a) to make a conjecture n as increases without bound. about the value of 5n Synthesis True or False? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. 107. If a < b, 108. Because , < then where a b 0. , then c a b u v and . c c b a u v, 109. Exploration Consider where u v 0. (a) Are the values of the expressions always equal? If not, under what conditions are they unequal? (b) If the two expressions are not equal for certain values is one of the expressions always greater and v, u of than the other? Explain. 110. Think About It Is there a difference between saying that a real number is positive and saying that a real number is nonnegative? Explain. 111. Think About It Because every even number is divisible is it possible that there exist any even prime by 2, numbers? Explain. 112. Writing Describe the differences among the sets of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. In Exercises 113 and 114, use the real numbers A, B, and C shown on the number line. Determine the sign of each expression. C B 0 A 113. (a) (b) A B A 114. (a) (b) C A C a a 115. Writing Can it ever be true that Explain. number a? for a real 333202_0A02.qxd 12/6/05 2:12 PM Page A11 A.2 Exponents and Radicals Appendix A.2 Exponents and Radicals A11 What you should learn • Use properties of exponents. • Use scientific notation to represent real numbers. • Use properties of radicals. • Simplify and combine radicals. • Rationalize denominators and numerators. • Use properties of rational exponents. Why you should learn it Real numbers and algebraic expressions are often written with exponents and radicals. For instance, in Exercise 105 on page A22, you will use an expression involving rational exponents to find the time required for a funnel to empty for different water heights. Te c h n o l o g y You can use a calculator to evaluate exponential expressions. When doing so, it is important to know when to use parentheses because the calculator follows the order of operations. For instance, evaluate 24 as follows Scientific: 2 y x 4 Graphing: 2 > 4 ENTER The display will be 16. If you omit the parentheses, the display will be 16. Integer Exponents Repeated multiplication can be written in exponential form. Repeated Multiplication a a a a a 444 2x2x2x2x Exponential Form a5 43 2x4 Exponential Notation n is a real number and If an a a a . . . a a is a positive integer, then n factors is the exponent and n where the th power.” n a is the base. The expression an is read “ to a An exponent can also be negative. In Property 3 below, be sure you see how to use a negative exponent. Properties of Exponents Let and be real numbers, variables, or algebraic expressions, and let and be integers. (All denominators and bases are nonzero.) a n b m Property aman amn Example 32 34 324 36 729 1. 2. 3. 4. 5. amn am an an 1 an a0 1, abm ambm n 1 a a 0 6. amn amn m a am bm b a2 a2 a2 7. 8. 4 x7 4 x3 x 7 x4 1 y4 1 y4 y x 2 10 1 5x3 53x3 125x3 y34 y3(4) y12 1 y12 3 23 x3 2 8 x3 x 22 22 22 4 333202_0A02.qxd 12/6/05 2:12 PM Page A12 A12 Appendix A Review of Fundamental Concepts of Algebra In and 24. It is important to recognize the difference between expressions such as the parentheses indicate that the exponent applies to the exponent applies 24 24, the negative sign as well as to the 2, but in only to the 2. So, 24 16. The properties of exponents listed on the preceding page apply to all integers 24 24, 24 16 and m and n, not just to positive integers as shown in the examples in this section. Example 1 Using Properties of Exponents Use the properties of exponents to simplify each expression. a. 3ab44ab3 b. 2xy 23 c. 3a4a20 d. 5x3 y 2 Solution a. 3ab44ab3 34aab4b3 12a2b 2xy 23 23x3y 23 8x3y6 3a4a20 3a1 3a, 52x32 5x3 25x . c. d. Now try Exercise 25. Example 2 Rewriting with Positive Exponents Rewrite each expression with positive exponents. a. x1 b. 1 3x2 c. 12a3b4 4a2b d. 3x2 y 2 c. a. b. 1 3x2 12a3b4 4a2b 2 3x2 y Solution x1 1 x 1x2 x 2 3 3 12a3 a2 4b b4 32x22 y2 32x4 y2 y2 32x4 y2 9x 4 d. Property 3 The exponent 2 does not apply to 3. 3a5 b5 Properties 3 and 1 Properties 5 and 7 Property 6 Property 3 Simplify. Now try Exercise 33. Rarely in algebra is there only one way to solve a problem. Don’t be concerned if the steps you use to solve a problem are not exactly the same as the steps presented in this text. The important thing is to use steps that you understand and, of course, steps that are justified by the rules of alge
bra. For instance, you might prefer the following steps for Example 2(d). 3x 2 y 2 y 3x 22 y 2 9x4 Note how Property 3 is used in the first step of this solution. The fractional form of this property is m a b b a m . 333202_0A02.qxd 12/6/05 2:12 PM Page A13 Appendix A.2 Exponents and Radicals A13 Scientific Notation Exponents provide an efficient way of writing and computing with very large (or very small) numbers. For instance, there are about 359 billion billion gallons of water on Earth—that is, 359 followed by 18 zeros. 359,000,000,000,000,000,000 It is convenient to write such numbers in scientific notation. This notation has is an integer. So, the number of the form gallons of water on Earth can be written in scientific notation as 1 ≤ c < 10 ± c 10n, where and n 3.59 100,000,000,000,000,000,000 3.59 1020. The positive exponent 20 indicates that the number is large (10 or more) and that the decimal point has been moved 20 places. A negative exponent indicates that the number is small (less than 1). For instance, the mass (in grams) of one electron is approximately 9.0 1028 0.0000000000000000000000000009. 28 decimal places Example 3 Scientific Notation Write each number in scientific notation. a. 0.0000782 b. 836,100,000 Solution a. 0.0000782 7.82 105 Now try Exercise 37. b. 836,100,000 8.361 108 Example 4 Decimal Notation Write each number in decimal notation. a. 9.36 106 b. 1.345 102 Solution a. 9.36 106 0.00000936 Now try Exercise 41. b. 1.345 102 134.5 Te c h n o l o g y Most calculators automatically switch to scientific notation when they are showing large (or small) numbers that exceed the display range. To enter numbers in scientific notation, your calculator should have an expo- nential entry key labeled EE or EXP . Consult the user’s guide for your calculator for instructions on keystrokes and how numbers in scientific notation are displayed. 333202_0A02.qxd 12/6/05 2:12 PM Page A14 A14 Appendix A Review of Fundamental Concepts of Algebra Radicals and Their Properties A square root of a number is one of its two equal factors. For example, 5 is a square root of 25 because 5 is one of the two equal factors of 25. In a similar way, a cube root of a number is one of its three equal factors, as in 125 53. Definition of nth Root of a Number Let a and b be real numbers and let n ≥ 2 be a positive integer. If a bn then b is an nth root of a. If root is a cube root. n 2, the root is a square root. If n 3, the Some numbers have more than one nth root. For example, both 5 and 25, square roots of 25. The principal square root of 25, written as root, 5. The principal nth root of a number is defined as follows. 5 are is the positive Principal nth Root of a Number Let a be a real number that has at least one nth root. The principal nth root of a is the nth root that has the same sign as a. It is denoted by a radical symbol na. Principal nth root The positive integer n is the index of the radical, and the number a is the radicand. If plural of index is indices.) omit the index and write rather than n 2, 2a. (The a A common misunderstanding is that the square root sign implies both negative and positive roots. This is not correct. The square root sign implies only a positive root. When a negative root is needed, you must use the negative sign with the square root sign. Incorrect: 4 ±2 Correct: 4 2 and 4 2 Example 5 Evaluating Expressions Involving Radicals a. b. c. d. e. because 62 36. 36 6 36 6 3125 5 4 64 532 2 481 to the fourth power to produce because because 36 62 6 6. because 5 3 53 125 43 64 4 25 32. . 81. is not a real number because there is no real number that can be raised Now try Exercise 51. 333202_0A02.qxd 12/6/05 2:12 PM Page A15 Appendix A.2 Exponents and Radicals A15 Here are some generalizations about the nth roots of real numbers. Generalizations About nth Roots of Real Numbers Real Number a Integer n a > 0 n > 0, is even. Root(s) of a na na, Example 481 3, 481 3 a > 0 or a < 0 n is odd. na 38 2 a < 0 a 0 n is even. No real roots 4 is not a real number. n is even or odd. n0 0 50 0 Integers such as 1, 4, 9, 16, 25, and 36 are called perfect squares because they have integer square roots. Similarly, integers such as 1, 8, 27, 64, and 125 are called perfect cubes because they have integer cube roots. Properties of Radicals Let a and b be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let m and n be positive integers. Property nam nam na nb nab na na nb b m na mna nan a , 1. 2. 3. 4. 5. b 0 6. For n even, For n odd, nan a. nan a. 43 Example 382 382 22 4 5 7 5 7 35 427 427 49 9 310 610 32 3 122 12 12 3123 12 A common special case of Property 6 is a2 a. Example 6 Using Properties of Radicals Use the properties of radicals to simplify each expression. a. 8 2 35 3 b. c. 3x3 d. 6y6 Solution a. 8 2 8 2 16 4 353 5 3x3 x 6y6 y b. c. d. Now try Exercise 61. 333202_0A02.qxd 12/6/05 2:12 PM Page A16 A16 Appendix A Review of Fundamental Concepts of Algebra When you simplify a radical, it is important that both expressions are defined for the same values of the variable. For instance, in Example 7(b), 75x3 5x3x are both defined only for nonnegative values of Similarly, in Example 7(c), and are both defined for all real values of x. 45x4 5x and x. Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied. 1. All possible factors have been removed from the radical. 2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator). 3. The index of the radical is reduced. To simplify a radical, factor the radicand into factors whose exponents are multiples of the index. The roots of these factors are written outside the radical, and the “leftover” factors make up the new radicand. Example 7 Simplifying Even Roots Perfect 4th power Leftover factor a. 448 416 3 424 3 2 43 Perfect square Leftover factor b. c. 75x3 25x 2 3x 5x2 3x 5x3x 45x4 5x 5x Find largest square factor. Find root of perfect square. Now try Exercise 63(a). Example 8 Simplifying Odd Roots Perfect cube Leftover factor a. 324 38 3 323 3 2 33 Perfect cube Leftover factor b. c. 324a4 38a3 3a 32a3 3a 2a 33a 340x6 38x6 5 32x23 5 2x2 35 Find largest cube factor. Find root of perfect cube. Find largest cube factor. Find root of perfect cube. Now try Exercise 63(b). 333202_0A02.qxd 12/6/05 2:12 PM Page A17 Appendix A.2 Exponents and Radicals A17 Radical expressions can be combined (added or subtracted) if they are like 2, are unlike radicals. To determine radicals—that is, if they have the same index and radicand. For instance, 32, 3 whether two radicals can be combined, you should first simplify each radical. are like radicals, but 1 and 2 and 2 2 Example 9 Combining Radicals a. 248 327 216 3 39 3 83 93 8 93 3 b. 316x 354x4 38 2x 327 x3 2x 2 32x 3x 32x 2 3x 32x Now try Exercise 71. Find square factors. Find square roots and multiply by coefficients. Combine like terms. Simplify. Find cube factors. Find cube roots. Combine like terms. Rationalizing Denominators and Numerators a bm To rationalize a denominator or numerator of the form multiply both numerator and denominator by a conjugate: a bm m is itself, perfect cube. a bm, and then the rationalizing factor for For cube roots, choose a rationalizing factor that generates a are conjugates of each other. If a bm a 0, m. or Example 10 Rationalizing Single-Term Denominators Rationalize the denominator of each expression. a. 5 23 b. 2 35 Solution 5 23 a. 5 23 53 23 53 6 3 3 3 is rationalizing factor. Multiply. Simplify. b. 2 35 2 35 2 352 353 2 325 5 352 352 352 is rationalizing factor. Multiply. Simplify. Now try Exercise 79. 333202_0A02.qxd 12/6/05 2:13 PM Page A18 A18 Appendix A Review of Fundamental Concepts of Algebra Example 11 Rationalizing a Denominator with Two Terms 23 7 33 37 73 77 23 7 32 72 23 7 9 7 23 7 2 3 7 Multiply numerator and denominator by conjugate of denominator. Use Distributive Property. Simplify. Square terms of denominator. Simplify. Now try Exercise 81. Sometimes it is necessary to rationalize the numerator of an expression. For instance, in Appendix A.4 you will use the technique shown in the next example to rationalize the numerator of an expression from calculus. Do not confuse the expression 5 7 with the expression 5 7. does not equal Similarly, equal x y. x y. x2 y2 In general, x y does not Example 12 Rationalizing a Numerator Multiply numerator and denominator by conjugate of numerator. Simplify. Square terms of numerator. 5 2 7 2 25 7 5 7 25 7 2 25 7 1 5 7 Simplify. Now try Exercise 85. Rational Exponents Definition of Rational Exponents If a is a real number and n is a positive integer such that the principal nth root of a exists, then a1n na, is the rational exponent of a. is defined as where a1n 1n Moreover, if m is a positive integer that has no common factor with n, then amn a1nm nam and amn am1n nam. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. 333202_0A02.qxd 12/6/05 2:13 PM Page A19 Appendix A.2 Exponents and Radicals A19 The numerator of a rational exponent denotes the power to which the base is raised, and the denominator denotes the index or the root to be taken. Power Index bmn nbm nbm When you are working with rational exponents, the properties of integer exponents still apply. For instance, 212213 2(12)(13) 256. Example 13 Changing from Radical to Exponential Form is not bmn is a real Rational exponents can be tricky, and you must remember that the expression nb defined unless number. This restriction produces some unusual-looking results. For instance, the number 813 is defined because 38 2, 826 68 but the number is undefined because is not a real number. a. b. c. 3 312 3xy5 23xy5 3xy(52) 2x 4x3 2xx34 2x1(34) 2x74 Now try Exercise 87. Te c h n o l o g y . For other There are four methods of evaluating radicals on most graphing calculato
rs. For square roots, you can use the square root key . For cube roots, you can use the cube root key 3 roots, you can first convert the radical to exponential form and then use the exponential key or you can use the xth root key . Consult the user’s guide x for your calculator for specific keystrokes. > , Example 14 Changing from Exponential to Radical Form a. b. c. x2 y 232 x2 y 23 x 2 y 23 2y34z14 2y3z14 2 4y3z a32 1 1 a32 a3 d. x 0.2 x15 5x Now try Exercise 89. Rational exponents are useful for evaluating roots of numbers on a calculator, for reducing the index of a radical, and for simplifying expressions in calculus. Example 15 Simplifying with Rational Exponents x 0 Reduce index. a. b. c. d. e. f. 3245 5324 24 1 24 1 16 5x533x34 15x(53)(34) 15x1112, 9a3 a39 a13 3a 3125 6125 653 536 512 5 2x 1432x 113 2x 1(43)(13) x 1 2 2x 1, x 1 x 112 x 1 x 112 x 132 x 10 x 132, x 112 x 112 x 1 Now try Exercise 99. 333202_0A02.qxd 12/6/05 2:13 PM Page A20 A20 Appendix A Review of Fundamental Concepts of Algebra A.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. In the exponential form nan, is the ________ and a is the ________. 2. A convenient way of writing very large or very small numbers is called ________ ________. 3. One of the two equal factors of a number is called a __________ __________ of the number. 4. The ________ ________ ________ of a number is the th root that has the same sign as n a, and is denoted by 5. In the radical form, and the number a na. na is called the ________. the positive integer n is called the ________ of the radical 6. When an expression involving radicals has all possible factors removed, radical-free denominators, and a reduced index, it is in ________ ________. 7. The expressions a bm and a bm are ________ of each other. 8. The process used to create a radical-free denominator is know as ________ the denominator. 9. In the expression bmn, m denotes the ________ to which the base is raised and denotes n the ________ or root to be taken. In Exercises 1 and 2, write the expression as a repeated multiplication problem. In Exercises 17–24, evaluate the expression for the given value of x. 1. 85 2. 27 In Exercises 3 and 4, write the expression using exponential notation. 3. 4. 4.94.94.94.94.94.9 1010101010 In Exercises 5–12, evaluate each expression. 5. (a) 6. (a) 7. (a) 8. (a) 9. (a) 10. (a) 11. (a) 12. (a) 32 3 55 52 330 23 322 3 44 34 41 4 32 22 31 21 31 31 22 (b) (b) (b) (b) 3 33 32 34 32 3 35 2 3 5 (b) 3225 (b) 20 (b) (b) 212 322 In Exercises 13–16, use a calculator to evaluate the expression. (If necessary, round your answer to three decimal places.) 4352 36 73 84103 43 34 13. 15. 16. 14. Expression 3x3 7x2 6x0 5x3 2x3 3x 4 4x2 5x3 17. 18. 19. 20. 21. 22. 23. 24. Value x 2 x 4 x 10 In Exercises 25–30, simplify each expression. 25. (a) 26. (a) 5z3 3x2 27. (a) 6y22y02 28. (a) z33z4 29. (a) 30. (a) 7x 2 x3 r 4 r 6 (b) (b) (b) (b) (b) (b) 5x4x2 4x30 3x5 x3 25y8 10y4 12x y3 9x y 4 33 4 y y In Exercises 31–36, rewrite each expression with positive exponents and simplify. x 50, 2x50, (b) (b) z 23z 21 x 5 2x 22 32. (a) 31. (a) x 0 333202_0A02.qxd 12/6/05 2:13 PM Page A21 Appendix A.2 Exponents and Radicals A21 33. (a) 2x234x31 34. (a) 4y28y4 35. (a) 3n 32n 36. (a) x 2 xn x3 xn (b) (b) (b) (b) 1 x 10 x3y4 3 5 a2 b2b a3 b3a a b 3 3 In Exercises 37– 40, write the number in scientific notation. 37. Land area of Earth: 57,300,000 square miles 38. Light year: 9,460,000,000,000 kilometers 39. Relative density of hydrogen: 0.0000899 gram per cubic centimeter 40. One micron (millionth of a meter): 0.00003937 inch In Exercises 41– 44, write the number in decimal notation. 41. Worldwide daily consumption of Coca-Cola: (Source: The Coca-Cola Company) ounces 4.568 109 54. (a) 55. (a) 56. (a) 10032 1 64 125 27 13 13 (b) (b) (b) 12 9 4 1 32 1 125 25 43 In Exercises 57– 60, use a calculator to approximate the number. (Round your answer to three decimal places.) 57. (a) 58. (a) 59. (a) 60. (a) 57 3452 12.41.8 7 4.13.2 2 (b) (b) (b) (b) 5273 6125 532.5 13 32 3 133 3 2 In Exercises 61 and 62, use the properties of radicals to simplify each expression. 61. (a) 62. (a) 343 12 3 (b) (b) 596x5 43x24 42. Interior temperature of the sun: 1.5 107 degrees Celsius In Exercises 63–74, simplify each radical expression. 43. Charge of an electron: 44. Width of a human hair: 1.6022 1019 9.0 105 meter coulomb In Exercises 45 and 46, evaluate each expression without using a calculator. 45. (a) 25 108 46. (a) 1.2 1075 103 (b) (b) 38 1015 6.0 108 3.0 103 In Exercises 47–50, use a calculator to evaluate each expression. (Round your answer to three decimal places.) 47. (a) (b) 48. (a) (b) 49. (a) 50. (a) 800 7501 0.11 365 67,000,000 93,000,000 0.0052 9.3 10636.1 104 2.414 1046 1.68 1055 4.5 109 2.65 10413 (b) (b) 36.3 104 9 104 63. (a) 64. (a) 8 316 27 65. (a) 72x3 66. (a) 54xy4 67. (a) 68. (a) 69. (a) 70. (a) 71. (a) 72. (a) 73. (a) 74. (a) 316x5 43x4y2 250 128 427 75 5x 3x 849x 14100x 3x 1 10x 1 x 3 7 5x3 7 (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) z3 354 75 4 182 32a4 b2 75x2y4 5160x8z4 1032 618 316 3 354 29y 10y 348x2 7 75x2 780x 2125x 11245x 3 945x 3 In Exercises 75–78, complete the statement with <, =, or >. 75. 77. 5 3 5 3 532 22 3 3 11 532 42 11 76. 78. In Exercises 51–56, evaluate each expression without using a calculator. In Exercises 79–82, rationalize the denominator of the expression. Then simplify your answer. 51. (a) 52. (a) 53. (a) 9 2713 3235 (b) (b) (b) 327 8 3632 16 81 34 79. 1 3 80. 5 10 333202_0A02.qxd 12/6/05 2:13 PM Page A22 A22 Appendix A Review of Fundamental Concepts of Algebra 104. Erosion A stream of water moving at the rate of v feet inches. Find per second can carry particles of size the size of the largest particle that can be carried by a 3 stream flowing at the rate of 4 foot per second. 0.03v 105. Mathematical Modeling A funnel is filled with water to h a height of centimeters. The formula t 0.031252 12 h52, 0 ≤ h ≤ 12 represents the amount of time take for the funnel to empty. t (in seconds) that it will (a) Use the table feature of a graphing utility to find the times required for the funnel to empty for water h 12 heights of centimeters. . . . h 0, h 1, h 2, (b) What value does appear to be approaching as the height of the water becomes closer and closer to 12 centimeters? t 106. Speed of Light The speed of light is approximately 11,180,000 miles per minute. The distance from the sun to Earth is approximately 93,000,000 miles. Find the time for light to travel from the sun to Earth. Synthesis True or False? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. 107. x k1 x x k 108. ank ank 109. Verify that exponents a0 1, a 0. aman amn. ) (Hint: Use the property of 110. Explain why each of the following pairs is not equal. 3x1 3 x a2b34 a6b7 4x2 2x (a) (c) (e) (b) y3 y2 y6 (d) (f) a b2 a2 b2 2 3 5 111. Exploration List all possible digits that occur in the units place of the square of a positive integer. Use that list to determine whether is an integer. 5233 112. Think About It Square 25 and note that the radical is eliminated from the denominator. Is this equivalent to rationalizing the denominator? Why or why not? the real number 81. 2 5 3 82. 3 5 6 In Exercises 83– 86, rationalize the numerator of the expression. Then simplify your answer. 83. 85. 8 2 5 3 3 84. 86. 2 3 7 3 4 In Exercises 87–94, fill in the missing form of the expression. Rational Exponent Form Radical Form 9 364 3215 14412 3216 24315 4813 1654 87. 88. 89. 90. 91. 92. 93. 94. In Exercises 95–98, perform the operations and simplify. 95. 97. 2x232 212x4 x3 x12 x32 x1 96. 98. x43y23 xy13 512 5x52 5x32 In Exercises 99 and 100, reduce the index of each radical. 99. (a) 100. (a) 432 6x3 (b) (b) 6(x 1)4 4(3x2)4 In Exercises 101 and 102, write each expression as a single radical. Then simplify your answer. 101. (a) 102. (a) 32 243x 1 (b) (b) 42x 310a7b T 103. Period of a Pendulum The period (in seconds) of a pendulum is T 2 L 32 L is the length of the pendulum (in feet). Find the where period of a pendulum whose length is 2 feet. The symbol indicates an example or exercise that highlights algebraic techniques specifically used in calculus. The symbol indicates an exercise or a part of an exercise in which you are instructed to use a graphing utility. 333202_0A03.qxd 12/6/05 2:14 PM Page A23 A.3 Polynomials and Factoring Appendix A.3 Polynomials and Factoring A23 What you should learn • Write polynomials in standard form. • Add, subtract, and multiply polynomials. • Use special products to multiply polynomials. • Remove common factors from polynomials. Polynomials 2x 5, The most common type of algebraic expression is the polynomial. Some examples are The first two are polynomials in x and the third is a polynomial in x and y. The terms of a polynomial in x have the form where a is the coefficient and k is the degree of the term. For instance, the polynomial 3x 4 7x 2 2x 4, 5x 2y 2 xy 3. ax k, and 2x 3 5x 2 1 2x 3 5x 2 0x 1 • Factor special polynomial has coefficients 2, 5, 0, and 1. forms. • Factor trinomials as the product of two binomials. • Factor polynomials by grouping. Why you should learn it Polynomials can be used to model and solve real-life problems. For instance, in Exercise 210 on page A34, a polynomial is used to model the stopping distance of an automobile. be real numbers and let n be a nonnegative integer. a0, a1, a2, . . . , an Definition of a Polynomial in x Let A polynomial in x is an expression of the form anx n an1x n1 . . . a1x a 0 The polynomial is of degree 0. n, where a0 and an is the constant term. an is the leading coefficient, Polynomials with one, two, and three terms are called monomials, binomials, and trinomials, respectively. In standard form, a polynomial is written with descending powers of x. Example 1 Writing Polynomials in Standard Form Polynomial 4x 2 5x 7 2 3x 4 9x 2 a. b. c. 8 Standard Form 5x 7 4x 2 3x 2 9x 2 4 8 8 8x 0 Degree 7 2 0 Now try
Exercise 11. A polynomial that has all zero coefficients is called the zero polynomial, denoted by 0. No degree is assigned to this particular polynomial. For polynomials in more than one variable, the degree of a term is the sum of the exponents of the variables in the term. The degree of the polynomial is the highest degree is of its terms. For instance, the degree of the polynomial 11 because the sum of the exponents in the last term is the greatest. The leading coefficient of the polynomial is the coefficient of the highest-degree term. Expressions are not polynomials if a variable is underneath a radical or if a polynomial expression (with degree greater than 0) is in the denominator of a term. The following expressions are not polynomials. 2x3y6 4xy x7y4 x3 3x x3 3x12 x2 5 x x2 5x1 The exponent “ 12 ” is not an integer. The exponent “ 1 ” is not a nonnegative integer. 333202_0A03.qxd 12/6/05 2:14 PM Page A24 A24 Appendix A Review of Fundamental Concepts of Algebra Operations with Polynomials You can add and subtract polynomials in much the same way you add and subtract real numbers. Simply add or subtract the like terms (terms having the same variables to the same powers) by adding their coefficients. For instance, 3xy 2 are like terms and their sum is 5xy 2 and 3xy 2 5xy 2 3 5xy 2 2xy 2. When an expression inside parentheses is preceded by a negative sign, remember to distribute the negative sign to each term inside the parentheses, as shown. x 2 x 3 x 2 x 3 Example 2 Sums and Differences of Polynomials a. 5x 3 7x 2 3 x 3 2x 2 x 8 5x 3 x 3 7x2 2x2 x 3 8 6x 3 5x 2 x 5 Group like terms. Combine like terms. b. 7x4 x 2 4x 2 3x4 4x 2 3x 7x4 x 2 4x 2 3x4 4x 2 3x 7x4 3x4 x2 4x2 4x 3x 2 4x4 3x 2 7x 2 Distributive Property Group like terms. Combine like terms. Now try Exercise 33. To find the product of two polynomials, use the left and right Distributive as a single quantity, you can 5x 7 Properties. For example, if you treat as follows. multiply 5x 7 3x 2 by 3x 25x 7 3x5x 7 25x 7 3x5x 3x7 25x 27 15x 2 21x 10x 14 Product of First terms Product of Outer terms Product of Inner terms Product of Last terms 15x 2 11x 14 Note in this FOIL Method (which can only be used to multiply two binomials) that the outer (O) and inner (I) terms are like terms and can be combined. Example 3 Finding a Product by the FOIL Method Use the FOIL Method to find the product of 2x 4 and x 5. Solution L F 2x 4x 5 2x2 10x 4x 20 O I 2x2 6x 20 Now try Exercise 47. 333202_0A03.qxd 12/6/05 2:14 PM Page A25 Appendix A.3 Polynomials and Factoring A25 Special Products Some binomial products have special forms that occur frequently in algebra. You do not need to memorize these formulas because you can use the Distributive Property to multiply. However, becoming familiar with these formulas will enable you to manipulate the algebra more quickly. Special Products v Let and be real numbers, variables, or algebraic expressions. u Special Product Example Sum and Difference of Same Terms u vu v u 2 v 2 Square of a Binomial u v2 u 2 2uv v 2 u v2 u 2 2uv v 2 Cube of a Binomial u v3 u 3 3u 2v 3uv 2 v 3 u v3 u 3 3u 2v 3uv2 v3 x 4x 4 x 2 42 x 2 16 x 32 x 2 2x3 32 x 2 6x 9 3x 22 3x2 23x2 22 9x 2 12x 4 x 23 x3 3x 22 3x22 23 x 3 6x 2 12x 8 x 13 x33x 213x1213 x 3 3x 2 3x 1 Example 4 Special Products Find each product. a. 5x 9 and 5x 9 b. x y 2 and x y 2 Solution a. The product of a sum and a difference of the same two terms has no middle u vu v u 2 v 2. 25x 2 81 5x 95x 9 5x2 9 2 term and takes the form b. By grouping x y as a special product. in parentheses, you can write the product of the trinomials Difference Sum x y 2x y 2 x y 2x y 2 x y 2 22 x 2 2xy y 2 4 Sum and difference of same terms Now try Exercise 67. 333202_0A03.qxd 12/6/05 2:14 PM Page A26 A26 Appendix A Review of Fundamental Concepts of Algebra Polynomials with Common Factors The process of writing a polynomial as a product is called factoring. It is an important tool for solving equations and for simplifying rational expressions. Unless noted otherwise, when you are asked to factor a polynomial, you can assume that you are looking for factors with integer coefficients. If a polynomial cannot be factored using integer coefficients, then it is prime or irreducible over the integers. For instance, the polynomial is irreducible over the integers. Over the real numbers, this polynomial can be factored as x2 3 x2 3 x 3x 3. A polynomial is completely factored when each of its factors is prime. For instance x3 x2 4x 4 x 1x2 4 Completely factored is completely factored, but x 3 x2 4x 4 x 1x2 4 Not completely factored is not completely factored. Its complete factorization is x 3 x2 4x 4 x 1x 2x 2. The simplest type of factoring involves a polynomial that can be written as the product of a monomial and another polynomial. The technique used here is the Distributive Property, ab ac ab c ab c ab ac, in the reverse direction. a is a common factor. Removing (factoring out) any common factors is the first step in completely factoring a polynomial. Example 5 Removing Common Factors Factor each expression. a. b. c. 6x3 4x 4x2 12x 16 x 22x x 23 Solution a. 6x 3 4x 2x3x2 2x2 2x3x2 2 2x is a common factor. b. 4x2 12x 16 4x2 43x 44 4 is a common factor. 4x2 3x 4 c. x 22x x 23 x 22x 3 x 2 is a common factor. Now try Exercise 91. 333202_0A03.qxd 12/6/05 2:14 PM Page A27 Appendix A.3 Polynomials and Factoring A27 Factoring Special Polynomial Forms Some polynomials have special forms that arise from the special product forms on page A25. You should learn to recognize these forms so that you can factor such polynomials easily. Factoring Special Polynomial Forms Factored Form Difference of Two Squares u 2 v 2 u vu v Perfect Square Trinomial u 2 2uv v 2 u v2 u 2 2uv v 2 u v2 Sum or Difference of Two Cubes u 3 v3 u vu 2 uv v2 u3 v3 u vu2 uv v2 Example 9x 2 4 3x 2 2 2 3x 23x 2 x 2 6x 9 x 2 2x3 32 x 2 6x 9 x 2 2x3 32 x 32 x 32 x 3 8 x 3 23 27x3 1 3x 3 13 x 2x 2 2x 4 3x 19x 2 3x 1 In Example 6, note that the first step in factoring a polynomial is to check for any common factors. Once the common factors are removed, it is often possible to recognize patterns that were not immediately obvious. One of the easiest special polynomial forms to factor is the difference of two squares. The factored form is always a set of conjugate pairs. u 2 v 2 u vu v Conjugate pairs Difference Opposite signs To recognize perfect square terms, look for coefficients that are squares of integers and variables raised to even powers. Example 6 Removing a Common Factor First 3 12x2 31 4x2 3 is a common factor. 312 2x2 31 2x1 2x Now try Exercise 105. Difference of two squares Example 7 Factoring the Difference of Two Squares a. x 22 y2 x 2 yx 2 y x 2 yx 2 y b. 16x 4 81 4x22 92 4x2 94x2 9 4x2 92x2 32 4x2 92x 32x 3 Now try Exercise 109. Difference of two squares Difference of two squares 333202_0A03.qxd 12/6/05 2:14 PM Page A28 A28 Appendix A Review of Fundamental Concepts of Algebra A perfect square trinomial is the square of a binomial, and it has the following form. u 2 2uv v 2 u v 2 or u 2 2uv v 2 u v 2 Like signs Like signs Note that the first and last terms are squares and the middle term is twice the product of and v. u Example 8 Factoring Perfect Square Trinomials Factor each trinomial. x2 10x 25 16x2 24x 9 b. a. Solution a. x 2 10x 25 x 2 2x5 5 2 16x2 24x 9 (4x2 24x3 32 4x 32 x 52 b. Now try Exercise 115. The next two formulas show the sums and differences of cubes. Pay special attention to the signs of the terms. Like signs Like signs u 3 v 3 u vu 2 uv v 2 u 3 v 3 u vu 2 uv v 2 Unlike signs Unlike signs Example 9 Factoring the Difference of Cubes Factor x 3 27. Solution x3 27 x3 33 Rewrite 27 as 33. x 3x 2 3x 9 Factor. Now try Exercise 123. Example 10 Factoring the Sum of Cubes a. y 3 8 y 3 23 y 2y 2 2y 4 b. 3x 3 64 3x 3 43 Rewrite 8 as 23. Factor. Rewrite 64 as 43. 3x 4x 2 4x 16 Factor. Now try Exercise 125. 333202_0A03.qxd 12/6/05 2:14 PM Page A29 Appendix A.3 Polynomials and Factoring A29 Trinomials with Binomial Factors ax 2 bx c, To factor a trinomial of the form use the following pattern. Factors of a ax2 bx c x x Factors of c inner products add up to the middle term 6x 2 17x 5, one has outer and inner products that add up to a The goal is to find a combination of factors of and such that the outer and For instance, in the trinomial you can write all possible factorizations and determine which 17x. 2x 13x 5, 2x 53x 1 6x 5x 1, 6x 1x 5, bx. c 2x 53x 1 You can see that (O) and inner (I) products add up to is the correct factorization because the outer 17x. F O I L O I 2x 53x 1 6x 2 2x 15x 5 6x2 17x 5. Example 11 Factoring a Trinomial: Leading Coefficient Is 1 Factor x 2 7x 12. Solution The possible factorizations are x 2x 6, x 1x 12, and x 3x 4. Testing the middle term, you will find the correct factorization to be x 2 7x 12 x 3x 4. Now try Exercise 131. Example 12 Factoring a Trinomial: Leading Coefficient Is Not 1 Factoring a trinomial can involve trial and error. However, once you have produced the factored form, it is an easy matter to check your answer. For instance, you can verify the factorization in Example 11 by multiplying out the expression x 3x 4 to see that you obtain the original trinomial, x2 7x 12. Factor 2x 2 x 15. Solution The eight possible factorizations are as follows. 2x 1x 15 2x 3x 5 2x 5x 3 2x 15x 1 2x 1x 15 2x 3x 5 2x 5x 3 2x 15x 1 Testing the middle term, you will find the correct factorization to be 2x 2 x 15 2x 5x 3. O I 6x 5x x Now try Exercise 139. 333202_0A03.qxd 12/6/05 2:14 PM Page A30 A30 Appendix A Review of Fundamental Concepts of Algebra Factoring by Grouping Sometimes polynomials with more than three terms can be factored by a method called factoring by grouping. It is not always obvious which terms to group, and sometimes several different groupings will work. Example 13 Factoring by Grouping Use factoring by grouping to factor x 3 2x2 3x 6. Solution x 3 2x 2 3x 6 x 3 2x 2 3x 6 x 2x 2 3x 2 x 2x 2 3 Group
terms. Factor each group. Distributive Property Now try Exercise 147. Factoring a trinomial can involve quite a bit of trial and error. Some of this trial and error can be lessened by using factoring by grouping. The key to this method of factoring is knowing how to rewrite the middle term. In general, to ac factor a trinomial b that add up to and use these factors to rewrite the middle term. This technique is illustrated in Example 14. by grouping, choose factors of the product ax2 bx c Example 14 Factoring a Trinomial by Grouping Use factoring by grouping to factor 2x 2 5x 3. Another way to factor the polynomial in Example 13 is to group the terms as follows. x3 2x2 3x 6 x3 3x 2x2 6 xx2 3 2x2 3 x2 3x 2 As you can see, you obtain the same result as in Example 13. Solution 2x 2 5x 3, In the trinomial 6. 6 ac Now, product is rewrite the middle term as factors as 5x 6x x. 2x 2 5x 3 2x 2 6x x 3 a 2 and 61 c 3, and which implies that the So, you can 6 1 5 b. This produces the following. 2x 2 6x x 3 2xx 3 x 3 x 32x 1 Rewrite middle term. Group terms. Factor groups. Distributive Property So, the trinomial factors as 2x 2 5x 3 x 32x 1. Now try Exercise 153. Guidelines for Factoring Polynomials 1. Factor out any common factors using the Distributive Property. 2. Factor according to one of the special polynomial forms. ax2 bx c mx rnx s. 3. Factor as 4. Factor by grouping. 333202_0A03.qxd 12/6/05 2:14 PM Page A31 Appendix A.3 Polynomials and Factoring A31 A.3 Exercises VOCABULARY CHECK: Fill in the blanks. 1. For the polynomial anxn an1xn1 . . . a1x a0, ________, and the constant term is ________. the degree is ________, the leading coefficient is 2. A polynomial in x in standard form is written with ________ powers of x. 3. A polynomial with one term is called a ________, while a polynomial with two terms is called a ________, and a polynomial with three terms is called a ________. 4. To add or subtract polynomials, add or subtract the ________ ________ by adding their coefficients. 5. The letters in “FOIL” stand for the following. F ________ O ________ I ________ L ________ 6. The process of writing a polynomial as a product is called ________. 7. A polynomial is ________ ________ when each of its factors is prime. In Exercises 1–6, match the polynomial with its description. [The polynomials are labeled (a), (b), (c), (d), (e), and (f).] In Exercises 23–28, determine whether the expression is a polynomial. If so, write the polynomial in standard form. (a) (c) (e) 3x 2 x 3 3x 2 3x 1 3x 5 2x 3 x (b) (d) (f) 1 2x 3 12 2 3x 4 x 2 10 1. A polynomial of degree 0 2. A trinomial of degree 5 3. A binomial with leading coefficient 2 4. A monomial of positive degree 5. A trinomial with leading coefficient 2 3 6. A third-degree polynomial with leading coefficient 1 In Exercises 7–10, write a polynomial that fits the description. (There are many correct answers.) 7. A third-degree polynomial with leading coefficient 2 8. A fifth-degree polynomial with leading coefficient 6 9. A fourth-degree binomial with a negative leading coefficient 10. A third-degree binomial with an even leading coefficient In Exercises 11–22, (a) write the polynomial in standard form, (b) identify the degree and leading coefficient of the polynomial, and (c) state whether the polynomial is a monomial, a binomial, or a trinomial. 11. 13. 15. 17. 19. 21. 14x 1 2x5 3x 4 2x 2 5 x 5 1 3 1 6x4 4x5 4x3y 12. 14. 16. 18. 20. 22. 2x 2 x 1 7x y 25y2 1 t2 9 3 2x x5y 2x2y2 xy 4 23. 25. 27. 28. 2x 3x 3 8 3x 4 x y2 y4 y 3 y 2 y4 24. 26. 2x 3 x 3x1 x2 2x 3 2 33. 34. 30. 32. 29. 36. 35. 31. In Exercises 29– 46, perform the operation and write the result in standard form. 6x 5 8x 15 2x 2 1 x 2 2x 1 x 3 2 4x 3 2x 5x 2 1 3x 2 5 15x 2 6 8.3x 3 14.7x 2 17 15.2x4 18x 19.1 13.9x4 9.6x 15 5z 3z 10z 8 y 3 1 y 2 1 3y 7 3xx 2 2x 1 y24y2 2y 3 5z3z 1 3x5x 2 1 x 34x 4x3 x 3 2.5x2 33x 2 3.5y2y3 4x1 8x 3 45. 8 y 46. 2y4 7 41. 44. 43. 38. 37. 42. 40. 39. 333202_0A03.qxd 12/6/05 2:14 PM Page A32 A32 Appendix A Review of Fundamental Concepts of Algebra In Exercises 47–84, multiply or find the special product. 47. 48. 49. 50. 51. 52. 53. 55. 57. 59. 61. 63. 65. 67. 68. 69. 71. 73. 75. 77. 79. 81. 82. 83. 84. x 3x 4 x 5x 10 3x 52x 1 7x 24x 3 x 2 x 1x 2 x 1 x 2 3x 2x 2 3x 2 x 10x 10 x 2yx 2y 2x 3 2 2x 5y2 x 13 2x y3 4x3 32 m 3 nm 3 n x y 1x y 1 x 3 y2 2r 2 52r 2 5 1 2x 32 1 3x 21 1.2x 32 1.5x 41.5x 4 5xx 1 3xx 1 2x 1x 3 3x 3 u 2u 2u 2 4 x yx yx 2 y 2 3x 2 54. 56. 58. 60. 62. 64. 66. 70. 72. 74. 76. 78. 80. 2x 32x 3 2x 3y2x 3y 4x 52 5 8x 2 x 23 3x 2y3 8x 32 x 1 y2 3a3 4b23a3 4b2 3t 52 2 2x 1 5 1.5y 32 2.5y 32.5y 3 2x 1 5 In Exercises 85–88, find the product. (The expressions are not polynomials, but the formulas can still be used.) 85. 87. x yx y x 5 2 86. 88. 5 x5 x x 32 In Exercises 89–96, factor out the common factor. 89. 91. 93. 95. 3x 6 2x 3 6x xx 1 6x 1 x 32 4x 3 90. 92. 94. 96. 5y 30 4x 3 6x 2 12x 3xx 2 4x 2 3x 12 3x 1 In Exercises 103 –112, completely factor the difference of two squares. 103. 104. 105. 106. 107. 108. 109. 111. x2 81 x 2 49 32y2 18 4 36y2 16x2 1 9 4 25 y2 64 x 1 2 4 9u2 4v2 110. 112. 25 z 5 2 25x2 16y2 In Exercises 113 –122, factor the perfect square trinomial. 113. 115. 117. 119. 121. x 2 4x 4 4t 2 4t 1 25y 2 10y 1 9u2 24uv 16v2 x2 4 3x 4 9 114. 116. 118. 120. 122. x 2 10x 25 9x 2 12x 4 36y2 108y 81 4x2 4xy y2 z 2 z 1 4 In Exercises 123 –130, factor the sum or difference of cubes. 123. 125. 127. 129. x 3 8 y 3 64 8t 3 1 u3 27v3 124. 126. 128. 130. x 3 27 z 3 125 27x 3 8 64x3 y3 In Exercises 131–144, factor the trinomial. 131. 133. 135. 137. 139. 141. 143. x 2 x 2 s 2 5s 6 20 y y 2 x 2 30x 200 3x 2 5x 2 5x 2 26x 5 9z 2 3z 2 132. 134. 136. 138. 140. 142. 144. x 2 5x 6 t 2 t 6 24 5z z 2 x 2 13x 42 2x 2 x 1 12x 2 7x 1 5u 2 13u 6 In Exercises 145–152, factor by grouping. 145. 147. 149. 151. x 3 x 2 2x 2 2x 3 x 2 6x 3 6 2x 3x3 x4 6x3 2x 3x 2 1 146. 148. 150. 152. x 3 5x 2 5x 25 5x 3 10x 2 3x 6 x 5 2x 3 x 2 2 8x5 6x2 12x3 9 In Exercises 97–102, find the greatest common factor such that the remaining factors have only integer coefficients. In Exercises 153–158, factor the trinomial by grouping. 97. 99. 101. 1 1 2x 4 2 x3 2x2 5x 3 xx 3 4x 3 2 98. 100. 102. 1 1 3 y 5 3 y 4 5y2 2y 5 yy 1 2 y 1 4 153. 155. 157. 3x 2 10x 8 6x 2 x 2 15x 2 11x 2 154. 2x 2 9x 9 6x 2 x 15 156. 158. 12x2 13x 1 333202_0A03.qxd 12/6/05 2:14 PM Page A33 In Exercises 159–192, completely factor the expression. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 1 16 6x 2 54 12x 2 48 x 3 4x 2 x 3 9x x 2 2x 1 16 6x x 2 1 4x 4x 2 9x 2 6x 1 2x 2 4x 2x 3 2y 3 7y 2 15y 9x 2 10x 1 13x 6 5x 2 1 81x2 2 9 x 8 96x 1 8x2 1 3x 3 x 2 15x 5 5 x 5x 2 x 3 x 4 4x 3 x 2 4x 3u 2u2 6 u3 1 4 x3 3x2 3 4 x 9 5 x3 x2 x 5 t 1 2 49 x 2 1 2 4x 2 x 2 8 2 36x 2 2t 3 16 5x 3 40 4x2x 1 2x 1 2 53 4x 2 83 4x5x 1 2x 1x 3 2 3x 1 2x 3 73x 2 21 x 2 3x 21 x3 7x2x 2 12x x2 1 27 3x 22x 14 x 2 34x 1 3 2xx 54 x 24x 5 3 5x6 146x53x 23 33x 223x6 15 x2 2 x2 14 x 2 15 1 In Exercises 193–196, find all values of b for which the trinomial can be factored. 193. 194. 195. 196. x 2 bx 15 x2 bx 50 x 2 bx 12 x2 bx 24 Appendix A.3 Polynomials and Factoring A33 In Exercises 197–200, find two integer values of c such that the trinomial can be factored. (There are many correct answers.) 197. 199. 2x 2 5x c 3x2 x c 198. 200. 3x 2 10x c 2x2 9x c 201. Cost, Revenue, and Profit An electronics manufacturer C radios per week. The total cost x can produce and sell (in dollars) of producing x radios is C 73x 25,000 and the total revenue R (in dollars) is R 95x. (a) Find the profit P in terms of x. (b) Find the profit obtained by selling 5000 radios per week. 202. Cost, Revenue, and Profit An artisan can produce and (in dollars) of hats per month. The total cost C x sell producing hats is x C 460 12x and the total revenue R (in dollars) is R 36x. (a) Find the profit P in terms of x. (b) Find the profit obtained by selling 42 hats per month. 203. Compound Interest After 2 years, an investment of $500 compounded annually at an interest rate will yield an amount of 5001 r2. r (a) Write this polynomial in standard form. (b) Use a calculator to evaluate the polynomial for the values of shown in the table. r r 5001 r2 21 2 % 3% 4% 41 2 % 5% (c) What conclusion can you make from the table? 204. Compound Interest After 3 years, an investment of $1200 compounded annually at an interest rate will yield an amount of 12001 r3. r (a) Write this polynomial in standard form. (b) Use a calculator to evaluate the polynomial for the values of shown in the table. r r 12001 r3 2% 3% 31 2 % 4% 41 2 % (c) What conclusion can you make from the table? 333202_0A03.qxd 12/6/05 2:14 PM Page A34 A34 A34 Appendix A Appendix A Review of Fundamental Concepts of Algebra Review of Fundamental Concepts of Algebra 205. Volume of a Box A take-out fast-food restaurant is constructing an open box by cutting squares from the corners of a piece of cardboard that is 18 centimeters by 26 centimeters (see figure). The edge of each cut-out square is centimeters. x (a) Find the volume of the box in terms of x. (b) Find the volume when x 1, x 2, and x 3. x x 2 − 8 2− x 1 x x 18 cm x x 26 26 cm 26 2− x 18 2− x 206. Volume of a Box An overnight shipping company is designing a closed box by cutting along the solid lines and folding along the broken lines on the rectangular piece of corrugated cardboard shown in the figure. The length and width of the rectangle are 45 centimeters and 15 centimeters, respectively. (a) Find the volume of the shipping box in terms of x. (b) Find the volume when x 3, x 5, and x 7. 45 cm x m c 5 1 Geometry In Exercises 207 and 208, find a polynomial that represents the total number of square feet for the floor plan shown in the figure. 207. x x 14 ft 209. Geometry Find the area of the shaded region in each figure. Write your result as a polynomial in standard form. 2 + 6 x x + 4 2
x x (a) (b) 12x 8x 6x 9x 210. Stopping Distance The stopping distance of an automobile is the distance traveled during the driver’s reaction time plus the distance traveled after the brakes are applied. In an experiment, these distances were measured (in feet) when the automobile was traveling at a speed of x miles per hour on dry, level pavement, as shown in the bar graph. The distance traveled during the reaction time R was R 1.1x and the braking distance was B B 0.0475x 2 0.001x 0.23. (a) Determine the polynomial that represents the total stopping distance T. (b) Use the result of part (a) to estimate the total stopping miles per x 30, x 40, x 55 and distance when hour. (c) Use the bar graph to make a statement about the total stopping distance required for increasing speeds. Reaction time distance Braking distance 22 ft ) 250 225 200 175 150 125 100 75 50 25 20 30 40 50 60 Speed (in miles per hour) x 208. 14 ft x x 18 ft x 333202_0A03.qxd 12/6/05 2:14 PM Page A35 Geometric Modeling In Exercises 211–214, draw a “geometric factoring model” to represent the factorization. For instance, a factoring model for 2x2 3x 1 2x 1x 1 is shown in the figure 211. 212. 213. 214. 3x 2 7x 2 3x 1x 2 x 2 4x 3 x 3x 1 2x 2 7x 3 2x 1x 3 x 2 3x 2 x 2x 1 1 x x x 1 Geometry In Exercises 215–218, write an expression in factored form for the area of the shaded portion of the figure. 215. 216. Appendix A.3 Polynomials and Factoring A35 (a) Factor the expression for the volume. (b) From the result of part (a), show that the volume of concrete is 2 (average radius)(thickness of the tank) h. 220. Chemistry The rate of change of an autocatalytic kQx kx 2, is the amount of x is the amount of substance is a constant of proportionality. Factor the chemical reaction is the original substance, formed, and expression. where Q k Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 221–224, determine whether 221. The product of two binomials is always a second-degree polynomial. 222. The sum of two binomials is always a binomial. 223. The difference of two perfect squares can be factored as the product of conjugate pairs. 224. The sum of two perfect squares can be factored as the binomial sum squared. 225. Find the degree of the product of two polynomials of degrees m n and . 226. Find the degree of the sum of two polynomials of degrees r r + 2 18 217. x x 8 x x 218x + 3) 219. Geometry The volume V cylindrical concrete storage figure is r radius, storage tank. V R 2h r 2h is the inside radius, and of concrete used to make the the is the outside is the height of the tank shown R where h in R h r r m and n if m < n. 227. Think About It When the polynomial x3 3x2 2x 1 is subtracted from an unknown polynomial, the difference is x + 3 5x 2 8. If it is possible, find the unknown polynomial. 228. Logical Reasoning Verify that by letting x2 y2 expressions. Are there any values of x y2 x2 y2? Explain. x 3 y 4 and x y2 is not equal to and evaluating both for which and y x 229. Factor 230. Factor 231. Factor x2n y2n x3n y3n x 3n y2n completely. completely. completely. 232. Writing Explain what is meant when it is said that a polynomial is in factored form. 233. Give an example of a polynomial that is prime with respect to the integers. 333202_0A04.qxd 12/6/05 2:16 PM Page A36 A36 Appendix A Review of Fundamental Concepts of Algebra A.4 Rational Expressions What you should learn • Find domains of algebraic expressions. • Simplify rational expressions. • Add, subtract, multiply, and divide rational expressions. • Simplify complex fractions and rewrite difference quotients. Why you should learn it Rational expressions can be used to solve real-life problems. For instance, in Exercise 84 on page A45, a rational expression is used to model the projected number of households banking and paying bills online from 2002 through 2007. Domain of an Algebraic Expression The set of real numbers for which an algebraic expression is defined is the domain of the expression. Two algebraic expressions are equivalent if they have the same domain and yield the same values for all numbers in their domain. For instance, x 1 x 2 are equivalent because 2x 3 and 2x 3. Example 1 Finding the Domain of an Algebraic Expression a. The domain of the polynomial 2x 3 3x 4 is the set of all real numbers. In fact, the domain of any polynomial is the set of all real numbers, unless the domain is specifically restricted. b. The domain of the radical expression x 2 is the set of real numbers greater than or equal to 2, because the square root of a negative number is not a real number. c. The domain of the expression x 2 x 3 is the set of all real numbers except zero, which is undefined. Now try Exercise 1. x 3, which would result in division by The quotient of two algebraic expressions is a fractional expression. Moreover, the quotient of two polynomials such as 1 x , 2x 1 x 1 , or x 2 1 x 2 1 is a rational expression. Recall that a fraction is in simplest form if its numerator and denominator have no factors in common aside from To write a fraction in simplest form, divide out common factors. ±1 The key to success in simplifying rational expressions lies in your ability to factor polynomials. 333202_0A04.qxd 12/6/05 2:16 PM Page A37 In Example 2, do not make the mistake of trying to simplify further by dividing out terms. x 6 3 x 6 3 x 2 Remember that to simplify fractions, divide out common factors, not terms. Appendix A.4 Rational Expressions A37 Simplifying Rational Expressions When simplifying rational expressions, be sure to factor each polynomial completely before concluding that the numerator and denominator have no factors in common. In this text, when a rational expression is written, the domain is usually not listed with the expression. It is implied that the real numbers that make the denominator zero are excluded from the expression. Also, when performing operations with rational expressions, this text follows the convention of listing by the simplified expression all values of that must be specifically excluded from the domain in order to make the domains of the simplified and original expressions agree. x Example 2 Simplifying a Rational Expression Write x 2 4x 12 3x 6 Solution in simplest form. x2 4x 12 3x 6 x 6x 2 3x 2 x 6 3 , x 2 Factor completely. Divide out common factors. (because division by Note that the original expression is undefined when zero is undefined). To make sure that the simplified expression is equivalent to the original expression, you must restrict the domain of the simplified expression by excluding the value x 2. x 2 Now try Exercise 19. Sometimes it may be necessary to change the sign of a factor to simplify a rational expression, as shown in Example 3. Example 3 Simplifying Rational Expressions Write 12 x x2 2x2 9x 4 Solution in simplest form. 12 x x2 2x2 9x 4 4 x3 x 2x 1x 4 x 43 x 2x 1x 4 Factor completely. 4 x x 4 3 x 2x 1 , x 4 Divide out common factors. Now try Exercise 25. 333202_0A04.qxd 12/6/05 2:16 PM Page A38 A38 Appendix A Review of Fundamental Concepts of Algebra Operations with Rational Expressions To multiply or divide rational expressions, use the properties of fractions discussed in Appendix A.1. Recall that to divide fractions, you invert the divisor and multiply. Example 4 Multiplying Rational Expressions 2x2 x 6 x2 4x 5 x3 3x2 2x 4x2 6x 2x 3x 2 x 5x 1 x 2x 2 2x 5 , xx 2x 1 2x2x 3 x 0, x 1, x 3 2 Now try Exercise 39. In Example 4 the restrictions x 0, are listed with the simplified expression in order to make the two domains agree. Note that the value x 5 is excluded from both domains, so it is not necessary to list this value. x 1, and x 3 2 Example 5 Dividing Rational Expressions x 3 8 x 2 4 x 2 2x 2x 4 x 2 4 x 2x2 2x 4 x 2x 2 Invert and multiply. x 2x2 2x 4 x2 2x 4 Divide out common factors. x ±2 x2 2x 4, Now try Exercise 41. To add or subtract rational expressions, you can use the LCD (least common denominator) method or the basic definition a b ± c d ad ± bc bd , b 0, d 0. Basic definition This definition provides an efficient way of adding or subtracting two fractions that have no common factors in their denominators. Example 6 Subtracting Rational Expressions x x 3 When subtracting rational expressions, remember to distribute the negative sign to all the terms in the quantity that is being subtracted. 2 3x 4 x3x 4 2x 3 x 33x 4 3x 2 4x 2x 6 x 33x 4 3x 2 2x 6 x 33x 4 Now try Exercise 49. Basic definition Distributive Property Combine like terms. 333202_0A04.qxd 12/6/05 2:16 PM Page A39 Appendix A.4 Rational Expressions A39 For three or more fractions, or for fractions with a repeated factor in the denominators, the LCD method works well. Recall that the least common denominator of several fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. Here is a numerical example The LCD is 12. 9 12 8 12 2 12 3 12 1 4 Sometimes the numerator of the answer has a factor in common with the denominator. In such cases the answer should be simplified. For instance, in the 3 example above, was simplified to 12 1 4. Example 7 Combining Rational Expressions: The LCD Method Perform the operations and simplify, x, and x 1x 1, you can see that Solution Using the factored denominators the LCD is 3 x 1 2 x xx 1x 1. x 3 x 1x 1 3xx 1 xx 1x 1 2x 1x 1 xx 1x 1 x 3x xx 1x 1 3xx 1 2x 1x 1 x 3x xx 1x 1 3x 2 3x 2x 2 2 x 2 3x xx 1x 1 3x2 2x2 x2 3x 3x 2 xx 1x 1 Distributive Property Group like terms. 2x2 6x 2 xx 1x 1 2x 2 3x 1 xx 1x 1 Now try Exercise 51. Combine like terms. Factor. 333202_0A04.qxd 12/6/05 2:16 PM Page A40 A40 Appendix A Review of Fundamental Concepts of Algebra Complex Fractions and the Difference Quotient Fractional expressions with separate fractions in the numerator, denominator, or both are called complex fractions. Here are two examples. 1 x x 2 1 and 1 x 1 x 2 1 A complex fraction can be simplified by combi
ning the fractions in its numerator into a single fraction and then combining the fractions in its denominator into a single fraction. Then invert the denominator and multiply. Example 8 Simplifying a Complex Fraction 3 2 x 1 1 x 1 x 2 3x 1x 1 1 x 1 2 3x x x 2 x 1 2 3x x x 1 x 2 2 3xx 1 xx 2 Combine fractions. Simplify. Invert and multiply. , x 1 Now try Exercise 57. Another way to simplify a complex fraction is to multiply its numerator and denominator by the LCD of all fractions in its numerator and denominator. This method is applied to the fraction in Example 8 as follows xx 1 xx 1 xx 1. LCD is 2 3x x x 2 x 1 xx 1 xx 1 2 3xx 1 xx 2 , x 1 333202_0A04.qxd 12/6/05 2:16 PM Page A41 Appendix A.4 Rational Expressions A41 The next three examples illustrate some methods for simplifying rational expressions involving negative exponents and radicals. These types of expressions occur frequently in calculus. To simplify an expression with negative exponents, one method is to begin by factoring out the common factor with the smaller exponent. Remember that when factoring, you subtract exponents. For instance, in the smaller exponent is and the common factor is 3x52 2x32 x52. 5 2 3x52 2x32 x5231 2x3252 x523 2x1 3 2x x52 Example 9 Simplifying an Expression Simplify the following expression containing negative exponents. x1 2x32 1 2x12 Solution Begin by factoring out the common factor with the smaller exponent. x1 2x32 1 2x12 1 2x32x 1 2x(12)(32) 1 2x32x 1 2x1 1 x 1 2x32 Now try Exercise 65. A second method for simplifying an expression with negative exponents is shown in the next example. Example 10 Simplifying an Expression with Negative Exponents (4 x 2)12 x 2(4 x 2)12 4 x 2 4 x 212 x 24 x 212 4 x 2 4 x 212 4 x 212 4 x 21 x 24 x 20 4 x 232 4 x 2 x 2 4 x 232 4 4 x 232 Now try Exercise 67. 333202_0A04.qxd 12/6/05 2:16 PM Page A42 A42 Appendix A Review of Fundamental Concepts of Algebra Example 11 Rewriting a Difference Quotient The following expression from calculus is an example of a difference quotient. x h x h Rewrite this expression by rationalizing its numerator. Solution x2 hx h x h hx h x 1 x h x , h 0 Notice that the original expression is undefined when h 0 exclude sions are equivalent. So, you must from the domain of the simplified expression so that the expres- h 0. Now try Exercise 73. Difference quotients, such as that in Example 11, occur frequently in calculus. Often, they need to be rewritten in an equivalent form that can be evaluated when h 0. Note that the equivalent form is not simpler than the original form, but it has the advantage that it is defined when h 0. A.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The set of real numbers for which an algebraic expression is defined is the ________ of the expression. 2. The quotient of two algebraic expressions is a fractional expression and the quotient of two polynomials is a ________ ________. 3. Fractional expressions with separate fractions in the numerator, denominator, or both are called ________ fractions. 4. To simplify an expression with negative exponents, it is possible to begin by factoring out the common factor with the ________ exponent. 5. Two algebraic expressions that have the same domain and yield the same values for all numbers in their domains are called ________. 6. An important rational expression, such as a ________ ________. x h2 x2 h , that occurs in calculus is called 333202_0A04.qxd 12/6/05 2:16 PM Page A43 Appendix A.4 Rational Expressions A43 In Exercises 1–8, find the domain of the expression. 30. 3. 5. 7. x ≥ 0 3x 2 4x 7 4x3 3, 1 x 2 x 1 2. 4. 6. 8. x > 0 2x 2 5x 2 6x 2 9, x 1 2x 1 6 x x 3 x2 x 6 1 x 2 In Exercises 9 and 10, find the missing factor in the numerator such that the two fractions are equivalent. 5 6x 2 5 2x 9. 10. 3 4x 1 3 4 In Exercises 11–28, write the rational expression in simplest form. 12. 14. 16. 18. 20. 22. 24. 18y 2 60y 5 2x2y xy y 9x 2 9x 2x 2 12 4x x 3 x 2 25 5 x x 2 8x 20 x 2 11x 10 x2 7x 6 x 2 11x 10 11. 13. 15. 17. 19. 21. 23. 25. 26. 27. 28. 15x2 10x 3xy xy x 4y 8y2 10y 5 x 5 10 2x y2 16 y 4 x 3 5x 2 6x x 2 4 y 2 7y 12 y 2 3y 18 2 x 2x 2 x3 x2 4 x 2 9 x 3 x 2 9x 9 z 3 8 z 2 2z 4 y 3 2y 2 3y y 3 1 In Exercises 29 and 30, complete the table. What can you conclude? 29. x 0 1 2 3 4 5 6 x2 2x 3 x 3 x 1 31. Error Analysis Describe the error. 5x3 2x3 4 5x3 2x3 4 5 2 4 5 6 32. Error Analysis Describe the error. x 3 25x x2 2x 15 xx2 25 x 5x 3 xx 5x 5 x 5x 3 xx 5 x 3 Geometry In Exercises 33 and 34, find the ratio of the area of the shaded portion of the figure to the total area of the figure. 33. 34 2x + 3 x + 5 In Exercises 35– 42, perform the multiplication or division and simplify. 36. 38. x 13 x 33 x 4y 16 5y 15 xx 3 5 2y 6 4 y x 1 5 25x 6t 9 x 2 xy 2y 2 x 3 x 2y t 3 t 2 4 35. 37. 39. 40. 41. x x 2 3xy 2y 2 x2 36 x x3 6x2 x2 x 42. x2 14x 49 x2 49 3x 21 x 7 333202_0A04.qxd 12/6/05 2:16 PM Page A44 A44 Appendix A Review of Fundamental Concepts of Algebra In Exercises 43–52, perform the addition or subtraction and simplify. In Exercises 61– 66, factor the expression by removing the common factor with the smaller exponent. 43. 45. 47. 48. 49. 50. 51. 52 44. 46. 2x 2x 5x 6 10 x 2 2x Error Analysis In Exercises 53 and 54, describe the error. 53. x 4 x 2 3x 8 x 2 x 4 3x 8 x 2 2x 4 x 2 8 x 2x 2 54. 6 x xx 2 x 2 x 2 2x 2 x 2 2 x6 x x 22 8 x 2x 2 6x x 2 x 2 4 8 x 2x 2 6 x 2 6x 2 x2x 2 In Exercises 55– 60, simplify the complex fraction. x 1 2 x 2 55. x2 x x 12 x 13 x 1 2x x 57. 59. 56. 58. 60. x 4 x 4 x 4 x2 1 x 12 61. 62. 63. 64. 65. 66. x 5 2x2 x5 5x3 x 2x 2 15 x 2 14 2xx 53 4x 2x 54 2x2x 112 5x 112 4x 32x 132 2x2x 112 In Exercises 67 and 68, simplify the expression. 67. 68. 3x13 x23 3x23 x 31 x 212 2x1 x 212 x4 In Exercises 69–72, simplify the difference quotient 69. 71. 70. 72. 1 (x h) In Exercises 73–76, simplify the difference quotient by rationalizing the numerator. 73. 74. 75. 76 Probability In Exercises 77 and 78, consider an experiment in which a marble is tossed into a box whose base is shown in the figure. The probability that the marble will come to rest in the shaded portion of the box is equal to the ratio of the shaded area to the total area of the figure. Find the probability. 77. 78x + 2) 333202_0A04.qxd 12/6/05 2:16 PM Page A45 79. Rate A photocopier copies at a rate of 16 pages per minute. (a) Find the time required to copy one page. (b) Find the time required to copy pages. x (c) Find the time required to copy 60 pages. 80. Rate After working together for hours on a common task, two workers have done fractional parts of t3 the job equal to respectively. What fractional part of the task has been completed? t5, and t Finance In Exercises 81 and 82, the formula that approximates the annual interest rate r of a monthly installment loan is given by r ] [24NM P N P NM 12 Appendix A.4 Rational Expressions A45 84. Interactive Money Management The table shows the projected numbers of U.S. households (in millions) banking online and paying bills online for the years 2002 through 2007. (Source: eMarketer; Forrester Research) Year Banking Paying Bills 2002 2003 2004 2005 2006 2007 21.9 26.8 31.5 35.0 40.0 45.0 13.7 17.4 20.9 23.9 26.7 29.1 Mathematical models for these data are Number banking online 0.728t2 23.81t 0.3 0.049t2 0.61t 1.0 where N is the total number of payments, M is the monthly payment, and P is the amount financed. and 81. (a) Approximate the annual interest rate for a four-year car loan of $16,000 that has monthly payments of $400. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 82. (a) Approximate the annual interest rate for a five-year car loan of $28,000 that has monthly payments of $525. (b) Simplify the expression for the annual interest rate r, and then rework part (a). 83. Refrigeration When food (at room temperature) is placed in a refrigerator, the time required for the food to cool depends on the amount of food, the air circulation in the refrigerator, the original temperature of the food, and the temperature of the refrigerator. The model that gives the temperature of food that has an original temperature of 75 40 F and is placed in a T 104t 2 16t 75 t 2 4t 10 F refrigerator is Number paying bills online 4.39t 5.5 0.002t2 0.01t 1.0 t represents the year, with t 2 corresponding to where 2002. (a) Using the models, create a table to estimate the projected number of households banking online and the projected number of households paying bills online for the given years. (b) Compare the values given by the models with the actual data. (c) Determine a model for the ratio of the projected number of households paying bills online to the projected number of households banking online. (d) Use the model from part (c) to find the ratio over the given years. Interpret your results. Synthesis is the temperature (in degrees Fahrenheit) and T where the time (in hours). t is True or False? the statement is true or false. Justify your answer. In Exercises 85 and 86, determine whether (a) Complete the table. 0 2 4 6 8 10 85. 86. xn 1n x 2n 12n xn 1n x2 3x 2 x 1 x 2, for all values of x. 12 14 16 18 20 22 87. Think About It How do you determine whether a rational expression is in simplest form? t T t T (b) What value of T does the mathematical model appear to be approaching? 333202_0A05.qxd 12/6/05 2:45 PM Page A46 A46 Appendix A Review of Fundamental Concepts of Algebra A.5 Solving Equations What you should learn • Identify different types of equations. • Solve linear equations in one variable and equations that lead to linear equations. • Solve quadratic equations by factoring, extracting square roots, completing the square, and using the Quadratic Formula. • Solve polynomial equations of degree three or greater. • Solve equations involving radicals. • Solve equations with absolute values. Why you should learn it Linear equations are used in many real-life applications. For example, in Exercise 185 on page A58, linear equations can be used to model
the relationship between the length of a thighbone and the height of a person, helping researchers learn about ancient cultures. Equations and Solutions of Equations An equation in example x is a statement that two algebraic expressions are equal. For 3x 5 7, x2 x 6 0, and 2x 4 are equations. To solve an equation in means to find all values of x 4 the equation is true. Such values are solutions. For instance, of the equation 3x 5 7 x x for which is a solution because 34 5 7 is a true statement. The solutions of an equation depend on the kinds of numbers being considered. For instance, in the set of rational numbers, has no solution because there is no rational number whose square is 10. However, in the set of real numbers, the equation has the two solutions x 10. x 10 x2 10 and An equation that is true for every real number in the domain of the variable is called an identity. The domain is the set of all real numbers for which the equation is defined. For example x2 9 x 3x 3 Identity is an identity because it is a true statement for any real value of x. The equation x 3x2 1 3x x 0, where Identity is an identity because it is true for any nonzero real value of x. An equation that is true for just some (or even none) of the real numbers in the domain of the variable is called a conditional equation. For example, the equation x2 9 0 Conditional equation x 3 is conditional because and satisfy the equation. The equation are no real values of conditional equations is the primary focus of this section. are the only values in the domain that is conditional because there for which the equation is true. Learning to solve x 3 2x 4 2x 1 x Linear Equations in One Variable Definition of a Linear Equation A linear equation in one variable the standard form ax b 0 x is an equation that can be written in where and are real numbers with a 0. a b 333202_0A05.qxd 12/6/05 2:45 PM Page A47 Appendix A.5 Solving Equations A47 A linear equation has exactly one solution. To see this, consider the follow- ing steps. (Remember that a 0. ) ax b 0 Write original equation. ax b x b a Subtract b from each side. Divide each side by a. To solve a conditional equation in x, isolate on one side of the equation by a sequence of equivalent (and usually simpler) equations, each having the same solution(s) as the original equation. The operations that yield equivalent equations come from the Substitution Principle (see Appendix A.1) and simplification techniques. x Generating Equivalent Equations An equation can be transformed into an equivalent equation by one or more of the following steps. 1. Remove symbols of grouping, combine like terms, or simplify fractions on one or both sides of the equation. 2. Add (or subtract) the same quantity to (from) each side of the equation. 3. Multiply (or divide) each side of the equation by the same nonzero quantity. 4. Interchange the two sides of the equation. Given Equation 2x x 4 Equivalent Equation x 4 x 1 6 x 5 2x 6 x 3 2 x x 2 After solving an equation, you should check each solution in the original equation. For instance, you can check the solution to Example 1(a) as follows. 3x 6 0 32 6 ? 0 0 0 Write original equation. Substitute 2 for x. Solution checks. ✓ Try checking the solution to Example 1(b). Example 1 Solving a Linear Equation a. b. 3x 6 0 3x 6 x 2 5x 4 3x 8 2x 4 8 2x 12 x 6 Original equation Add 6 to each side. Divide each side by 3. Original equation Subtract 3x from each side. Subtract 4 from each side. Divide each side by 2. Now try Exercise 13. 333202_0A05.qxd 12/6/05 2:45 PM Page A48 A48 Appendix A Review of Fundamental Concepts of Algebra An equation with a single fraction on each side can be cleared of denominators by cross multiplying, which is equivalent to multiplying by the LCD and then dividing out. To do this, multiply the left numerator by the right denominator and the right numerator by the left denominator as follows. c a d b bd c d ad cb bd a b LCD is bd. Multiply by LCD. Divide out common factors. Recall that the least common denominator of two or more fractions consists of the product of all prime factors in the denominators, with each factor given the highest power of its occurrence in any denominator. For instance, in Example 3, by factoring each denominator you can determine that the LCD is x 2x 2. To solve an equation involving fractional expressions, find the least common denominator (LCD) of all terms and multiply every term by the LCD. This process will clear the original equation of fractions and produce a simpler equation to work with. Example 2 An Equation Involving Fractional Expressions Solve x 3 3x 4 2. Solution 12 x 3 122 2 3x x 4 3 12 3x 4 4x 9x 24 13x 24 x 24 13 Write original equation. Multiply each term by the LCD of 12. Divide out and multiply. Combine like terms. Divide each side by 13. The solution is x 24 13. Check this in the original equation. Now try Exercise 21. When multiplying or dividing an equation by a variable quantity, it is possible to introduce an extraneous solution. An extraneous solution is one that does not satisfy the original equation. Therefore, it is essential that you check your solutions. Example 3 An Equation with an Extraneous Solution Solve 1 x 2 3 x 2 6x x2 4 . Solution The LCD is 1 x 2 x 2x 2. x 2 4, or x 2x 2 3 Multiply each term by this LCD. x 2x 2 6x x 2x 2 x2 4 x ±2 In the original equation, extraneous solution, and the original equation has no solution. yields a denominator of zero. So, x 2 is an x 2 Extraneous solution x 2 x 2 3x 2 6x, x 2 3x 6 6x x 2 3x 6 4x 8 x 2 Now try Exercise 37. 333202_0A05.qxd 12/6/05 2:45 PM Page A49 Appendix A.5 Solving Equations A49 Quadratic Equations A quadratic equation in ax2 bx c 0 x is an equation that can be written in the general form a, a 0. where known as a second-degree polynomial equation in and are real numbers, with b, c A quadratic equation in x. x is also You should be familiar with the following four methods of solving quadratic equations. The Square Root Principle is also referred to as extracting square roots. Solving a Quadratic Equation Factoring: If ab 0, a 0 then b 0. Example: or x2 x 6 0 x 3x , Square Root Principle: If where Example: x 32 16 x 3 ±4 x 3 x 2 c > 0, then u ± c. x 7 x 3 ± 4 x 1 or x2 bx c, 2 then Add 2 b 2 to each side. Completing the Square: If x 2 bx b 2 x b 2 c b 2 c b2 4 2 2 . Example: x2 6x 5 Add 2 6 2 to each side. x2 6x 32 5 32 x 32 14 x 3 ± 14 x 3 ± 14 Quadratic Formula: If ax2 bx c 0, then x b ± b2 4ac 2a . Example: 2x2 3x 1 0 x 3 ± 32 421 22 3 ± 17 4 You can solve every quadratic equation by completing the square or using the Quadratic Formula. 333202_0A05.qxd 12/6/05 2:45 PM Page A50 A50 Appendix A Review of Fundamental Concepts of Algebra Example 4 Solving a Quadratic Equation by Factoring a. 2x 2 9x 7 3 2x2 9x 4 0 2x 1x 4 0 2x Original equation Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. b. The solutions are 6x 2 3x 0 3x2x 1 0 3x 0 2x 1 0 x 1 2 and x 4. Check these in the original equation. Original equation Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. Check these in the original equation. x 0 x 1 2 x 1 2. The solutions are x 0 and Now try Exercise 57. Note that the method of solution in Example 4 is based on the Zero-Factor Property from Appendix A.1. Be sure you see that this property works only for equations written in general form (in which the right side of the equation is zero). So, all terms must be collected on one side before factoring. For instance, in the x 5x 2 8, it is incorrect to set each factor equal to 8. Try to equation solve this equation correctly. Example 5 Extracting Square Roots Solve each equation by extracting square roots. a. 4x2 12 b. x 32 7 Solution a. 4x 2 12 x 2 3 x ± 3 Write original equation. Divide each side by 4. Extract square roots. When you take the square root of a variable expression, you must account for both positive and negative solutions. So, and x 3. x 32 7 Check these in the original equation. the solutions are Write original equation. x 3 b. x 3 ± 7 x 3 ± 7 Extract square roots. Add 3 to each side. The solutions are x 3 ± 7. Check these in the original equation. Now try Exercise 77. 333202_0A05.qxd 12/6/05 2:45 PM Page A51 Appendix A.5 Solving Equations A51 When solving quadratic equations by completing the square, you must add b22 to each side in order to maintain equality. If the leading coefficient is not 1, you must divide each side of the equation by the leading coefficient before completing the square, as shown in Example 7. Example 6 Completing the Square: Leading Coefficient Is 1 Solve x2 2x 6 0 by completing the square. Solution x2 2x 6 0 x2 2x 6 x2 2x 12 6 12 half of 22 x 12 7 x 1 ± 7 The solutions are x 1 ± 7 x 1 ± 7. Write original equation. Add 6 to each side. Add 12 to each side. Simplify. Take square root of each side. Subtract 1 from each side. Check these in the original equation. Now try Exercise 85. Example 7 Completing the Square: Leading Coefficient Is Not 1 3x2 4x 5 0 3x2 4x 5 x2 x2 4 3 Original equation Add 5 to each side. Divide each side by 3. 2 3 2 Add 2 3 2 to each side. 3 half of 4 x2 19 9 19 9 x 2 3 ± 19 3 x 2 3 ± 19 3 Now try Exercise 91. Simplify. Perfect square trinomial. Extract square roots. Solutions 333202_0A05.qxd 12/6/05 2:45 PM Page A52 A52 Appendix A Review of Fundamental Concepts of Algebra When using the Quadratic Formula, remember that before the formula can be applied, you must first write the quadratic equation in general form. Example 8 The Quadratic Formula: Two Distinct Solutions Use the Quadratic Formula to solve x 2 3x 9. Solution x2 3x 9 x 2 3x 9 0 x b ± b2 4ac 2a x 3 ± 32 419 21 x x 3 ± 45 2 3 ± 35 2 Write original equation. Write in general form. Quadratic Formula Substitute b 3, a 1, and c 9. Simplify. Simplify. The equation has two solutions: 3 35 2 x and x 3 35 2 . Check these in the original equation. Now try Exercise 101. Example 9 The Quadratic Formula: One Solution Use the Quadrat
ic Formula to solve 8x2 24x 18 0. Solution 8x2 24x 18 0 4x2 12x 9 0 x b ± b2 4ac 2a Write original equation. Divide out common factor of 2. Quadratic Formula x 12 ± 122 449 24 Substitute b 12, a 4, and c 9. x 12 ± 0 8 3 2 Simplify. This quadratic equation has only one solution: equation. x 3 2. Check this in the original Now try Exercise 105. Note that Example 9 could have been solved without first dividing out a common a 8, b 24, factor of 2. Substituting into the Quadratic Formula produces the same result. c 18 and 333202_0A05.qxd 12/6/05 2:45 PM Page A53 A common mistake that is made in solving an equation such as that in Example 10 is to divide each side of the equation by the x 2. variable factor This loses the x 0. solution When solving an equation, always write the equation in general form, then factor the equation and set each factor equal to zero. Do not divide each side of an equation by a variable factor in an attempt to simplify the equation. Appendix A.5 Solving Equations A53 Polynomial Equations of Higher Degree The methods used to solve quadratic equations can sometimes be extended to solve polynomial equations of higher degree. Example 10 Solving a Polynomial Equation by Factoring Solve 3x4 48x2. Solution First write the polynomial equation in general form with zero on one side, factor the other side, and then set each factor equal to zero and solve. 3x4 48x2 3x4 48x 2 0 3x2x2 16 0 3x2x 4x 4 0 3x2 0 x 4 0 x 4 0 Write original equation. Write in general form. Factor out common factor. Write in factored form. Set 1st factor equal to 0. Set 2nd factor equal to 0. Set 3rd factor equal to 0. x 0 x 4 x 4 You can check these solutions by substituting in the original equation, as follows. Check 304 4802 344 4842 344 4842 0 checks. ✓ checks. ✓ 4 4 checks. ✓ So, you can conclude that the solutions are x 0, x 4, and x 4. Now try Exercise 135. Example 11 Solving a Polynomial Equation by Factoring Solve x3 3x 2 3x 9 0. Solution x3 3x 2 3x 9 0 x2x 3 3x 3 0 x 3x2 3 0 x 3 0 x2 3 0 x 3, x 3, The solutions are equation. Write original equation. Factor by grouping. Distributive Property Set 1st factor equal to 0. Set 2nd factor equal to 0. x 3 x ± 3 and x 3. Check these in the original Now try Exercise 143. 333202_0A05.qxd 12/6/05 2:45 PM Page A54 A54 Appendix A Review of Fundamental Concepts of Algebra Equations Involving Radicals Operations such as squaring each side of an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity all can introduce extraneous solutions. So, when you use any of these operations, checking your solutions is crucial. Example 12 Solving Equations Involving Radicals a. 2x 7 x 2 2x 7 x 2 2x 7 x2 4x 4 0 x2 2x 3 0 x 3x Original equation Isolate radical. Square each side. Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. By checking these values, you can determine that the only solution is 2x 5 x 3 1 Original equation b. x 1. 2x 5 x 3 1 2x 5 x 3 2x 3 1 2x 5 x 2 2x 3 x 3 2x 3 x2 6x 9 4x 3 Isolate 2x 5. Square each side. Combine like terms. Isolate 2x 3. Square each side. x2 10x 21 0 x 3x The solutions are and Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. x 3 x 7 x 7. Check these in the original equation. Now try Exercise 155. Example 13 Solving an Equation Involving a Rational Exponent x 423 25 3x 42 25 x 42 15,625 x 4 ±125 x 129, x 121 Now try Exercise 163. Original equation Rewrite in radical form. Cube each side. Take square root of each side. Add 4 to each side. When an equation contains two radicals, it may not be possible to isolate both. In such cases, you may have to raise each side of the equation to a power at two different stages in the solution, as shown in Example 12(b). 333202_0A05.qxd 12/6/05 2:45 PM Page A55 Appendix A.5 Solving Equations A55 Equations with Absolute Values To solve an equation involving an absolute value, remember that the expression inside the absolute value signs can be positive or negative. This results in two separate equations, each of which must be solved. For instance, the equation x 2 3 results in the two equations the equation has two solutions: x 2 3 x 5 and x 1. and x 2 3, which implies that Example 14 Solving an Equation Involving Absolute Value Solve x2 3x 4x 6. Solution Because the variable expression inside the absolute value signs can be positive or negative, you must solve the following two equations. First Equation x2 3x 4x 6 x2 x 6 0 x 3x 2 0 x 3 0 x 2 0 Second Equation x2 3x 4x 6 x2 7x 6 0 x 1x 6 0 x 1 0 x 6 0 Check x 3 x 2 x 1 x 6 32 33 ? 43 6 18 18 22 32 ? 42 6 2 2 12 31 ? 41 6 2 2 62 36 ? 46 6 18 18 x 3 The solutions are and x 1. Now try Exercise 181. Use positive expression. Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. Use negative expression. Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. Substitute 3 checks. ✓ 3 for x. Substitute 2 for x. 2 does not check. x. Substitute 1 for 1 checks. ✓ Substitute 6 for x. 6 does not check. 333202_0A05.qxd 12/6/05 2:45 PM Page A56 A56 Appendix A Review of Fundamental Concepts of Algebra A.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. An ________ is a statement that equates two algebraic expressions. 2. To find all values that satisfy an equation is to ________ the equation. 3. There are two types of equations, ________ and ________ equations. 4. A linear equation in one variable is an equation that can be written in the standard from ________. 5. When solving an equation, it is possible to introduce an ________ solution, which is a value that does not satisfy the original equation. ax2 bx c 0, 6. An equation of the form x. polynomial equation in a 0 is a ________ ________, or a second-degree 7. The four methods that can be used to solve a quadratic equation are ________, ________, ________, and the ________. In Exercises 1–10, determine whether the equation is an identity or a conditional equation. In Exercises 27– 48, solve the equation and check your solution. (If not possible, explain why.) 1. 2. 3. 4. 5. 6. 7. 8. 9. 2x 1 2x 2 3x 2 5x 4 6x 3 5 2x 10 3x 2 5 3x 1 4x 1 2x 2x 2 7x 3 4x 37 x x2 8x 5 x 42 11 x2 23x 2 x2 6x 4 4x 3 1 x 1 x 1 10. 5 x 3 x 24 In Exercises 11–26, solve the equation and check your solution. 12. 14. 16. 7 x 19 7x 2 23 7x 3 3x 17 22. x 5 x 2 3 3x 10 11. 13. 15. 17. 18. 19. 20. 21. 23 24. 25. 26. x 11 15 7 2x 25 8x 5 3x 20 2x 5 7 3x 2 3x 3 51 x 1 x 32x 3 8 5x 9x 10 5x 22x 5 5x 3x 1 2 4 0.25x 0.7510 x 3 0.60x 0.40100 x 50 x 2 10 z 24 0 27. 28. 29. 30. 31. 33. 35. 37. 38. 39. 40. 41. 42. 43. 44. 6 x 8 2x 2 x 8x 2 32x 1 2x 5 100 4x 5x 6 4 3 17 y 32 y y y 5x 4 5x 4 10 13 x 3 2 2 2 3 4 5 x 100 36. 32. 34. z 2 10x 3 5x 6 15 15 3x 1 3x 1 2 0 4 2x 1 4 x 4 8x x x 4 7 2x 1 2 x 4x x2 3x 2 6 x 1 x 3 10 3 1 x 3 4 x 3 x 3x 5 x2 3x x 3 x2 9 4 x2 x 6 333202_0A05.qxd 12/6/05 2:45 PM Page A57 45. 46. 47. 48. x 22 5 x 32 x 12 2x 2 x 1x 2 x 22 x2 4x 1 2x 12 4x 2 x 1 In Exercises 49–54, write the quadratic equation in general form. 49. 51. 53. 2x 2 3 8x x 32 3 3x 2 10 18x 1 5 50. 52. 54. x 2 16x 13 3x 72 0 xx 2 5x 2 1 In Exercises 55– 68, solve the quadratic equation by factoring. 55. 57. 59. 61. 63. 65. 67. 68. 6x 2 3x 0 x 2 2x 8 0 x 2 10x 25 0 3 5x 2x 2 0 x 2 4x 12 3 4 x2 8x 20 0 x 2 2ax a 2 0, x a2 b 2 0, a a 56. 58. 60. 62. 64. 66. 9x 2 1 0 x 2 10x 9 0 4x 2 12x 9 0 2x 2 19x 33 x 2 8x 12 1 8 x2 x 16 0 is a real number and are real numbers b In Exercises 69–82, solve the equation by extracting square roots. 69. 71. 73. 75. 77. 79. 81. x 2 49 x 2 11 3x 2 81 x 122 16 x 22 14 2x 12 18 x 72 x 32 70. 72. 74. 76. 78. 80. 82. x 2 169 x 2 32 9x 2 36 x 132 25 x 52 30 4x 72 44 x 52 x 42 In Exercises 83–92, solve the quadratic equation by completing the square. 83. 85. 87. 89. 91. x 2 4x 32 0 x2 12x 25 0 9x 2 18x 3 8 4x x 2 0 2x2 5x 8 0 84. 86. 88. 90. 92. x 2 2x 3 0 x 2 8x 14 0 9x 2 12x 14 x2 x 1 0 4x 2 4x 99 0 In Exercises 93– 116, use the Quadratic Formula to solve the equation. 93. 95. 97. 2x 2 x 1 0 16x 2 8x 3 0 2 2x x 2 0 94. 96. 98. 2x 2 x 1 0 25x 2 20x 3 0 x 2 10x 22 0 Appendix A.5 Solving Equations A57 99. 101. 103. 105. 107. 109. 111. 113. 115. x 2 14x 44 0 x 2 8x 4 0 12x 9x 2 3 9x2 24x 16 0 4x 2 4x 7 28x 49x 2 4 8t 5 2t 2 y 52 2y 1 2x2 3 8x 2 100. 102. 104. 106. 108. 110. 112. 114. 116. 6x 4 x 2 4x 2 4x 4 0 16x 2 22 40x 36x 2 24x 7 0 16x 2 40x 5 0 3x x2 1 0 25h2 80h 61 0 z 62 2z 5 7x 142 8x In Exercises 117–124, use the Quadratic Formula to solve the equation. (Round your answer to three decimal places.) 117. 118. 119. 120. 121. 122. 123. 124. 5.1x 2 1.7x 3.2 0 2x 2 2.50x 0.42 0 0.067x2 0.852x 1.277 0 0.005x 2 0.101x 0.193 0 422x 2 506x 347 0 1100x2 326x 715 0 12.67x2 31.55x 8.09 0 3.22x2 0.08x 28.651 0 In Exercises 125–134, solve the equation using any convenient method. 125. 127. 129. 131. 132. 133. 134. 0 x 2 2x 1 0 x 32 81 x2 x 11 4 x 12 x 2 a 2x 2 b 2 0, a 3x 4 2x2 7 4x 2 2x 4 2x 8 b 126. 128. 130. 11x 2 33x 0 x2 14x 49 0 x2 3x 3 4 0 and are real numbers In Exercises 135–152, find all solutions of the equation. Check your solutions in the original equation. 135. 137. 139. 141. 142. 143. 144. 145. 146. 147. 149. 151. 4x4 18x2 0 x4 81 0 x 3 216 0 5x3 30x 2 45x 0 9x4 24x3 16x 2 0 x3 3x 2 x 3 0 x3 2x 2 3x 6 0 x4 x3 x 1 0 x4 2x3 8x 16 0 x4 4x2 3 0 4x4 65x 2 16 0 x6 7x3 8 0 136. 138. 140. 20x3 125x 0 x6 64 0 27x 3 512 0 148. x4 5x 2 36 0 36t4 29t 2 7 0 150. 152. x6 3x3 2 0 333202_0A05.qxd 12/6/05 2:45 PM Page A58 A58 Appendix A Review of Fundamental Concepts of Algebra In Exercises 153–184, find all solutions of the equation. Check your solutions in the original equation. 153. 155. 157. 159. 161. 163. 165. 167. 169. 170. 171. 173. 175. 177. 179. 181. 183. 158. 160. 156. 154. 162. 2x 10 0 x 10 4 0 32x 5 3 0 26 11x 4 x x 1 3x 1 x 532 8 x 323 8 x2 532 27 168. 3xx 112 2x 132 0 4x2x 113 6xx 143 0 4 x 3 x x 1 1 2 166. 164. 172. 3 174. 1 x 20 2x 1 5 x x2 x 3 x 1 x 2 5 4x 3 0 5
x 3 0 33x 1 5 0 x 31 9x 5 x 5 x 5 x 332 8 x 223 9 x2 x 2232 27 4x 3x 2 7 x 2 6x 3x 18 x 10 x 2 10x 176. 178. 180. 182. 184. 185. Anthropology The relationship between the length of an adult’s femur (thigh bone) and the height of the adult can be approximated by the linear equations y 0.432 x 10.44 Female y 0.449x 12.15 Male y is the length of the femur in inches and where height of the adult in inches (see figure). x is the discovers a male adult femur that is 19 inches long. Is it likely that both the foot bones and the thigh bone came from the same person? (c) Complete the table to determine if there is a height of an adult for which an anthropologist would not be able to determine whether the femur belonged to a male or a female. Height, x Female femur length, y Male femur length, y 60 70 80 90 100 110 186. Operating Cost A delivery company has a fleet of vans. The annual operating cost per van is C C 0.32m 2500 m where is the number of miles traveled by a van in a year. What number of miles will yield an annual operating cost of $10,000? 187. Flood Control A river has risen 8 feet above its flood stage. The water begins to recede at a rate of 3 inches per hour. Write a mathematical model that shows the number hours. If the water of feet above flood stage after continually recedes at this rate, when will the river be 1 foot above its flood stage? t 188. Floor Space The floor of a one-story building is 14 feet longer than it is wide. The building has 1632 square feet of floor space. (a) Draw a diagram that gives a visual representation of and show the floor space. Represent the width as the length in terms of w. w y in. femur x in. (c) Find the length and width of the floor of the building. (b) Write a quadratic equation in terms of w. 189. Packaging An open box with a square base (see figure) is to be constructed from 84 square inches of material. The height of the box is 2 inches. What are the dimensions of the box? (Hint: The surface area is S x 2 4xh. ) (a) An anthropologist discovers a femur belonging to an adult human female. The bone is 16 inches long. Estimate the height of the female. (b) From the foot bones of an adult human male, an anthropologist estimates that the person’s height was 69 inches. A few feet away from the site where the the anthropologist foot bones were discovered, 2 in. x x 333202_0A05.qxd 12/6/05 2:45 PM Page A59 190. Geometry The hypotenuse of an isosceles right triangle is 5 centimeters long. How long are its sides? Synthesis Appendix A.5 Solving Equations A59 191. Geometry An equilateral triangle has a height of 10 inches. How long is one of its sides? (Hint: Use the height of the triangle to partition the triangle into two congruent right triangles.) 192. Flying Speed Two planes leave simultaneously from Chicago’s O’Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane. In Exercises 197–200, determine whether True or False? the statement is true or false. Justify your answer. x3 x 10 197. The equation is a linear equation. 198. If 2x 3x 5 8, then either 2x 3 8 or x 5 8. 199. An equation can never have more than one extraneous solution. 200. When solving an absolute value equation, you will always have to check more than one solution. 201. Think About It What is meant by equivalent equations? Give an example of two equivalent equations. N S W 2440 mi 202. Writing Describe the steps used to transform an equa- tion into an equivalent equation. E 203. To solve the equation 2 x2 3x 15x, each side by and solves the equation x 6 resulting solution Is there an error? Explain. a student divides 2x 3 15. The satisfies the original equation. x 193. Voting Population The total voting-age population P (in millions) in the United States from 1990 to 2002 can be modeled by P 182.45 3.189t 1.00 0.026t 0 ≤ t ≤ 12 , t where 1990. t 0 represents the year, with (Source: U.S. Census Bureau) corresponding to (a) In which year did the total voting-age population reach 200 million? (b) Use the model to predict when the total voting-age population will reach 230 million. Is this prediction reasonable? Explain. 194. Airline Passengers An airline offers daily flights C between Chicago and Denver. The total monthly cost C 0.2x 1 (in millions of dollars) of these flights is is the number of passengers (in thousands). The where total cost of the flights for June is 2.5 million dollars. How many passengers flew in June? x 195. Demand The demand equation for a video game is p 40 0.01x 1 where is the nummodeled by p is the price per unit. ber of units demanded per day and Approximate the demand when the price is $37.55. x 196. Demand The demand equation for a high definition television set is modeled by p 800 0.01x 1 is the number of units demanded per month and is the price per unit. Approximate the demand when the x where p price is $750. 204. Solve 3x 42 x 4 2 0 u x 4, (a) Let and solve the resulting equation for u Then solve the -solution for x. u. in two ways. (b) Expand and collect like terms in the equation, and solve the resulting equation for x. (c) Which method is easier? Explain. Think About It In Exercises 205–210, write a quadratic equation that has the given solutions. (There are many correct answers.) 11 and 206. 205. and 6 3 4 207. 8 and 14 2 1 and 6 5 1 2 3 5 208. 209. 210. and 1 2 and 3 5 In Exercises 211 and 212, consider an equation of the form x x a b, b where and are constants. a 211. Find and when the solution of the equation is b a x 9. (There are many correct answers.) 212. Writing Write a short paragraph listing the steps required to solve this equation involving absolute values and explain why it is important to check your solutions. 213. Solve each equation, given that and are not zero. b a ax2 bx 0 (a) (b) ax2 ax 0 333202_0A06.qxd 12/6/05 2:21 PM Page A60 A60 Appendix A Review of Fundamental Concepts of Algebra A.6 Linear Inequalities in One Variable What you should learn • Represent solutions of linear inequalities in one variable. • Solve linear inequalities in one variable. • Solve inequalities involving absolute values. • Use inequalities to model and solve real-life problems. Why you should learn it Inequalities can be used to model and solve real-life problems. For instance, in Exercise 101 on page A68, you will use a linear inequality to analyze the average salary for elementary school teachers. Introduction Simple inequalities were discussed in Appendix A.1. There, you used the >, inequality symbols to compare two numbers and to denote subsets of real numbers. For instance, the simple inequality and ≤, <, ≥ x ≥ 3 denotes all real numbers x that are greater than or equal to 3. Now, you will expand your work with inequalities to include more involved statements such as 5x 7 < 3x 9 and 3 ≤ 6x 1 < 3. As with an equation, you solve an inequality in the variable by finding all values of for which the inequality is true. Such values are solutions and are said to satisfy the inequality. The set of all real numbers that are solutions of an inequality is the solution set of the inequality. For instance, the solution set of x x x 1 < 4 is all real numbers that are less than 3. The set of all points on the real number line that represent the solution set is the graph of the inequality. Graphs of many types of inequalities consist of intervals on the real number line. See Appendix A.1 to review the nine basic types of intervals on the real number line. Note that each type of interval can be classified as bounded or unbounded. Example 1 Intervals and Inequalities Write an inequality to represent each interval, and state whether the interval is bounded or unbounded. a. b. c. d. 3, 5 3, 0, 2 , Solution 3, 5 a. 3, 0, 2 , d. b. c. corresponds to corresponds to 3 < x ≤ 5. 3 < x. 0 ≤ x ≤ 2. corresponds to corresponds to < x < . Now try Exercise 1. Bounded Unbounded Bounded Unbounded 333202_0A06.qxd 12/6/05 2:21 PM Page A61 Appendix A.6 Linear Inequalities in One Variable A61 Properties of Inequalities The procedures for solving linear inequalities in one variable are much like those for solving linear equations. To isolate the variable, you can make use of the Properties of Inequalities. These properties are similar to the properties of equality, but there are two important exceptions. When each side of an inequality is multiplied or divided by a negative number, the direction of the inequality symbol must be reversed. Here is an example. 2 < 5 32 > 35 Original inequality Multiply each side by 3 and reverse inequality. 6 > 15 Simplify. Notice that if the inequality was not reversed you would obtain the false statement 6 < 15. Two inequalities that have the same solution set are equivalent. For instance, the inequalities x 2 < 5 and x < 3 are equivalent. To obtain the second inequality from the first, you can subtract 2 from each side of the inequality. The following list describes the operations that can be used to create equivalent inequalities. Properties of Inequalities d and be real numbers. Let c,b, a, 1. Transitive Property a < b and b < c a < c 2. Addition of Inequalities a < b and c < d a c < b d 3. Addition of a Constant a < b a c < b c 4. Multiplication by a Constant For c > 0, a < b For c < 0, a < b ac < bc ac > bc Reverse the inequality. Each of the properties above is true if the symbol < is replaced by and For instance, another form of the multiplication ≤ ≥. the symbol > is replaced by property would be as follows. For c > 0, a ≤ b For c < 0, a ≤ b ac ≤ bc ac ≥ bc 333202_0A06.qxd 12/6/05 2:21 PM Page A62 A62 Appendix A Review of Fundamental Concepts of Algebra Solving a Linear Inequality in One Variable The simplest type of inequality is a linear inequality in one variable. For instance, 2x 3 > 4 is a linear inequality in x. In the following examples, pay special attention to the steps in which the in
equality symbol is reversed. Remember that when you multiply or divide by a negative number, you must reverse the inequality symbol. Example 2 Solving Linear Inequalities Checking the solution set of an inequality is not as simple as checking the solutions of an equation. You can, however, get an indication of the validity of a solution set by substituting a few convenient values of x. a. Solve each inequality. 5x 7 > 3x 9 1 3x 2 ≥ x 4 b. Solution a. 5x 7 > 3x 9 2x 7 > 9 2x > 16 x > 8 Write original inequality. Subtract 3x from each side. Add 7 to each side. Divide each side by 2. 8, . The solution set is all real numbers that are greater than 8, which is denoted by The graph of this solution set is shown in Figure A.8. Note that a parenthesis at 8 on the real number line indicates that 8 is not part of the solution set. 6 7 8 9 10 x Solution interval: FIGURE A.8 8, b. 1 3x 2 ≥ x 4 2 3x ≥ 2x 8 2 5x ≥ 8 5x ≥ 10 Write original inequality. Multiply each side by 2. Subtract 2x from each side. Subtract 2 from each side. x ≤ 2 Divide each side by 5 and reverse the inequality. The solution set is all real numbers that are less than or equal to 2, which is The graph of this solution set is shown in Figure A.9. denoted by Note that a bracket at 2 on the real number line indicates that 2 is part of the solution set. , 2. 0 1 2 3 4 x Solution interval: FIGURE A.9 , 2 Now try Exercise 25. 333202_0A06.qxd 12/6/05 2:21 PM Page A63 Appendix A.6 Linear Inequalities in One Variable A63 Sometimes it is possible to write two inequalities as a double inequality. 5x 2 < 7 4 ≤ 5x 2 and For instance, you can write the two inequalities more simply as 4 ≤ 5x 2 < 7. Double inequality This form allows you to solve the two inequalities together, as demonstrated in Example 3. Example 3 Solving a Double Inequality To solve a double inequality, you can isolate as the middle term. x 3 ≤ 6x 1 < 3 3 1 ≤ 6x 1 1 < 3 1 ≤ 6x 6 2 ≤ 6x < < Original inequality Add 1 to each part. Simplify. Divide each part by 6. Simplify. The solution set is all real numbers that are greater than or equal to . 3, 2 2 than which is denoted by 3, Figure A.10. and less The graph of this solution set is shown in 1 0 1 x Solution interval: 1 3, 2 3 FIGURE A.10 Now try Exercise 37. The double inequality in Example 3 could have been solved in two parts as follows. 3 ≤ 6x 1 2 ≤ 6x 1 3 ≤ x and 6x 1 < 3 6x < 4 2 3 x < The solution set consists of all real numbers that satisfy both inequalities. In other words, the solution set is the set of all values of for which x 1 3 ≤ x < 2 3 . When combining two inequalities to form a double inequality, be sure that the inequalities satisfy the Transitive Property. For instance, it is incorrect to x ≤ 1 This “inequality” combine the inequalities is wrong because 3 is not less than 1. 3 < x ≤ 1. 3 < x and as 333202_0A06.qxd 12/6/05 2:21 PM Page A64 A64 Appendix A Review of Fundamental Concepts of Algebra Te c h n o l o g y Inequalities Involving Absolute Values A graphing utility can be used to identify the solution set of the graph of an inequality. For instance, to find the solution set of rewrite the inequality as x 52 < 0, x 5 < 2 (see Example 4), enter Y1 abs X 5 2, and press the graph key. The graph should look like the one shown below. Solving an Absolute Value Inequality x Let be a variable or an algebraic expression and let be a real number such that a ≥ 0. a x < a if and only if x > a 1. The solutions of x < a 2. The solutions of greater than x > a a. are all values of x that lie between a and a. a < x < a. Double inequality are all values of x that are less than a or if and only if x < a or x > a. Compound inequality These rules are also valid if < is replaced by and > is replaced by ≥. ≤ 6 −4 −1 Example 4 Solving an Absolute Value Inequality 10 Solve each inequality. x 5 < 2 a. b. x 3 ≥ 7 Notice that the graph is below the -axis on the interval 3, 7. x Solution a Write original inequality. Write equivalent inequalities. Add 5 to each part. Simplify. The solution set is all real numbers that are greater than 3 and less than 7, which is denoted by The graph of this solution set is shown in Figure A.11. 3, 7. b Write original inequality. or Write equivalent inequalities. Subtract 3 from each side. x ≤ 10 x ≥ 4 Simplify. x 5 < 2 Note that the graph of the inequality can be described as all real numbers within two units of 5, as shown in Figure A.11. or The solution set is all real numbers that are less than or equal to greater than or equal to 4. The interval notation for this solution set is , 10 4, . is called a union symbol and is used to denote the combining of two sets. The graph of this solution set is shown in Figure A.12. The symbol 10 2 units 2 units 7 units 7 units : FIGURE A.11 Solutions lie inside x 8 3, 7 Now try Exercise 49. −12 −10 −8 −6 −4 −2 x 3 ≥ 7: FIGURE A.12 Solutions lie outside 0 2 4 x 6 10, 4 333202_0A06.qxd 12/6/05 2:21 PM Page A65 Appendix A.6 Linear Inequalities in One Variable A65 Applications A problem-solving plan can be used to model and solve real-life problems that involve inequalities, as illustrated in Example 5. Example 5 Comparative Shopping You are choosing between two different cell phone plans. Plan A costs $49.99 per month for 500 minutes plus $0.40 for each additional minute. Plan B costs $45.99 per month for 500 minutes plus $0.45 for each additional minute. How many additional minutes must you use in one month for plan B to cost more than plan A? Solution Verbal Model: Labels: Inequality: Monthly cost for plan B > Monthly cost for plan A Minutes used (over 500) in one month Monthly cost for plan A Monthly cost for plan B 0.45m 45.99 > 0.40m 49.99 m 0.40m 49.99 0.45m 45.99 0.05m > 4 m > 80 minutes (minutes) (dollars) (dollars) Plan B costs more if you use more than 80 additional minutes in one month. Now try Exercise 91. Example 6 Accuracy of a Measurement You go to a candy store to buy chocolates that cost $9.89 per pound. The scale that is used in the store has a state seal of approval that indicates the scale is accurate to within half an ounce (or of a pound). According to the scale, your purchase weighs one-half pound and costs $4.95. How much might you have been undercharged or overcharged as a result of inaccuracy in the scale? 1 32 Solution x represent the true weight of the candy. Because the scale is accurate Let to within half an ounce (or of a pound), the difference between the exact weight x of a pound. That is, x 1 is less than or equal to and the scale weight 1 32 1 2 1 32 2 ≤ 1 32. 1 32 15 32 You can solve this inequality as follows. ≤ x 1 2 ≤ x ≤ 17 32 ≤ 1 32 0.46875 ≤ x ≤ 0.53125 In other words, your “one-half pound” of candy could have weighed as little as 0.46875 pound (which would have cost $4.64) or as much as 0.53125 pound (which would have cost $5.25). So, you could have been overcharged by as much as $0.31 or undercharged by as much as $0.30. Now try Exercise 105. 333202_0A06.qxd 12/6/05 2:21 PM Page A66 A66 Appendix A Review of Fundamental Concepts of Algebra A.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The set of all real numbers that are solutions to an inequality is the ________ ________ of the inequality. 2. The set of all points on the real number line that represent the solution set of an inequality is the ________ of the inequality. 3. To solve a linear inequality in one variable, you can use the properties of inequalities, which are identical to those used to solve equations, with the exception of multiplying or dividing each side by a ________ number. 4. Two inequalities that have the same solution set are ________ ________. 5. It is sometimes possible to write two inequalities as one inequality, called a ________ inequality. 6. The symbol is called a ________ symbol and is used to denote the combining of two sets. In Exercises 1– 6, (a) write an inequality that represents the interval and (b) state whether the interval is bounded or unbounded. 1, 5 11, , 2 2, 10 5, , 7 1. 5. 3. 2. 6. 4. In Exercises 7–12, match the inequality with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (d) (e) (f) −5 −4 −3 −2 −1 0 1 2 3 4 5 −3 −2 −1 0 1 2 3 4 5 6 −5 −4 −3 −2 −. 9. 11. 10. 12. In Exercises 13–18, determine whether each value of solution of the inequality. x is a Inequality 5x 12 > 0 13. 14. 2x 1 < 3 Values (a) (c) (a) (cb) (d) (b) (d Inequality x 2 4 3 x 2 15. 0 < < 2 16. 1 < ≤ 1 17. x 10 ≥ 3 18. 2x 3 < 15 Values (a) (c) (a) (c) (a) (c) (a) (c 13 x 14 x 6 x 12 (b) (d) (b) (d) (b) (d) (b) (d) x 10 In Exercises 19–44, solve the inequality and sketch the solution on the real number line. (Some inequalities have no solutions.) 20. 22. 10x < 40 6x > 15 26. 19. 27. 24. 29. 23. 28. 25. 21. 4x < 12 2x > 3 x 5 ≥ 7 x 7 ≤ 12 2x 7 < 3 4x 3x 1 ≥ 2 x 2x 1 ≥ 1 5x 6x 4 ≤ 2 8x 4 2x < 33 x 4x 1 < 2x 3 3 4x 8x 1 ≥ 3x 5 1 2 2 9x 1 < 3 4 3.6x 11 ≥ 3.4 15.6 1.3x < 5.2 1 < 2x 3 < 9 37. 38. 8 ≤ 3x 5 < 13 16x 2 32. 36. 34. 33. 31. 35. 30. 333202_0A06.qxd 12/6/05 2:21 PM Page A67 39. 40. 41. 42. 43. 44. < 4 0 ≤ 4 < 2x .2 ≤ 0.4x 1 ≤ 4.4 1.5x 6 2 4.5 > > 10.5 In Exercises 45–60, solve the inequality and sketch the solution on the real number line. (Some inequalities have no solution.) x < 6 x > 4 46. 45 20 ≤ 6 x 8 ≥ 0 3 4x ≥ 9 1 2x < 5 2 ≥ 4 x 3 1 2x 3 < 1 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 9 2x 2 < 1 x 14 3 > 17 2x 10 ≥ 9 34 5x ≤ 9 Graphical Analysis In Exercises 61–68, use a graphing utility to graph the inequality and identify the solution set. 61. 62. 63. 64. 65. 66. 67. 68. 6x > 12 3x 1 ≤ 5 5 2x ≥ 1 3x 1 < x 7 x 8 ≤ 14 2x 9 > 13 2x 7 ≥ 13 2x 1 ≤ 3 1 Appendix A.6 Linear Inequalities in One Variable A67 Graphical Analysis In Exercises 69–74, use a graphing utility to graph the equation. Use the graph to approximate that satisfy each inequality. the values of x Equation y 2x 3 y 2 3x 1 y 1 2x 2 y 3x 8 y x 3 2x 1 y 1 69. 70. 71. 72. 73. 74. Inequalities (a) (a) (a) (a) (a) (ab) (b) (b) (b) (b) (b In Exercises 75– 80, find th
e interval(s) on the real number line for which the radicand is nonnegative. 75. 77. 79. x 5 x 3 47 2x 76. 78. 80. x 10 3 x 46x 15 81. Think About It The graph of can be described as all real numbers within three units of 5. Give x 10 < 8. a similar description of x 5 < 3 82. Think About It The graph of can be described as all real numbers more than five units from 2. Give a similar description of x 8 > 4. x 2 > 5 In Exercises 83–90, use absolute value notation to define the interval (or pair of intervals) on the real number line. 83. 84. 85. 86. −3 −2 −1 −3 −2 − 10 11 12 13 14 x x x x −7 − 6 −5 −4 −3 −2 −1 0 1 2 3 87. All real numbers within 10 units of 12 88. All real numbers at least five units from 8 89. All real numbers more than four units from 3 90. All real numbers no more than seven units from 6 91. Checking Account You can choose between two types of checking accounts at your local bank. Type A charges a monthly service fee of $6 plus $0.25 for each check written. Type B charges a monthly service fee of $4.50 plus $0.50 for each check written. How many checks must you write in a month in order for the monthly charges for type A to be less than that for type B? 333202_0A06.qxd 12/6/05 2:21 PM Page A68 A68 Appendix A Review of Fundamental Concepts of Algebra 92. Copying Costs Your department sends its copying to the photocopy center of your company. The center bills your department $0.10 per page. You have investigated the possibility of buying a departmental copier for $3000. With your own copier, the cost per page would be $0.03. The expected life of the copier is 4 years. How many copies must you make in the four-year period to justify buying the copier? 93. Investment In order for an investment of $1000 to grow to more than $1062.50 in 2 years, what must the annual interest rate be? A P1 rt 94. Investment In order for an investment of $750 to grow to more than $825 in 2 years, what must the annual interest rate be? A P1 rt 95. Cost, Revenue, and Profit The revenue for selling units The cost of producing units R 115.95x. x x of a product is is C 95x 750. To obtain a profit, the revenue must be greater than the cost. For what values of will this product return a profit? x 96. Cost, Revenue, and Profit The revenue for selling units The cost of producing units R 24.55x. x x of a product is is C 15.4x 150,000. To obtain a profit, the revenue must be greater than the cost. For what values of will this product return a profit? x 97. Daily Sales A doughnut shop sells a dozen doughnuts for $2.95. Beyond the fixed costs (rent, utilities, and insurance) of $150 per day, it costs $1.45 for enough materials (flour, sugar, and so on) and labor to produce a dozen doughnuts. The daily profit from doughnut sales varies between $50 and $200. Between what levels (in dozens) do the daily sales vary? 98. Weight Loss Program A person enrolls in a diet and 11 exercise program that guarantees a loss of at least 2 pounds per week. The person’s weight at the beginning of the program is 164 pounds. Find the maximum number of weeks before the person attains a goal weight of 128 pounds. 99. Data Analysis: IQ Scores and GPA The admissions office of a college wants to determine whether there is and grade-point a relationship between IQ scores averages after the first year of school. An equation that models the data the admissions office obtained is x y y 0.067x 5.638. (a) Use a graphing utility to graph the model. (b) Use the graph to estimate the values of grade-point average of at least 3.0. x that predict a 100. Data Analysis: Weightlifting You want to determine whether there is a relationship between an athlete’s (in pounds) and the athlete’s maximum benchweight press weight (in pounds). The table shows a sample of data from 12 athletes. y x (a) Use a graphing utility to plot the data. Athlete’s weight, x Bench-press weight, y 165 184 150 210 196 240 202 170 185 190 230 160 170 185 200 255 205 295 190 175 195 185 250 155 (b) A model for the data is Use a graphing utility to graph the model in the same viewing window used in part (a). y 1.3x 36. (c) Use the graph to estimate the values of that predict a maximum bench-press weight of at least 200 pounds. x (d) Verify your estimate from part (c) algebraically. (e) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete’s weight is not a particularly good indicator of the athlete’s maximum bench-press weight, list other factors that might influence an individual’s maximum bench-press weight. 101. Teachers’ Salaries The average salary (in thousands of dollars) for elementary school teachers in the United States from 1990 to 2002 is approximated by the model S 1.05t 31.0, 0 ≤ t ≤ 12 S t where 1990. represents the year, with (Source: National Education Association) corresponding to t 0 (a) According to this model, when was the average salary at least $32,000, but not more than $42,000? (b) According to this model, when will the average salary exceed $48,000? 333202_0A06.qxd 12/6/05 2:21 PM Page A69 102. Egg Production The number of eggs (in billions) produced in the United States from 1990 to 2002 can be modeled by E 1.64t 67.2, 0 ≤ t ≤ 12 E Appendix A.6 Linear Inequalities in One Variable A69 110. Music Michael Kasha of Florida State University used physics and mathematics to design a new classical guitar. He used the model for the frequency of the vibrations on a circular plate t where 1990. represents the year, with (Source: U.S. Department of Agriculture) corresponding to t 0 (a) According to this model, when was the annual egg production 70 billion, but no more than 80 billion? (b) According to this model, when will the annual egg production exceed 95 billion? 103. Geometry The side of a square is measured as 10.4 inches with a possible error of inch. Using these measurements, determine the interval containing the possible areas of the square. 1 16 104. Geometry The side of a square is measured as 24.2 centimeters with a possible error of 0.25 centimeter. Using these measurements, determine the interval containing the possible areas of the square. 105. Accuracy of Measurement You stop at a self-service gas station to buy 15 gallons of 87-octane gasoline at $1.89 a gallon. The gas pump is accurate to within of a gallon. How much might you be undercharged or overcharged? 1 10 106. Accuracy of Measurement You buy six T-bone steaks that cost $14.99 per pound. The weight that is listed on the package is 5.72 pounds. The scale that weighed the package is accurate to within ounce. How much might you be undercharged or overcharged? 1 2 107. Time Study A time study was conducted to determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two-thirds of the workers in the study satisfied the inequality t 15.6 1.9 < 1 t is time in minutes. Determine the interval on the where real number line in which these times lie. 108. Height The heights of two-thirds of the members of a h population satisfy the inequality h 68.5 2.7 ≤ 1 h is measured in inches. Determine the interval on where the real number line in which these heights lie. 109. Meteorology An electronic device is to be operated in an environment with relative humidity in the interval h 50 ≤ 30. defined by What are the minimum and maximum relative humidities for the operation of this device? h v 2.6t d2 E is the frequency (in vibrations per second), v where plate thickness (in millimeters), plate, density of the plate material. For fixed values of , the graph of the equation is a line (see figure). is the elasticity of the plate material, and d, is the is the diameter of the is the E, and ( 700 600 500 400 300 200 100 t 1 4 Plate thickness (in millimeters) 2 3 (a) Estimate the frequency when the plate thickness is 2 millimeters. (b) Estimate the plate thickness when the frequency is 600 vibrations per second. (c) Approximate the interval for the plate thickness when the frequency is between 200 and 400 vibrations per second. (d) Approximate the interval for the frequency when the plate thickness is less than 3 millimeters. Synthesis True or False? In Exercises 111 and 112, determine whether the statement is true or false. Justify your answer. 111. If 112. If c b, a, 10 ≤ x ≤ 8, and are real numbers, and 10 ≥ x then 113. Identify the graph of the inequality ac ≤ bc. a ≤ b, then x ≥ 8. and x a ≥ 2. (a) (cbd) x x 114. Find sets of values of and such that a solution of the inequality ax b ≤ c. b, a, c 0 ≤ x ≤ 10 is 333202_0A07.qxd 12/6/05 2:22 PM Page A70 A70 Appendix A Review of Fundamental Concepts of Algebra A.7 Errors and the Algebra of Calculus What you should learn • Avoid common algebraic errors. • Recognize and use algebraic techniques that are common in calculus. Why you should learn it An efficient command of algebra is critical in mastering this course and in the study of calculus. Algebraic Errors to Avoid This section contains five lists of common algebraic errors: errors involving parentheses, errors involving fractions, errors involving exponents, errors involving radicals, and errors involving dividing out. Many of these errors are made because they seem to be the easiest things to do. For instance, the operations of subtraction and division are often believed to be commutative and associative. The following examples illustrate the fact that subtraction and division are neither commutative nor associative. Not commutative 4 3 3 4 15 5 5 15 Not associative 8 6 2 8 6 2 20 4 2 20 4 2 Errors Involving Parentheses Potential Error a x b a x b a b2 a 2 b 2 b 1 1 2 2 3x 6 2 3x 2 2 a1 2 ab Errors Involving Fractions Potential Error bx 3x 3 13x 1 3x 1x 2 1 x 2 Correct Form 2ab b 2 b 1 1 ab 4 2 4 3x 6 2 3x 2 2 32x 22 a1 2 ab Comment Change all signs when distributing minus sign. Remember the middle term when squaring binomials. 1 2 occurs twice as a factor. When factoring, apply exponents to all fact
ors. Correct Form Comment Leave as a x b . Do not add denominators when adding fractions ab 1 a 1 b x ab 1 1 1 x 3x 3 13x 1 3 x x 3 Multiply by the reciprocal when dividing fractions. Use the property for adding fractions. Use the property for multiplying fractions. Be careful when using a slash to denote division. 1x 2 1 x 2 1 2x x Be careful when using a slash to denote division and be sure to find a common denominator before you add fractions. 333202_0A07.qxd 12/6/05 2:22 PM Page A71 Errors Involving Exponents Potential Error 2x 3 2x3 Correct Form x 2 3 x 23 x 6 x 2 x 3 x 23 x 5 2x 3 2x 3 1 x 2 x 3 x2 x3 Leave as 1 x 2 x 3 . Errors Involving Radicals Appendix A.7 Errors and the Algebra of Calculus A71 Comment Multiply exponents when raising a power to a power. Add exponents when multiplying powers with like bases. Exponents have priority over coefficients. Do not move term-by-term from denominator to numerator. Potential Error 5x 5x Correct Form 5x 5x Leave as Leave as x 2 a 2. x a. 1 bx Errors Involving Dividing Out Potential Error a bx a a ax a 1 x 2x a bx a a ax a 1 x 2x Correct Form bx a a a a1 x a 1 1 2 3 2 Comment Radicals apply to every factor inside the radical. Do not apply radicals term-by-term. Do not factor minus signs out of square roots. Comment x Divide out common factors, not common terms. Factor before dividing out. Divide out common factors. A good way to avoid errors is to work slowly, write neatly, and talk to yourself. Each time you write a step, ask yourself why the step is algebraically legitimate. You can justify the step below because dividing the numerator and denominator by the same nonzero number produces an equivalent fraction. 2x 6 2 x 2 3 x 3 Example 1 Using the Property for Adding Fractions Describe and correct the error. 1 2x 1 3x 1 5x Solution When adding fractions, use the property for adding fractions: 1 a 1 b b a ab . 1 2x 1 3x 3x 2x 6x2 5x 6x2 5 6x Now try Exercise 17. 333202_0A07.qxd 12/6/05 2:22 PM Page A72 A72 Appendix A Review of Fundamental Concepts of Algebra Some Algebra of Calculus In calculus it is often necessary to take a simplified algebraic expression and “unsimplify” it. See the following lists, taken from a standard calculus text. Useful Calculus Form Comment 5 8 x 4 x 2 3x 1 6 2x2 x 2 3 2 Write with fractional coefficient. Write with fractional coefficient. Factor out the leading coefficient. Unusual Factoring Expression 5x 4 8 x 2 3x 6 2x 2 x 3 x 112 x 112 x 2 x 112 2 x 2x 1 Factor out factor with lowest power. Writing with Negative Exponents Expression Useful Calculus Form Comment 9 5x3 7 2x 3 9 5 x3 72x 312 Move the factor to the numerator and change the sign of the exponent. Move the factor to the numerator and change the sign of the exponent. Writing a Fraction as a Sum Expression x 2x2 1 x Useful Calculus Form x12 2x 32 x12 1 x x 2 1 2x x 2 2x Comment Divide each term by x12. Rewrite the fraction as the sum of fractions. Add and subtract the same term. x 2 1 x 1 x 2 1 2x 2 2 x 2 2x 1 2x 2 x 2 2x 1 2 x 1 2 Rewrite the fraction as the difference of fractions Use long division. (See Section 2.3.) Use the method of partial fractions. (See Section 7.4.) 333202_0A07.qxd 12/6/05 2:22 PM Page A73 Appendix A.7 Errors and the Algebra of Calculus A73 Inserting Factors and Terms Expression Useful Calculus Form 2x 13 7x 24x 3 512 4x 2 9 4y 2 1 x x 1 2x 1 32 1 2 7 12 4x 3 51212x 2 y 2 x 2 14 94 Comment Multiply and divide by 2. Multiply and divide by 12. Write with fractional denominators. Add and subtract the same term. The next five examples demonstrate many of the steps in the preceding lists. Example 2 Factors Involving Negative Exponents Factor xx 112 x 112. Solution When multiplying factors with like bases, you add exponents. When factoring, you are undoing multiplication, and so you subtract exponents. xx 112 x 112 x 112xx 10 x 11 x 112x x 1 x 1122x 1 Now try Exercise 23. Another way to simplify the expression in Example 2 is to multiply the expression by a fractional form of 1 and then use the Distributive Property. xx 112 x 112 xx 112 x 112 x 112 x 112 xx 10 x 11 x 112 2x 1 x 1 Example 3 Inserting Factors in an Expression Insert the required factor: x 2 x2 4x 32 1 x2 4x 32 2x 4. Solution The expression on the right side of the equation is twice the expression on the left side. To make both sides equal, insert a factor of 1 2 x 2 x2 4x 32 1 x2 4x 32 Right side is multiplied and divided by 2. 2x 4 1 2. Now try Exercise 25. 333202_0A07.qxd 12/6/05 2:22 PM Page A74 A74 Appendix A Review of Fundamental Concepts of Algebra Example 4 Rewriting Fractions Explain why the two expressions are equivalent. y 2 1 4 4y 2 x 2 9 4 4x 2 9 Solution To write the expression on the left side of the equation in the form given on the right side, multiply the numerators and denominators of both terms by 1 4. 4y2 4x2 91 4 1 4 4y21 4 1 4 x2 9 4 y2 1 4 4x2 9 Now try Exercise 29. Example 5 Rewriting with Negative Exponents Rewrite each expression using negative exponents. a. 4x 1 2x22 Solution 4x 1 2x22 a. b. 2 5x 3 1 x 3 54x 2 4x1 2x22 b. Begin by writing the second term in exponential form. 2 5x 3 1 x 3 54x 2 3 1 2 x 12 5x 3 x3 x12 3 5 2 5 54x 2 4x2 Now try Exercise 39. Example 6 Writing a Fraction as a Sum of Terms Rewrite each fraction as the sum of three terms. a. x2 4x 8 2x Solution b. x 2x2 1 x a. x2 4x 8 2x 8 2x x2 4x 2x 2x x 2 4 x 2 b. x 2x2 1 x x x12 2x2 x12 x12 2x 32 x12 1 x12 Now try Exercise 43. 333202_0A07.qxd 12/8/05 11:27 AM Page A75 Appendix A.7 Errors and the Algebra of Calculus A75 A.7 Exercises VOCABULARY CHECK: Fill in the blanks. 1. To write the expression 2 x with negative exponents, move x to the ________ and change the sign of the exponent. 2. When dividing fractions, multiply by the ________. In Exercises 1–18, describe and correct the error. 1. 2. 3. 5. 7. 9. 11. 13. 15. 17. 2x 3y 4 2x 3y 4 5z 3x 2 5z 3x 2 14x 1 4 4 16x 2x 1 5z6z 30z ax ax ay y x 9 x 3 2x 2 1 2x 1 5x 5 1 1 1 a b a1 b1 x 2 5x12 xx 512 xx xyz xyxz x 1 xx 5 4. 6. 8. 4x 2 4x 2 10. 12. 14. 16. 18. x y x y y 25 x2 5 x 6x y 6x y 1 x y1 x2x 1 2 2x 2 x 2 1 2y 12y x 1 In Exercises 19–38, insert the required factor in the parentheses. 7x2 10 3 4x 1 7 10 1 4 2 19. 21. 23. 24. 25. 26. 27. 28. 29. 30. 2 20. 1 5 3x 5 1 3x 2 5 3x 2 1 x2x3 14 x3 143x2 x1 2x 23 1 2x234x 22. 3 1 x2 3x 73 1 x 2 2x 32 3 x 6x 5 3x3 2 4x 6 x 2 3x 7 3 x 1 x 2 2x 32 5 3 2x2 x x 1 2 169 16y 2 49 9y2 16 9x 2 25 3x 169 y 5 2 2x 3 2x 2 31. 32. 33. 34. 35. 36. 37. 38. y 2 23 y2 78 3y 2 12x 2 8y2 9x2 x 2 112 x2 49 x 13 5x 43 x 13 32x 1x 12 4x 32 x 12 1 3x43 4x1 3x13 1 3x13 1 2x 1 10 t 173 3 4 2x 1 32 2x 1 52 1 15 6 t 1 43 3t 143 5x 32 10x 52 1 2x 2x 1 32 3 7 28 In Exercises 39 – 42, write the expression using negative exponents. 39. 41. 3x2 2x 13 4 4 x4 3x 7x 32x 40. 42. x 1 x6 x12 1 x2 x x 2 8 39x3 In Exercises 43– 48, write the fraction as a sum of two or more terms. 43. 45. 47. 16 5x x 2 x 4x 3 7x 2 1 x13 3 5x 2 x 4 x 44. 46. 48. x 3 5x 2 4 x 2 2x 5 3x 3 5x 1 x 32 x 3 5x 4 3x 2 In Exercises 49 – 60, simplify the expression. 49. 50. 51. 2x2 332xx 13 3x 12x2 32 x 132 x 53x 2 142x x 2 135x 4 x 52 6x 1 327x 2 2 9x 3 2x36x 126 6x 13 2 333202_0A07.qxd 12/6/05 2:22 PM Page A76 A76 Appendix A Review of Fundamental Concepts of Algebra 52. 53. 54. 55. 56. 57. 58. 59. 60. 4x2 9128x 4x 2 9122 2x 31 4x 2 912 2 x 2 34x 323 x 313x 214 x 234 2 2 2x 112 x 22x 112 23x 113 2x 1 1 3 3x 1 23 3x 1233 x 1 1 2 2x 3x 2122 6x 2x 3x 212 x 1 2 2 1 x2 412 1 x2 4122x 2 2x 1 1 x2 6 x2 5123 2x 5 3x 2123 3x 2321 2 3x 2123x 6121 x 631 2 x2 5122x 3x 2323 2 61. Athletics An athlete has set up a course for training as part of her regimen in preparation for an upcoming triathlon. She is dropped off by a boat 2 miles from the nearest point on shore. The finish line is 4 miles down the coast and 2 miles inland (see figure). She can swim 2 miles per hour and run t (in hours) required for her to 6 miles per hour. The time reach the finish line can be approximated by the model t x2 4 2 4 x2 4 6 (c) The expression below was obtained using calculus. It can be used to find the minimum amount of time required for the triathlete to reach the finish line. Simplify the expression. 2xx2 412 1 6 y2 x 4x2 8x 2012 analytically. 1 y1 x 2 1232x x 2 1132x y1 62. (a) Verify that x 21 3 2x4x 2 3 3x 2 1 23 y2 (b) Complete the table and demonstrate the equality in part (a) numerically y1 y2 Synthesis True or False? statement is true or false. Justify your answer. In Exercises 63–66, determine whether the 63. 65. x1 y2 y2 x xy2 x 4 x 16 1 x 4 64. 66. 1 x2 y1 x2 9 x 3 x2 y x 3 x is the distance down the coast (in miles) to which where she swims and then leaves the water to start her run. In Exercises 67–70, find and correct any errors. If the problem is correct, state that it is correct. Start 2 mi Swim x 4 − x Run 2 mi Finish 67. x n x3n x3n2 69. x 2n y2n x n y n2 68. 70. x n2n x 2nn 2x 2n2 x 2n x 3n x 3n x2 x 5n x 3n x2 71. Think About It You are taking a course in calculus, and for one of the homework problems you obtain the following answer. 2x 152 1 6 2x 132 1 10 (a) Find the times required for the triathlete to finish x 1.0, . . . , when she swims to the points x 3.5, x 0.5, miles down the coast. x 4.0 and (b) Use your results from part (a) to determine the distance down the coast that will yield the minimum amount of time required for the triathlete to reach the finish line. in answer 2x 1323x 1. The the is 1 Show how the second answer can 15 be obtained from the first. Then use the same technique to simplify each of the following expressions. book back the of (a) (b) 2 3 2 3 x2x 332 2 15 x4 x32 2 15 2x 352 4 x52 333200_01_AN.qxd 12/12/05 11:14 AM Page A77 Answers to Odd-Numbered Exercises and Tests A77 Answers to Odd-Numbered Exercises and Tests Chapter 1 Section 1.1 (page 9) Vocabulary Check (page 9) 33. (a) y (− 4, 10) 10 8 6 2 (b) 17 0, 5 (c) 2 (b) vi 1. (a) v 2. Cartesian 4. Midpoint Formula 3. Distance Formula (c) i (d) iv (e) iii (f) ii − . A: 3. 2, 6, B: 6, 2, C: D: 3, 2 4, 4, 5
. 35. (a6 −4 −2 2 4 6 8 x − 4 −6 37. (a) 9. 3, 4 7. 13. Quadrant II 17. Quadrant III 21. y 11. Quadrant IV 5, 5 15. Quadrant III or IV 19. Quadrant I or III y 5 4 3 (− 1, 2) − 1 − 1 ( 5 ,− 2 4 3 ) −4 −6 (4, − 5) (b) (c) 210 2, 3 (5, 4 82 3 1, 7 6 (b) (c) C H A P T E R 1 5000 4500 4000 3500 3000 10 11 12 13 Year (6 ↔ 1996) 25. 5 23. 8 27. (a) 4, 3, 5 29. (a) (b) 10, 3, 109 42 32 52 102 32 1092 (b) (b) 10 (c) 5, 4 (9, 7) 31. (a) y 12 10 8 6 4 2 (1, 1) − 2 2 4 6 8 10 x − 5 2 −2 − 3 2 −1 − 1 2 1 2 39. (a) y 8 6 4 2 (− 3.7, 1.8) (b) (c) 110.97 1.25, 3.6 (6.2, 5.4 , 45. 41. 43. x1, 2ym x2 3y1 4 3x2 4 52 452 502 2xm 3x1 x1 2505 45 yards 47. 0, 1, 4, 2, 1, 4) 51. 55. $3.31 per pound; 2001 y1 y2 4 3y2 4 , x1 53. y1 , x2 2 y1 , y2 2 , 49. $3803.5 million 3, 6, 2, 10, 2, 4, 3, 4 57. 250% 333200_01_AN.qxd 12/9/05 1:22 PM Page A78 A78 Answers to Odd-Numbered Exercises and Tests 59. (a) The number of artists elected each year seems to be nearly steady except for the first few years. From six to eight artists will be elected in 2008. (b) The Rock and Roll Hall of Fame was opened in 1986. 61. 1998: $19,384.5 million; 2000: $20,223.0 million; 2002: $21,061.5 million 1.12 inches 34.47 Length of side 43 centimeters; area 800.64 square centimeters 63. 65. 67. (a) w (b) (c) l 1.5w; p 5w 7.5 meters 5 meters l 69. (a) y (b) 2002 210 205 200 195 190 185 180 ) 10 11 12 13 Year (6 ↔ 1996) x (c) Answers will vary. Sample answer: Technology now enables us to transport information in many ways other than by mail. The Internet is one example. 71. y 8 6 4 2 (−3, 5) (−2, 1) (3, 5) (2, 1 (−7, −3) − 4 − 6 − 8 (7, −3) (a) The point is reflected through the y-axis. (b) The point is reflected through the x-axis. (c) The point is reflected through the origin. 73. False. The Midpoint Formula would be used 15 times. 75. No. It depends on the magnitudes of the quantities measured. 77. b 83. x 1 81. 14 < x < 22 x 2 ± 11 79. d 85. 80. a 78. c 87. x < 3 5 Section 1.2 (page 22) Vocabulary Check (page 22) 1. solution or solution point 3. intercepts 6. numerical 4. y-axis 2. graph 5. circle; h, k; r (b) Yes 3. (a) No (b) Yes 1. (a) Yes 5, 0 5 x, y 1, 7 0, 5 1, 3 2, y 1, 4 0, 0 1, 2 2, 2 32 −1 −1 −2 13. 9. x-intercepts: y-intercept: xintercept: yintercept: 17. x-intercepts: y-intercept: y 21. ±2, 0 0, 16 4, 0 0, 2 0, 0, 2, 0 0, 0 4 3 2 1 11. 15. 19. 23. xintercept: y-intercept: xintercept: yintercept: xintercept: yintercepts: 5, 0 6 0, 6 7 3, 0 0, 7 6, 0 0, ± 6 y 4 3 2 1 –4 –3 –1 1 3 4 –4 –3 –2 1 2 3 4 x x −2 –2 –3 –4 25. y-axis symmetry 29. Origin symmetry 27. Origin symmetry 31. x-axis symmetry 333200_01_AN.qxd 12/9/05 1:23 PM Page A79 Answers to Odd-Numbered Exercises and Tests A79 35. y 4 3 2 (0, 10, 0) (2, 0) −2 −1 1 2 3 4 x −1 −2 33. 37. y 7 6 5 4 2 1 (0, 3) 391 3( −3, 0 ( −4 − 3 − 2 41. y 43. (3, 0 10 12 10 8 6 4 2 − 2 − 2 (0, 6) (−1, 0) (0, 1) – 2 1 2 3 4 (6, 0) 2 4 6 8 10 12 x (0, −1) –2 –3 45. 10 47. − 10 10 − 10 10 −10 Intercepts: − 10 6, 0, 0, 3 Intercepts: 3, 0, 1, 0, 0, 3 49. 10 51. 10 −10 10 −10 10 Intercept: 53. −10 0, 0 10 Intercept: 55. −10 0, 0 10 −10 10 −10 10 Intercepts: −10 0, 0, 6, 0 Intercepts: −10 3, 0, 0, 3 x 2 2 y 1 2 16 0, 0; Radius: 5 67. Center: 1, 3; Radius: 3 x2 y 2 16 25 59. 57. 61. 63. 65. Center: y 6 4 3 2 1 −1 −2 −3 −4 −6 −4 −3 −2 −3 −2 (0, 0) 1 2 3 4 6 x 69. Center: 1 2, 1 2 ; Radius: 3 2 71. 1 2 4 5 x (1, −3) y 1 −1 −2 −3 −4 −5 −6 −1 1 2 3 x 73. (a) y x 250,000 200,000 150,000 100,000 50,000 Year (b) Answers will varyc) 8000 (d) x 86 2 3, y 86 2 3 0 0 180 (e) A regulation NFL playing field is 120 yards long and yards wide. The actual area is 6400 square yards. 531 3 75. (a) and (b) y 100 80 60 40 20 40 60 80 20 Year (20 ↔ 1920) t 100 (c) 66.0 years (e) Answers will vary. (d) 2005: 77.0 years; 2010: 77.1 years 333200_01_AN.qxd 12/9/05 1:23 PM Page A80 A80 Answers to Odd-Numbered Exercises and Tests 77. False. A graph is symmetric with respect to the x-axis if, is also on the graph. 79. The viewing window is incorrect. Change the viewing is on the graph whenever x, y x, y window. Answers will vary. 9x5, 4x3, 7 107x x 83. 3t 87. 22x 81. 85. Section 1.3 (page 34) Vocabulary Check (page 34) 17. m 0; y -intercept: y 0, 3 (0, 3) 5 4 2 1 −3 −2 −1 −1 1 2 3 x 19. m is undefined. There is no y-intercept. y 4 3 2 1 −7 −6 −4 x 1 −3 −2 −1 −1 −2 −3 −4 2. slope 1. linear 4. perpendicular 6. linear extrapolation 7. a. iii b. i c. v d. ii e. iv 3. parallel 21. 5. rate or rate of change 23. (1, 6) (−6, 4) y 6 5 4 2 1 –5 –4 –3 –1 1 2 3 x –8 (−6, −1) –2 y 6 4 2 –2 x (−3, −2) m 2 25 is undefined. 27. (−5.2, 1.6) y 8 6 4 (4.8, 3.1) −6 − 4 −2 2 4 6 x x 4 5 6 ( 11 , 1, 3, 1, 1, 1 6, 5, 7, 4, 8, 3 8, 0, 8, 2, 8, 3 4, 6, 3, 8, 2, 10 9, 1, 11, 0, 13, 1 y 3x 2 41. 29. 31. 33. 35. 37. 39. y 2 1 –2 –1 1 2 3 4 x –1 –2 (0, − 2) − 2 − 4 m 0.15 y 2x (−3, 6) y 6 4 –6 –4 –2 2 4 6 x –2 –4 –6 L2 1. (a) y 3. (b) L3 (c) L1 (2, 33 x m = 2 1 2 5. 9. 4 3 7. 2 m 5; y -intercept: 0, 3 y 5 4 3 (0, 3) x x − 4 −3 − 2 − 1 1 2 3 13. is undefined. m y There is no -intercept. y 2 1 –1 1 2 3 –1 –2 11. m 1 2; y -intercept: 00, 4) − 15. −2 m 7 6; y -intercept: y 0, 5 (0, 51 −2 333200_01_AN.qxd 12/9/05 1:23 PM Page A81 434, 0) x –1 1 2 3 4 Answers to Odd-Numbered Exercises and Tests A81 45. x 6 y 6 4 2 –4 –2 2 4 x (6, −1) –2 –4 –6 632 −3 −4 −5 −6 −7 −8 ) , −8 ) 7 3 –1 –2 47 49. y 5x 27.3 y 3 2 1 (−5.1, 1.8) −7 −6 −4 −3 −2 −1 1 x ) 2) 4 51. y 3 5 x 2 (−5, 5) –6 –4 –2 y 8 6 4 –2 –4 55. y 1 2 x 3 2 2 (5, −1) 6 57. x x −2 −3 −4 −5 53. x 8 (−8, 7) (−8, 1) – 10 – 18 25 , 1 2 −1 1 2 3 −1 y 2 1 −2 −1 , − ( − 1 10 ( 3 5 x −2 x 1 2 ( 9 10 , − ( 9 5 59. y 0.4x 0.2 61. y 1 y 3 2 1 (1, 0.6) y 3 2 1 −3 (−2, −0.61) (2, −1) −2 −3 2 x 2 3 x 127 72 8 67. Parallel y 1 (b) y 4 y 2x 3 y 3 4 x 3 y 0 (b) x 2 (b) y x 4.3 3x 2y 6 0 x y 3 0 65. Perpendicular 69. (a) 71. (a) 73. (a) 75. (a) 77. (a) 79. 83. 85. Line (b) is perpendicular to line (c). (a) (b) x 1 y 5 (b) (b) 4 y x 9.3 81. 12x 3y 2 0 (c) 6 −6 −4 87. Line (a) is parallel to line (b). Line (c) is perpendicular to line (a) and line (b). x 8 (c10 −8 14 (b) (a) 3x 2y 1 0 89. 93. (a) Sales increasing 135 units per year 91. 80x 12y 139 0 (b) No change in sales (c) Sales decreasing 40 units per year 95. (a) Salary increased greatest from 1990 to 1992; Least from 1992 to 1994 (b) Slope of line from 1990 to 2002 is about 2351.83 (c) Salary increased an average of $2351.83 over the 12 years between 1990 and 2002 97. 12 feet 99. 101. b; The slope is Vt 3165 125t 20, amount of the loan each week. The y-intercept is which represents the original amount of the loan. which represents the decrease in the 0, 200 102. c; The slope is 2, which represents the hourly wage per which repre- 0, 8.50 unit produced. The y-intercept is sents the initial hourly wage. 103. a; The slope is 0.32, which represents the increase in 0, 30 travel cost for each mile driven. The y-intercept is which represents the amount per day for food. 333200_01_AN.qxd 12/12/05 11:15 AM Page A82 A82 Answers to Odd-Numbered Exercises and Tests 104. d; The slope is 100, which represents the decrease in the value of the word processor each year. The y-intercept is 0, 750 which represents the initial purchase price of the computer. y 0.4825t 2.2325; V 175t 875 y18 $6.45; y20 $7.42 yt 179.5t 40,571 y8 42,007; y10 42,366 (c) m 179.5 C 16.75t 36,500 P 10.25t 36,500 (b) (d) R 27t t 3561 hours (b) y 8x 50 105. 107. 109. (a) (b) S 0.85L 111. 113. (a) (c) 115. (a) 10 m x 15 m x (c) 150 (d) m 8, 8 meters 0 0 10 C 0.38x 120 117. 119. (a) and (b ( 150 125 100 75 50 25 x 4 8 10 12 2 6 Year (0 ↔ 1990) (c) Answers will vary. Sample answer: y 11.72x 14.1 (d) Answers will vary. Sample answer: The y-intercept indicates that initially there were million subscribers which doesn’t make sense in the context of this problem. Each year, the number of cellular phone subscribers increases by 11.72 million. 14.1 (e) The model is accurate. (f) Answers will vary. Sample answer: 196.9 million 121. False. The slope with the greatest magnitude corresponds to the steepest line. 123. Find the distance between each two points and use the Pythagorean Theorem. y 125. No. The slope cannot be determined without knowing the scale on the -axis. The slopes could be the same. 127. V -intercept: initial cost; Slope: annual depreciation 129. d 133. 137. No solution 130. c 135. 132. b 131. a 1 7 2, 7 139. Answers will vary. Section 1.4 (page 48) Vocabulary Check (page 48) 1. domain; range; function 2. verbally; numerically; graphically; algebraically 3. independent; dependent 5. implied domain 6. difference quotient 4. piecewise-defined 3. No 1. Yes 5. Yes, each input value has exactly one output value. 7. No, the input values of 7 and 10 each have two different output values. 9. (a) Function (b) Not a function, because the element 1 in A corre- sponds to two elements, 2 and 1, in B. (c) Function (d) Not a function, because not every element in A is matched with an element in B. 11. Each is a function. For each year there corresponds one and only one circulation. 13. Not a function 19. Not a function 1 9 25. (a) (b) 9 36 27. (a) (b) 2 29. (a) 1 (b) 2.5 15. Function 21. Function 2x 5 (c) 32 r3 (c) 3 3 2x (c) 17. Function 23. Not a function 31. (a) 1 9 (b) Undefined 1 y2 6y 33. (a) 1 (b) 1 (c) (c) x 1 x 1 1 7 35. (a) 37. (a) 39. x (b) 2 (b) 4 (c) 6 (c 41 43, ±1 4 3 ±3 51. 49. 47. 57. All real numbers 45. 5 55. 3, 0 59. All real numbers t except 61. All real numbers y such that 63. All real numbers x such that 65. All real numbers x except x 0 53. 2, 1 333200_01_AN.qxd 12/9/05 1:23 PM Page A83 Answers to Odd-Numbered Exercises and Tests A83 except s 4 101. (a) s ≥ 1 x > 0 67. All real numbers s such that 69. All real numbers x such that 71. 73. 75. 2, 4, 1, 1, 0, 0, 1, 1, 2, 4 2, 4, 1, 3, 0, 2, 1, 3, 2, 4 gx cx2; c 2 rx c x 3 h, h 0 x 3 9x2 , x 3 f x cx; c 1 76. 4 hx cx; c 3 3x 2 3xh h2 3, h 0 A P2 16 5x 5 x 5 77. 79. 83. ; c 32 78. 87. 81. 85. 89. (a) The maximum volume is 1024 cubic centimeters. (b) V 1200 1000 800 600 400 200 Height 5 6
x Yes, V is a function of x. V x24 2x2, 0 < x < 12 (c) 91. A x2 2x 2, x > 2 C 12.30x 98,000 R 17.98x P 5.68x 98,000 97. (a) (b) (c) 93. Yes, the ball will be at a height of 6 feet. 95. 1990: $27,300 1991: $28,052 1992: $29,168 1993: $30,648 1994: $32,492 1995: $34,700 1996: $37,272 1997: $40,208 1998: $41,300 1999: $43,800 2000: $46,300 2001: $48,800 2002: $51,300 99. (a) R 240n n2 20 , n ≥ 80 (b) n 90 100 110 120 130 140 150 Rn $675 $700 $715 $720 $715 $700 $675 The revenue is maximum when 120 people take the trip. d h 3000 ft (b) h d2 30002, d ≥ 3000 103. False. The range is 105. The domain is the set of inputs of the function, and the 1, . range is the set of outputs. 107. (a) Yes. The amount you pay in sales tax will increase as the price of the item purchased increases. (b) No. The length of time that you study will not neces- sarily determine how well you do on an exam. 109. 115. 15 111. 8 10x 9y 15 0 1 5 113. 2x 3y 11 0 Section 1.5 (page 61) Vocabulary Check (page 61) 2. Vertical Line Test 1. ordered pairs 3. zeros 5. maximum 7. odd 8. even 4. decreasing 6. average rate of change; secant , 1, 1, 1. Domain: Range: 0, (b) 1 (b) 0 5. (a) 0 3 7. (a) 11. Not a function 17. 0 25. 19. 6 0, ± 2 (d) (c) 0 (d) (c) 1 13. Function 2, 6 27. 21. ± 1 3. Domain: Range: 0, 4 4, 4 2 3 9. Function 5 2 1 2 , 6 15. 231 3 , −6 11 2 31. Increasing on 9 3 −9 5 3 29. −3 1 3 −6 2 −2 33. Increasing on , 0 and 2, Decreasing on 0, 2 333200_01_AN.qxd 12/9/05 1:23 PM Page A84 A84 Answers to Odd-Numbered Exercises and Tests 35. Increasing on 37. Increasing on Constant on 4 39. , 0 1, ; 1, 1 and 2, ; Decreasing on Constant on , 1 − 3 3 0 Constant on , 43. − 3 1 − 3 3 45. − 4 3 2 − 1 Decreasing on , 1 49. − 8 2 − 10 Relative minimum: 1, 9 53. − 12 10 −6 12 0, 2 59. y 61. y 41. −6 7 −1 6 Decreasing on Increasing on , 0 0, Increasing on Decreasing on , 0 0, 47. 4 –2 –2 –3 –4 5 4 3 2 1 −1 −1 1, f x < 0 0 x1 1 x1 1 x1 3 x1 for all 2. x2 is to x2 is 18. to x2 is 0. to x2 is to 73. Odd; origin symmetry x 3 5 3 11 1 4. 63. The average rate of change from 65. The average rate of change from 67. The average rate of change from 69. The average rate of change from 71. Even; -axis symmetry 75. Neither even nor odd; no symmetry 77. 81. 85. (a) h x 2 4x 3 L 1 L 4 y 2 2y 2 83. 79. 6000 y h 2x x2 (b) 30 watts 0 0 6 Increasing on 2 0, 20 0 90 6 87. (a) Ten thousands 89. (a) 2200 (b) Ten millions (c) Percents 51. 10 −3 −4 Relative maximum: 1.5, 0.25 2 0 7 (b) The average rate of change from 2002 to 2007 is 408.56. The estimated revenue is increasing each year at a fast pace. s 16t2 64t 6 91. (a) (b) 100 0 0 5 16 (c) Average rate of change (d) The slope of the secant line is positive. 16t 6 (e) Secant line: (f) 100 Relative maximum: Relative minimum: 1.79, 8.21 1.12, 4.06 55. y 57. 5 4 3 2 1 –3 −2 −1 −1 1 2 3 x , 4 , 1, 0, 0 0 5 333200_01_AN.qxd 12/9/05 1:23 PM Page A85 93. (a) (b) s 16t2 120t 270 0 0 8 8 (c) Average rate of change (d) The slope of the secant line is negative. 8t 240 (e) Secant line: (f) 270 Answers to Odd-Numbered Exercises and Tests A85 (e) −6 4 −4 (f) 6 −6 4 −4 6 x All the graphs pass through the origin. The graphs of the odd powers of are symmetric with respect to the origin, and the graphs of the even powers are symmetric with y respect to the -axis. As the powers increase, the graphs become flatter in the interval 109. 0, (b) (b) 28 (c) 27 9 1 < x < 1. 5x 43 ±1 107. 0, 10 111. (a) 37 9 113. (a) 95. (a) (b) 0 0 s 16t2 120 140 0 0 8 4 (c) The given value is not in the domain of the function. h 4, h 0 115. Section 1.6 (page 71) Vocabulary Check (page 71) 1. g 6. e 2. i 7. f 3. h 8. c 4. a 9. d 5. b 32 (c) Average rate of change (d) The slope of the secant line is negative. 32t 120 (e) Secant line: (f) 140 0 0 4 97. False. The function real numbers. f x x2 1 has a domain of all 99. (a) Even. The graph is a reflection in the -axis. (b) Even. The graph is a reflection in the -axis. (c) Even. The graph is a vertical translation of f. (d) Neither. The graph is a horizontal translation of f. x y 101. (a) 103. (a) 105. (a) 2, 4 3 4, 9 (b) 2, 4 3 4, 9 (b) 4 (b) −6 −6 (c) −4 4 −4 6 −6 (d) 6 −6 4 −4 4 −. (a) (b) y 12 10 8 6 4 2 7. (a) (b) y 1 −1 −2 −3 −4 −5 −6 −8 −9 −6 11. x x f x 2x 6 1. (a) (b) y 6 5 4 3 2 1 −. (a) (b) y 3 2 1 −3 −2 −1 1 2 3 −2 −3 4 9. −6 6 −4 f x 3x 11 2 6 8 10 12 f x 6 7 x 45 6 333200_01_AN.qxd 12/9/05 1:23 PM Page A86 Answers to Odd-Numbered Exercises and Tests 15. 18 45. − 10 20 4 −4 4 −4 4 −4 A86 13. −6 −6 17. 21. 12 −1 −1 25. −6 29. (a) 2 31. (a) 1 33. (a) 6 35. (a) 37. 10 6 6 9 6 4 (c) (c) 7 −2 7 −3 5 −5 19. 23. 27. −7 −5 −9 8 10 3 4 −4 (d) (d) 3 19 (d) 1 39. 22 (d) 41 y 2 1 −4 −3 −2 −1 1 2 3 4 x 11 (b) 4 (c) 6 (c) (b) 2 (b) 3 (b) y 4 3 2 x 3 4 43 412 −5 −6 y 4 3 1 –1 1 2 3 4 x –1 –2 y 5 4 3 1 –4 –3 –2 –1 1 2 3 4 x 49. −4 −3 –2 –3 y 5 4 3 2 1 −1 −1 −2 −3 1 2 3 4 x 47. y 10 8 4 2 –4 –2 2 4 6 8 –2 x 51. (a) 8 −9 9 −4 (b) Domain: Range: , ; 0, 2 (c) Sawtooth pattern gx x 2 1 gx x 13 2 gx 2 gx x 2 (b) (b) (b) (b) 53. (a) 55. (a) 57. (a) 59. (a) 61. (a) f x x f x x3 Time (in minutes) 8 10 12 63. (a ( 48 40 32 24 16 8 (b) $5.64 (b) $50.25 t x 4 2 10 6 Weight (in pounds) 8 65. (a) (b) W30 360; W40) 480; W45 570; W50) 660 Wh 12h, 18h 45 540, 0 < h ≤ 45 h > 45 67. (a) f x 0.505x2 1.97x 1.47x 26.3, 6.3, 1 ≤ x ≤ 6 6 < x ≤ 12 Answers will vary. Sample answer: The domain is determined by inspection of a graph of the data with the two models. 333200_01_AN.qxd 12/9/05 1:23 PM Page A87 (b) y 3. (a) (b) Answers to Odd-Numbered Exercises and Tests A87 e u n e v e R f o ) 20 s r 18 a l l 16 o d 14 12 10 Month (1 ↔ January) 7 8 x 10 11 12 (c) f 5 11.575, f 11 4.63; These values represent the revenue for the months of May and November, respectively. (d) These values are quite close to the actual data values. 69. False. A piecewise-defined function is a function that is defined by two or more equations over a specified domain. That domain may or may not include - and -intercepts 3x 6, 5x 16 75. Neither 71. 73 Section 1.7 (page 79) Vocabulary Check (page 79) 2. f x; f x 1. rigid 4. horizontal shrink; horizontal stretch 5. vertical stretch; vertical shrink 6. (a) iv (c) iii (b) ii (d) i 3. nonrigid 1. (a) (b1 x x c = 1 c = −1 c x x −4 −2 2 4 −2 (c2 3 4 x −4 (c) −4 −3 5. (a) y y (b) y 4 3 2 1 −1 −2 (df) (6, 2) (5, 1) x 4 5 6 (3, 0) 1 2 3 (2, −10, 1) (1, 0) x 1 3 4 5 (3, −1) (4, −2) (4, 4) (3, 3) 5 4 3 2 1 (1, 2) (0, 1) 1 2 3 4 5 x (c) y 4 3 2 1 (1, 0) (4, 4) (3, 20, −2) − 2 − 3 (e) y 3 2 (1, 2) (−4, 2) y y 3 2 (−2, 0) (0, 1) (−3, 1) (−1, 0) − 3 − 1 1 2 x −5 −4 −3 −2 −1 (− 3, −1) − 1 − 2 x (0, − 1) − 2 333200_01_AN.qxd 12/9/05 1:23 PM Page A88 A88 Answers to Odd-Numbered Exercises and Tests (c) (d) 21. (a) f x x3 (g8, 2) (6, 1) 3 4 5 6 7 8 9 (2, 0) 2 (0, −1) 7. (a) y y (− 2, 3) 3 1 −1 −2 − 2 − 1 (0, 2) (1, −1) 3 x x (b) y y (− 1, 4) 4 3 2 1 (1, 3) (3, − 2) −1 −1 1 (2, 0) x 4 (4, −1) (2, 4) (− 3, 4) (0, 3) (− 1, 3 (−1, 0) −3 −1 1 2 (−3, −1) − 1 x −3 −2 −1 (0, 0) x 2 −1 (2, −1) (f) (5, 1) (3, 0) 1 2 4 5 (−2, 2) x y y 3 2 1 ) 0, ) 3 2 (1, 0) −2 −1 1 −1 −2 x ) 3, − ) 1 2 x (eg) (0, −4) (−1, 4) (2, − 3) y 5 4 3 2 1 (0, 3) 2( , 01 ( − ( 3 ( , −1 y x 52 3 y 1 x 12 (d) (b) 9. (a) (c) 11. (a) (c) y x2 1 y x 22 6 y x 5 y x 3 (b) y x 2 4 (d) y x 3; 13. Horizontal shift of 15. Reflection in the -axis of x 17. Reflection in the y 1 x f x x2 19. (a) x y x 6 1 y x 23 y x2 ; y x2 -axis and vertical shift of y x ; (b) Reflection in the x-axis, and vertical shift 12 units upward, of f x x2 (c) y (d) gx 12 f x 12 4 − 4 − 8 − 12 − 12 − 8 x 8 12 (b) Vertical shift seven units upward, of (c) (d) y f x x3 gx f x 7 11 10 9 8 5 4 3 2 1 −6 −5 −4 −3 −1 1 2 3 4 5 6 x 23. (a) f x x2 (b) Vertical shrink of two-thirds, and vertical shift four f x x2 (d) gx 2 3 f x 4 units upward, of (c) y 7 6 5 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 x 25. (a) f x x2 (b) Reflection in the x-axis, horizontal shift five units to the left, and vertical shift two units upward, of f x x2 333200_01_AN.qxd 12/9/05 1:23 PM Page A89 Answers to Odd-Numbered Exercises and Tests A89 (c2 −1 1 x −2 −3 −4 (d) gx 2 f x 5 (c) (d) gx 27. (a) f x x 35. (a) f x x (b) Horizontal shrink of of (c) y 1 3, f x x (d) gx f 3x 29. (a) f x x3 (b) Vertical shift two units upward, and horizontal shift one f x x3 unit to the right, of y (d) gx f x 1 2 (c 31. (a) f x x (b) Reflection in the x-axis, and vertical shift two units f x x (d) gx f x 2 1 2 3 x downward, of y (c 33. (a) f x x (b) Reflection in the x-axis, horizontal shift four units to the f x x left, and vertical shift eight units upward, of (b) Reflection in the x-axis, and vertical shift three units f x x (d) gx 3 f x upward, of y (c3 −2 −1 1 2 3 x 6 −2 −3 37. (a) f x x (b) Horizontal shift of nine units to the right, of (c) gx f x 9 (d) y f x x 15 12 9 6 3 3 6 9 12 15 x 39. (a) f x x (b) Reflection in the y-axis, horizontal shift of seven units to the right, and vertical shift two units downward, of f x x (d) gx f7 x 2 (c 41. (a) f x x (b) Horizontal stretch, and vertical shift four units down- ward, of f x x 333200_01_AN.qxd 12/9/05 1:23 PM Page A90 A90 Answers to Odd-Numbered Exercises and Tests (c) y (d) gx f 1 2x 45. f x x 133 f x x 6 49. f x x 22 8 43. f x x 10 47. y 3x2 51. (a) (b) 2x y 1 53. (a) (b) y x 3 ; 55. Vertical stretch of 57. Reflection in the x 2 x2 y 1 y 4x2 3 y 3x 3 y 2 x3 -axis and vertical shrink of y x2 ; 59. Reflection in the -axis and vertical shrink of 23 2 61. 65. (a) 63. y x 3 (b) 67. (a) Horizontal stretch of 0.035 and a vertical shift of 20.6 units upward 30 35 ) 40 ( 20 25 15 t 8 4 12 20 Year (0 ↔ 1980) 16 (b) 0.77-billion-gallon increase in fuel usage by trucks The graph is shifted (c) each year f t 20.6 0.035t 102. 10 units to the left. (d) 52.1 billion gallons. Yes. 69. True. 71. (a) (c) x x 4 f t gt 3 gt f t 2 (b) gt f t 10,000 73. 2, 0 , 1, 1, 0, 2 75. 4 x1 x 77. 3x 2 xx 1 5x 3, 79. x 4x2 4 x2 4 x 3 81. 57 83. (a) 38 (b) 4 85. All real numbers x except 87. All real numbers x such that (c) x 1 x2 12x 38 4 −3 x x (d) −4 −3 −2 (f) −4 − 3 −c
e2 −3 −4 −5 −6 4 3 2 1 −2 −3 −4 −5 −6 −4 −2 −2 −4 −6 −8 x g 65 Section 1.8 (page 89) Vocabulary Check (page 89) 1. addition; subtraction; multiplication; division gx 4. inner; outer 2. composition 3. x 4 5 6 1 g 1. y 3. y 4 3 2 1 5. (a) (d) 7. (a) g 4 2 x 6 8 10 c) x2 4 all real numbers x except x 2 (b) 4 2x x 2 x 2 x2 4x 5 ; (d) x2 4x 5 ; (b) x2 4x 5 (c) all real numbers x except x 5 4 4 x3 5x2 333200_01_AN.qxd 12/9/05 1:23 PM Page A91 Answers to Odd-Numbered Exercises and Tests A91 (b) x2 6 1 x 55. T 3 4 x 1 15 x2 9. (a) (c) (d) x2 6 1 x x2 61 x x2 61 x 1 x ; all real numbers x such that 11. (a) x 1 x2 (b) x 1 x2 (c) (d) x; all real numbers x except 1 x 3 x 0 17. 9t 2 3t 5 19. 74 13. 3 21. 26 25. 15. 5 23 ( 300 250 200 150 100 50 T B R x 10 20 30 60 Speed (in miles per hour) 40 50 27. y 5 4 3 f 57. (a) (b) 59. (a) f + g x –3 –2 –1 3 4 −2 g (b 29. −15 10 f 15 f + g −10 g f (x), g(x) 31. (a) 33. (a) x 35. (a) (b) (b) x 1 x 12 (b) x (c) x2 4 (b) f Domains of and g Domains of and g f : f g: x2 1 g f : f g: x 6 f g, and 1 x g f : Domains of and Domains of Domains of Domains of and Domains of and x 6 (b) g, Domains of 1 x 3 f g: f g: 3 f g (b) f, 37. (a) 39. (a) 41. (a) 43. (a) 3 47. 49. f (x) x2, g(x) 2x 1 f (x) 3x, g(x) x2 4 f (x(x) x 2 gx x2 , 51. 53. (c) x2 1 x4 3 3x 1 1 x 4 x ≥ 4 all real numbers all real numbers all real numbers x such that x ≥ 0 g f : all real numbers all real numbers x except x 0 all real numbers all real numbers x except 45. (a) 0 (b) 4 x 3 (b) 0 pt 100 is the population change in the year 2005. ct pt bt dt c5 A Nt 5.31t2 102.0t 1338 A N4 1014.96 A N8 861.84 A N12 878.64 A Nt 1.41t2 17.6t 132 A N4 84.16 A N8 81.44 A N12 123.84 10.20t 92.7 y1 3.357t 2 26.46t 379.5 y2 0.465t2 9.71t 7.4 y3 y2 y1 represents the amount spent on health care in the United States. 720 2.892t 2 6.55t 479.6; y3 this amount y1 + y2 + y3 y2 y1 y3 10 61. (a) (b) (c) 63. (a) (d) In 2008, $1298.708 billion is estimated to be spent on health services and supplies, and in 2010, $1505.4 billion is estimated. r (x) x (b) 2 A rx x 2 circular base of the tank on the square foundation with side length NTt 303t2 2t 20 ber of bacteria in the food as a function of time. t 2.846 hours Ar r 2 2 This represents the num- represents the area of the A rx (c) x. ; represents 3 percent of an amount over $500,000. f gx 6x 1 and g f x 6x 6 65. (a) (b) g f x 67. 69. False. 71. Answers will vary. 73. 3 75. 4 xx h 333200_01_AN.qxd 12/9/05 1:23 PM Page A92 A92 Answers to Odd-Numbered Exercises and Tests 77. 3x y 10 0 79. 3x 2y 22 0 (b − 10 2 4 6 8 10 x (2, −4) y 10 8 6 4 2 −2 −2 2 4 6 (8, −1) 10 12 1 −3 −4 −4 −3 19. (a) f gx f x2 4, x ≥ 0 Section 1.9 (page 99) Vocabulary Check (page 99) 1. inverse; f-inverse 3. y x 4. one-to-one 2. range; domain 5. horizontal 1. 5. f 1x 1 6 x f 1x x 1 3 3. f 1x x 9 7. f 1x x 3 9. c 13. (a) (b) 10. b f gx fx 2 12. d x 11. a 2x 2 2x 2 g f x g2x x y 3 2 1 f g x –3 –2 1 2 3 –2 –3 15. (a) f gx fx 1 7 7x 1 7 1 x g f x g7x 1 7x 1 1 7 x (b) y 5 4 3 2 1 g f 17. (a) f gx f 38x 38x3 8 x g f x gx3 8 38x3 8 x x2 4 4 x g f x gx 4 x 42 4 x (b) y 10 8 6 4 2 g f 21. (a) (b) 2 4 6 8 10 x f gx f 9 x, x ≤ 9 9 9 x2 x g f x g9 x2, x ≥ 0 9 9 x2 x y 12 9 6 f g −12 –9 –6 –3 6 9 12 x –6 –9 –12 23. (a) f gx f5x 1 x 1 5x 1 x 1 5x 1 x 1 1 5 5x 1 x 1 5x 1 5x 5 x 5x 1 x 5 x 1 x 5 1 1 5x gx 1 x 5 333200_01_AN.qxd 12/9/05 1:23 PM Page A93 Answers to Odd-Numbered Exercises and Tests A93 (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) The domains and ranges of f and f 1 are all real f x 2 4 6 8 10 43. (a) (b) (b) y f − 10 − 8 − 6 g 25. No 27. 10 − 10 x f 1x 29. Yes 33. 31. No 4 g 0 1 2 2 35. 6 2 8 3 2 0 4 1 10 −4 The function has an inverse. 37. 20 −10 The function does not have an inverse. The function does not have an inverse. −12 12 45. (a) (b) numbers. f 1x x2, x ≥ 0 − numbers x such that f 1x 4 x2, 0 ≤ x ≤ 2 x ≥ 0. y 3 2 1 −1 f = f 1 2 3 x −4 8 −10 10 (d) The domains and ranges of f and f 1 are all real 1 2 3 4 5 (c) The graph of y x. the line f 1 is the reflection of the graph of f in C H A P T E R 1 39. (a) (b) −20 f 1x 1 f 2 4 6 8 x –2 − 2 f 1 (c) The graph of (d) The domains and ranges of f and is the same as the graph of f. f 1 are all real 47. (a) (b) 0 ≤ x ≤ 2. numbers x such that f 1x 4 x y 4 3 2 1 −1 f = f –2 –3 f 1 (c) The graph of (d) The domains and ranges of f and is the same as the graph of f. f 1 are all real numbers x except 49. (a) f 1x 2x 1 x 1 x 0. (c) The graph of y x. the line f 1 is the reflection of the graph of f in –3 –2 –1 1 2 3 4 x (d) The domains and ranges of f and f 1 are all real 41. (a) (b) numbers. f 1x 5x 1 x 2 3 333200_01_AN.qxd 12/9/05 1:23 PM Page A94 A94 (b) Answers to Odd-Numbered Exercises and Tests 1 − 6 f f −1 f 6 x (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) The domain of f and the range of x 2. numbers x except range of f are all real numbers x except f 1x x 3 1 f 1 The domain of are all real f 1 and the x 1. 51. (a) (b) y 6 4 2 −1 f f represents the year for a given number of house- (e) f 1117,022 17 the results are similar. 79. (a) (b) (c) (d) f 1108,209 11 f 1 holds in the United States. y 1578.68t 90,183.63 f 1 t 90,183.63 f 1108,209 11.418; 1578.68 (f) 81. (a) Yes f 1 (b) by motor vehicles. f 12632 8 f (t (c) (d) No. y x 245.50 x 0.03 degrees Fahrenheit; (b) 100 would not pass the Horizontal Line Test. 83. (a) , 245.5 < x < 545.5 % load y (c) 0 < x < 92.11 yields the year for a given number of miles traveled − 6 − 4 2 4 6 x 0 600 − 6 (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) The domains and ranges of f and f 1 are all real 53. (a) (b) numbers. f 1x 5x 4 6 4x 3 − 2 −1 f f 1 2 3 x −1 f (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) The domain of f and the range of numbers x except range of f are all real numbers x except x 5 4. f 1 The domain of are all real f 1 and the x 3 2. g1x 8x 59. No inverse 55. No inverse 61. 57. f 1x x 3 65. No inverse 67. 69. 32 71. 600 75. x 1 2 77. x 1 2 63. No inverse f 1x x2 3 2 2 3x 3 73. , x ≥ 0 f x x2 0 85. False. 87. Answers will vary. 89. x 1 3 4 has no inverse. 6 7 x f 1x 91 1x 3 –2 –1 1 2 3 4 5 6 x –2 –3 333200_01_AN.qxd 12/9/05 1:23 PM Page A95 93. 101. k 1 4 5, 10 3 95. ±8 103. 16, 18 97. 3 2 99. 3 ± 5 Section 1.10 (page 109) Vocabulary Check (page 109) 1. variation; regression 3. correlation coefficient 5. constant of variation 7. inverse 8. combined 2. sum of square differences 4. directly proportional 6. directly proportional 9. jointly proportional 1 ( 145,000 140,000 135,000 130,000 125,000 t Answers to Odd-Numbered Exercises and Tests A95 9. (a) 800 (b) S 38.4t 224 5 0 (c) 800 5 0 14 14 The model is a good fit. (d) 2005: $800 million; 2007: $876.8 million (e) Each year the annual gross ticket sales for Broadway shows in New York City increase by $38.4 million. 11. Inversely 13. x y k x2 2 4 4 6 8 10 16 36 64 100 4 6 2 Year (2 ↔ 1992) 8 10 12 The model is a good fit for the actual data. 3. y y 5 2x 3 y 1 4x 3 7. (a) and (b) y 240 220 200 180 160 140 ) 100 80 60 40 20 2 4 6 8 10 x 15. x x y k x2 2 2 4 8 6 8 18 32 10 50 C H A P T E R 1 y 50 40 30 20 10 t 2 4 6 8 10 x 108 12 36 84 60 Year (12 ↔ 1912) y 1.03t 130.27 (c) (e) Part (b): 238 feet; Part (c): 241.51 feet (f) Answers will vary. (d) The models are similar. 333200_01_AN.qxd 12/9/05 1:23 PM Page A96 Answers to Odd-Numbered Exercises and Tests 2 1 2 4 1 8 6 1 18 8 1 32 10 1 50 71. (a A96 17. x y k x2 y 5 10 4 10 3 10 2 10 1 10 2 4 6 8 10 xx 19. x y k x2 2 5 2 4 5 8 6 5 18 8 5 32 10 1 10 10 xx y 7 10 I 0.035P x 21. 23. y 5 x y 205x 29. 13 x; 27. 31. Model: y 33 y 0.0368x; $7360 (b) 33. 35. (a) 0.05 meter 1762 3 newtons 25. y 12 5 x 25.4 centimeters, 50.8 centimeters 37. 39.47 pounds 39. A k r2 41. y k x2 43. F kg r2 45. P k V 47. F km1m2 r2 49. The area of a triangle is jointly proportional to its base and height. 51. The volume of a sphere varies directly as the cube of its radius. 53. Average speed is directly proportional to the distance and inversely proportional to the time. 55. A r2 57. y 28 x 59. F 14rs 3 61. z 2x2 3y 63. 0.61 mile per hour 65. 506 feet 67. 1470 joules 69. The velocity is increased by one-third. 2000 4000 Depth (in meters) 4200, k2 4800, k5 3800, k3 4500 4200, (b) Yes. k1 k4 C 4300 d (c) (d) 6 (e) 1433 meters 0 0 73. (a) 0.2 25 0 (b) 0.2857 microwatt per square centimeter 6000 55 75. False. y will increase if k is positive and y will decrease if k is negative. 77. True. The closer the value of 79. The accuracy is questionable when based on such limited is to 1, the better the fit. r data. x > 5 81. 834 −3 −2 −c) 21 87. Answers will vary. (page 117) 3. Quadrant IV 85. (a) 5 7 3 3 Review Exercises (b) 1. y 6 4 2 −6 −4 −2 −2 2 4 6 8 x −4 −6 −8 333200_01_AN.qxd 12/9/05 1:23 PM Page A97 (b) 5 (c) 1, 13 2 (b) 9.9 (c) 2.8, 4.1 5. (a) (− 3, 8. (a) (0, 8.2) (1, 5) x 2 4 8 6 4 2 9. 13. 15. − 2 (5.6, 0) x 2 4 6 2, 5, 4, 5, 2, 0, 4, 0 Radius 22.5 centimeters 11. $656.45 million x y 2 11 3 –2 –1 1 2 3 x –1 –2 –3 –4 –5 1 4 y 5 4 173 –2 –1 1 2 4 5 x –2 –3 Answers to Odd-Numbered Exercises and Tests A97 19. 23. y 6 5 4 3 1 21. y 6 5 4 3 1 –2 –1 1 2 3 4 5 6 x –2 –5 –4 –3 –1 1 2 3 –2 y 1 –3 –2 –1 1 2 3 x x –2 –3 –4 –5 25. xy- intercept: intercept: 2, 0 7 0, 7 29. No symmetry y 1, 0, 5, 0 27. x-intercepts: 0, 5 intercept: 31. y-axis symmetry y- − 4 − 3 −2 4 1 − 1 − 1 −2 −3 −3 −1 −1 −2 C H A P T E R 1 33. No symmetry 35. No symmetry 6 −5 − 4 −3 −2 x x −1 −1 1 2 37. Center: 0, 0; Radius: 3 y 4 2 1 (0, 0) x –4 –2 –1 –1 1 2 4 –2 –4 39. Center: 2, 0; Radius: 4 y (−2, 0) –8 –4 –2 6 2 –2 –6 x 4 333200_01_AN.qxd 12/9/05 1:23 PM Page A98 A98 Answers to Odd-Numbered Exercises and Tests 41. Center: 2, 1; 1 y Radius: 6 −1 55. y 3 2x 5 57. y 1 2x 0, − 5 10 12 x (10, −3) x 22 y 32 13 43. 45. (a) x F 0 0 4 5 8 10 12 15 16 20 20 25 4 59. y 5 x 0 61. 4x 23 V 850t 7400, 3x 8 y 4 3 y 4 63. (a) (b) 6 ≤ t ≤ 11 65. t 4 1 71. (a) 5 (b) 17 (c) 73. All real numbers x such that y 5x 2 5 67
. No t2 2t 2 69. Yes (d) 5 ≤ x ≤ 5 (b) F 30 25 20 15 10 c) 12.5 pounds 47. slope: 0 y-intercept: 6 y 8 4 2 x 24 − 6 −4 −2 10 12 20 Length (in inches) 16 49. slope: 3 y-intercept: 13 y 12 6 3 −3 −6 3 6 9 x 75. All real numbers x except y x 3, 2 6 4 2 −2 −4 −6 x 4 6 77. (a) 16 feet per second (b) 1.5 seconds −9 −6 −3 51. (− 7, 1) − 8 − 53. (−4.5, 6) y 8 6 2 (2.1, 3) x 2 4 (3, −4) −6 −4 −2 2 4 6 x −2 −4 16 feet per second (c) 4x 2h 3, h 0 85. 79. 83. Not a function 89. Increasing on Decreasing on Constant on 3 91. 0, , 1 1, 0 81. Function 3 87. 8 7 3, 3 (1, 2) 93. 0.25 (0.12, 0.00) −0.75 0.75 m 1 2 m 5 11 −3 3 −1 −0.75 333200_01_AN.qxd 12/9/05 1:23 PM Page A99 95. 4 97. 1 2 2 101. Odd 103. f x 3x 99. Neither even nor odd 1056 −4 107. 109. y 6 4 −2 −4 − 111 x3 115. 117. (a) f x x2 −1 1 2 3 4 5 6 x 113. y 6 3 −12 −9 −6 −3 3 6 9 12 15 x −12 −15 (b) Vertical shift of nine units downward (c − 10 (d) 119. (a) hx f x 9 f x x (b) Horizontal shift of seven units to the right Answers to Odd-Numbered Exercises and Tests A99 (c) y 12 10 8 6 4 2 h 2 4 6 8 10 12 x x 4 6 (d) 121. (a) hx f x 7 f x x2 (b) Reflection in the x-axis, horizontal shift of three units to the left, and vertical shift of one unit upward C H A P T E R 1 (cd) 123. (a) hx f x 3 1 f x x (b) Reflection in the x-axis and vertical shift of six units upward y (c) 9 6 5 4 3 2 1 −3 −2 −1 −2 −d) 125. (a) hx f x 6 f x x (b) Reflections in the x-axis and the y-axis, horizontal shift of four units to the right, and vertical shift of six units upward (c) y 10 d) hx f x 4 6 333200_01_AN.qxd 12/9/05 1:23 PM Page A100 A100 Answers to Odd-Numbered Exercises and Tests 127. (a) f x x (b) Horizontal shift of nine units to the right and vertical 147. (a) (b) f 1x 2x 6 f −1 y 8 6 2 − 10 − 8 − 6 − 2 − 6 − 8 − 10 f 8 x is the reflection of the graph of f in f 1 (c) The graph of y x. the line f 1 (d) Both f and have domains and ranges that are all 149. (a) (b) real numbers. f 1x x2 11 f –1 –1 2 3 4 5 x (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) f has a domain of has a domain of 1, 0, and a range of and a range of 0, ; 1, . f 1 151. x ≥ 4; f 1x x 2 4 153. (a ( 65 60 55 50 45 t 5 6 7 8 9 10 11 12 Year (5 ↔ 1995) (b) The model is a good fit for the actual data. 3.2 kilometers, 16 kilometers 159. 2 hours, 26 minutes 155. Model: 157. A factor of 4 m 8 5k; stretch y (c) 25 20 15 10 5 − 2 − 5 − 10 − 15 h x 2 4 6 10 12 14 (d) 129. (a) hx 5 f x 9 f x x (b) Reflection in the x-axis, vertical stretch, and horizontal shift of four units to the right (c x2 2x 4 all real numbers x except x 1 2 hx 2 f x 4 x2 2x 2 (b) 2x3 x2 6x 3 x2 3 2x 1 x 8 3 Domains of and f x x3, gx 6x 5 (b) f, g, f g, x 8 (d) ; (d) 131. (a) (c) 133. (a) 135. 137. (a) (b) v dt 36.04t2 804.6t 1112 (v + d)(t) v(t) d(t) 4000 7 0 13 g f : all real numbers 139. 143. v d10 3330 (c) f 1x x 7 141. The function has an inverse. 6 −2 −4 145. −4 8 4 −4 8 The function has an inverse. The function has an inverse. 333200_01_AN.qxd 12/9/05 1:23 PM Page A101 161. False. The graph is reflected in the -axis, shifted nine units to the left, and then shifted 13 units downward. x 3 6 9 x − 12 − − 12 − 18 Answers to Odd-Numbered Exercises and Tests A101 7. 2x y 1 0 (b) 7x 4y 53 0 x x2 18x 6. 8. 9. (a) x 12 y 32 16 17x 10y 59 0 4x 7y 44 0 1 8 10 ≤ x ≤ 10 0, ±0.4314 1 28 (b) (c) 10. (a) 11. 12. (a) (b) 0.1 −1 1 y 163. True. If x 1ky. is directly proportional to x to a set A Therefore, 165. A function from a set A in the set to each element set B. B x x, is directly proportional to y kx, y. then is a relation that assigns in the exactly one element y so −0.1 Chapter Test (page 123) 1. y (− 2, 56, 0) x 1 2 3 4 5 6 2, 5 Midpoint: 11.937 centimeters ; Distance: 2 2. 3. No symmetry 89 4. y- axis symmetry (c) Increasing on Decreasing on 0.31, 0, 0.31, , 0.31, 0, 0.31 (d) Even 13. (a) 0, 3 (b) 10 −2 4 −10 , 2 (c) Increasing on 2, 3 Decreasing on (d) Neither even nor odd 5 14. (a) (b) 10 0, 3. y- axis symmetry y 6 5 4 3 2 1 x (−4, 0) −4 −3 −2 − 1 −1 −2 (0, 4) (4, 0) 1 2 3 4 x 15. y 4 3 2 1 (−1, 01, 0) x 1 2 (0, −1) 3 4 −12 6 −2 5, , 5 (c) Increasing on Decreasing on (d) Neither even nor odd y 30 20 10 − 2 − 10 − 20 − 30 − 6 2 4 6 x 333200_01_AN.qxd 12/9/05 1:23 PM Page A102 A102 Answers to Odd-Numbered Exercises and Tests 16. Reflection in the -axis of 17. Reflection in the -axis, horizontal shift, and vertical shift x of y x y 10 18. (a) (c) (d) (e) (f) 19. (a) (c) (e) 2x2 4x 2 3x 4 12x3 22 x2 28x 35 4x2 4x 12 (b) , x 5, 1 3x2 7 x2 4x 5 3x 4 24x3 18x2 120x 68 9x 4 30x2 16 1 2x32 x 2x x x 2x 1 2x32, x > 0 2x , x > 0 x 1 2x32 d) (b) (f) , x > 0 20. 22. 24. f 1x 3x 8 f 1x 1 A 25 6 3 x23, x ≥ 0 25. b 48 a xy 21. No inverse 23. v 6s Problem Solving (page 125) (c) 1. (a) (c) W1 5,000 2000 0.07S (b) W2 2300 0.05S (15,000, 3,050) 0 0 30,000 Both jobs pay the same monthly salary if sales equal $15,000. (d) No. Job 1 would pay $3400 and job 2 would pay $3300. 3. (a) The function will be even. (b) The function will be odd. (c) The function will be neither even nor odd. 5. f x a2n x2n a2n2 x2n2 . . . a2 x2 a0 f x a2n x2n a2n2 x2n2 . . . a2 x2 a0 miles per hour 7. (a) (c) f x hours 180 7 812 3 y Domain: (b) 255 7 x 3400 0 ≤ x ≤ 1190 9 0 ≤ y ≤ 3400 Range: y (d 4000 3500 3000 2500 2000 1500 1000 500 x 30 60 90 120 150 Hours 9. (a) (c) (d) (e) (b) f gx 4x 24 f 1x 1 4 x; g1x x 6 g1 f 1x 1 4 x 6 f gx 8x3 1; f g1x 1 2 f 1x 3x 1; g1x 1 g1 f 1x 1 2 (f) Answers will vary. 3x 1 (g) 2x; 11. (a) f g1x 1 4x 6 3x 1; y y 3 2 1 −1 −3 3 2 1 −1 −2 −3 −3 −2 −1 −3 −2 −1 f g1x g1 f 1x (bd) −3 −2 −1 −3 −2 −1 3 2 1 −1 −2 −3 y 3 2 −1 −2 −3 1 2 3 1 2 3 x x 333200_02_AN.qxd 12/9/05 1:27 PM Page A103 Answers to Odd-Numbered Exercises and Tests A103 (e) y (f) y (c3 −2 −1 3 1 −1 −2 −3 1 2 3 x 13. Proof 15. (a) x f f1x (b 1x 5 1 3 5 (c) x 3 2 f f 1x 4 0 (d) x f 1x Chapter 2 Section 2.1 (page 134) Vocabulary Check (page 134) 1. nonnegative integer; real 3. axis 5. negative; maximum 4. positive; minimum 2. quadratic; parabola 1. g 5. f 9. (a) 2. c 6. a 3. b 7. e y 4. h 8. d (b Vertical shrink −6 −4 6 4 2 −2 −4 −6 Vertical shrink and reflection in the x-axis y 5 4 3 2 1 (d) y 6 4 2 −6 −4 −2 2 4 6 x −3 −2 −1 −1 1 2 3 Vertical stretch 11. (a) y 5 4 3 −2 −1 −1 Vertical stretch and reflection in the x-axis y (b3 −2 −1 1 2 3 −1 x Horizontal shift Horizontal shrink and vertical shift (c) y (d) 8 6 4 2 −2 −4 −6 −2 2 6 x −8 −6 −4 −2 y 10 8 2 − Horizontal stretch and vertical shift Horizontal shift 13. y 2 1 15. −4 −3 −1 1 3 4 x −2 −3 y 3 2 1 −4 −3 −1 1 2 3 4 x −2 −3 x 4 6 −6 0, 5 Vertex: Axis of symmetry: y-axis x-intercepts: ± 5, 0 −5 0, 4 Vertex: Axis of symmetry: y-axis x-intercepts: ±22, 0 333200_02_AN.qxd 12/9/05 1:27 PM Page A104 Answers to Odd-Numbered Exercises and Tests 19. y 35. 4 A104 17. − 20 − 12 y 20 16 12 − 8 x 4 8 5, 6 Vertex: Axis of symmetry: x-intercepts: x 5 5 ± 6, 0 21. y 23. 20 16 12 8 4 −4 4 8 12 16 4, 0 Vertex: Axis of symmetry: 4, 0 x-intercept, 1 1 Vertex: Axis of symmetry: No x-intercept 2 − 4 1, 6 Vertex: Axis of symmetry: x-intercepts: x 1 1 ± 6, 0 25. y 27. y 4 − 8 4 8 16 x x x −8 4 2, 3 Vertex: Axis of symmetry: x-intercepts: x 2 2 ± 6, 0 y x 12 4 43. f x x 22 5 −4 y x 1 2 39. y 2x 22 2 f x 1 2 f x 24 49 ±4, 0 37. 41. 45. 49. 53. 57. 47. 51. x 32 4 x 1 2 3 2 4 5, 0, 1, 0 55. 59. f x 3 4 f x 16 3 x 52 12 x 5 2 2 12 −4 4 −4 0, 0, 4, 0 61. −5 10 8 10 −40 2, 0, 6, 0 5 f x x2 2x 3 gx x2 2x 3 f x 2x 2 7x 3 gx 2x2 7x 3 65. 69. 71. 55, 55 73. 12, 6 A 8x50 x 3 −8 16 −4 3, 0, 6, 0 10 63. −10 14 −6 7, 0, 1, 0 67. f x x2 10x gx x2 10x 20 10 4 − 8 − 4 2, 20 1 Vertex: Axis of symmetry: No x-intercept x 8 x 1 2 29. 5 −8 7 31. −5 14 −18 12 −6 33. 48 −6 12 −12 − 12 −16 −20 75. (a) (b) x 4 4, 16 Vertex: Axis of symmetry: x-intercepts: 4, 0, 12, 0 1, 4 Vertex: Axis of symmetry: x-intercepts: x 1 1, 0, 3, 0 4, 5 Vertex: Axis of symmetry: x-intercepts: x 4 4 ± 5, 0 4, 1 Vertex: Axis of symmetry: 4 ± 1 x-intercepts: x 4 2, 0 2 x A 5 10 15 20 25 30 600 1067 1400 1600 1667 1600 x 25 feet, y 33 1 3 feet (c) 2000 0 0 60 x 25 feet, y 33 1 3 feet x 252 5000 A 8 (d) 3 3 79. 20 fixtures 77. 16 feet 83. (a) $14,000,000; $14,375,000; $13,500,000 81. 350,000 units (e) They are identical. (b) 24; $14,400 85. (a) 5000 (b) 4299; answers will vary. (c) 8879; 24 0 0 43 333200_02_AN.qxd 12/9/05 1:27 PM Page A105 87. (a) 25 (b) 69.6 miles per hour (c) y (d) y Answers to Odd-Numbered Exercises and Tests A105 0 −5 100 89. True. The equation has no real solutions, so the graph has x no -intercepts. f x ax b 2a 91. 2 4ac b2 4a 93. Yes. A graph of a quadratic equation whose vertex is on −4 −3 −2 (e) 6 5 3 2 1 −1 −4 −3 −2 −1 1 2 3 4 (f) −4 −3 −2 −1−1 −2 1 2 3 4 x −4 −3 −1 −1 1 3 4 13. Falls to the left, rises to the right 15. Falls to the left, falls to the right 17. Rises to the left, falls to the right 19. Rises to the left, falls to the right 21. Falls to the left, falls to the right 25. 23. 8 g f −4 4 −8 27. (a) ±5 12 −8 8 g f −20 C H A P T E R 2 (b) odd multiplicity; number of turning points: 1 (c) 10 −30 30 the x-axis has only one x-intercept. y 1 4 x 3 1408 49 3 x 5 103. 109 y 5 97. 3 95. 101. 99. 27 105. Answers will vary. Section 2.2 (page 148) Vocabulary Check (page 148) 2. Leading Coefficient Test n ; 1. continuous n 1 3. 5. touches; crosses 7. Intermediate Value 4. (a) solution; (b) 6. standard x a; (c) x-intercept . c 5. a 9. (a) 2. g 6. e y 3. h 7. d 4. f 8. b (b) − 3 − 2 (cd) −3 − 11. (a) (b) y 6 5 4 3 2 1 −30 29. (a) 3 1 2 4 5 x (b) even multiplicity; number of turning points: 1 (c) 4 −18 18 31. (a) 2, 1 −20 (b) odd multiplicity; number of turning points: 1 (c) 4 3 2 1 −2 −3 −4 −5 y 4 3 2 1 −4 −3 −2 2 3 4 x −4 −4 333200_02_AN.qxd 12/9/05 1:27 PM Page A106 A106 Answers to Odd-Numbered Exercises and Tests 33. (a) 0, 2 ± 3 (b) odd multiplicity; number of turning points: 2 (c) 8 −6 6 −24 35. (a) 0, 2 (b) 0, odd multiplicity; 2, even multiplicity; number of turning points: 2 (c) −7 5 −5 8 37. (a) 0, ± 3 (b) 0, odd multiplicity; of turning points: 4 ± 3, even multiplicity; numbe
r (c) −9 6 −6 9 39. (a) No real zeros (b) number of turning points: 1 (c) 40 45. (a) −6 4 −4 6 49. 0, 0, ±1, 0, ±2, 0 x -intercepts: x 0, 1, 1, 2, 2 x f x x2 4x 12 (b) (c) (d) The answers in part (c) match the -intercepts. f x x2 10x f x x3 5x2 6x f x x 4 4x3 9x2 36x f x x2 2x 2 57. f x x3 2x2 3x f x x4 x3 15x2 23x 10 f x x5 16x4 96x3 256x2 256x 47. 51. 53. 55. 59. 63. 65. 67. (a) Falls to the left, rises to the right (c) Answers will vary. f x x2 4x 4 f x x3 3x 0, ±3 61. (b) (d) y 12 4 −4 −8 (−3, 0) −12 −8 −4 (0, 0) (3, 0) 4 8 12 x 69. (a) Rises to the left, rises to the right (b) No zeros (d) y (c) Answers will vary. −4 4 41. (a) ±2, 3 −5 (b) odd multiplicity; number of turning points: 2 (c) 4 8 6 2 −8 −16 43. (a) 12 −2 −4 7 6 x -intercepts: 2, 0 (b) (d) The answers in part (c) match the -intercepts. x 0, 5 2 x 0, 0, 5 (c) −4 −2 2 4 t 71. (a) Falls to the left, rises to the right (c) Answers will vary. (b) 0, 3 (d) y 1 (0, 0) −1 1 2 (3, 0) 4 x −3 −4 333200_02_AN.qxd 12/9/05 1:27 PM Page A107 Answers to Odd-Numbered Exercises and Tests A107 73. (a) Falls to the left, rises to the right (c) Answers will vary. (b) 0, 2, 3 (d) y (2, 0) (3, 00, 0) −3 − 2 − 1 − 1 − 2 75. (a) Rises to the left, falls to the right (c) Answers will vary. 5, 0 (b) (d) y 5 (−5, 0) − 10 − 15 (0, 0) 5 10 x − 20 77. (a) Falls to the left, rises to the right (b) 0, 4 (d) (c) Answers will vary. y 2 (0, 0) (4, 0) − 4 − 2 2 6 8 x 79. (a) Falls to the left, falls to the right (b) (d) ±2 (c) Answers will vary. (2, 0) 1 2 3 t (−2, 0 81. 6 83. 14 −9 9 −12 18 −6 0, ±2, Zeros: odd multiplicity −6 1, Zeros: even multiplicity; 3, 9 2, odd multiplicity 85. 1, 0, 1, 2, 2, 3; 0.879, 1.347, 2.532 87. 89. (a) 2, 1, 0, 1; 1.585, 0.779 V l w h 36 2x36 2xx x36 2x2 (b) Domain: (c) 0 < x < 18 x V 1 2 3 4 5 6 7 1156 2048 2700 3136 3380 3456 3388 24 inches 24 inches 6 inches (d) 3600 0 0 18 x 6 A 2x2 12x 0 inches < x < 6 (b) inches 91. (a) (c) (d) V 384x2 2304x x 3, the volume is maximum at When dimensions of gutter are 3 inches 6 inches 3 inches. 4000 V 3456; (e The maximum value is the same. (f) No. Answers will vary. 93. 200 7 140 13 The model is a good fit. 95. Region 1: 259,370 Region 2: 223,470 Answers will vary. 60 97. (a) −10 45 −5 t 15 15.22, 2.54 (b) (c) Vertex: (d) The results are approximately equal. 333200_02_AN.qxd 12/9/05 1:27 PM Page A108 A108 Answers to Odd-Numbered Exercises and Tests 99. False. A fifth-degree polynomial can have at most four 121. Vertical stretch by a factor of 2 and vertical translation turning points. 101. True. The degree of the function is odd and its leading coefficient is negative, so the graph rises to the left and falls to the right. 103. 105. 109. 115a) Vertical shift of two units; Even (b) Horizontal shift of two units; Neither even nor odd (c) Reflection in the y-axis; Even (d) Reflection in the x-axis; Even (e) Horizontal stretch; Even (f) Vertical shrink; Even gx x3; (g) gx x16; (h) 5x 8x 3 7 111. 5 ± 185 4 x24x 5x 3 1 ± 22 113. Neither odd nor even Even 4, 1 107. 2, 4 5 3 117. Horizontal translation four units to the left of y y x2 1 −1 x 1 119. Horizontal translation one unit left and vertical translation five units down of 5 nine units up of 6 −3 −2 −1 1 2 x −2 Section 2.3 (page 159) Vocabulary Check (page 159) 1. dividend; divisor; quotient; remainder 2. improper; proper 4. factor 5. remainder 3. synthetic division 1. Answers will vary. 3. 6 5. 2x 4 −9 9 7. 13. 17. 21. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 7 11 11. x 2 x2 2x 4 2x 11 x2 2x 3 3x2 2x 5 19. 15. x3 3x2 1 −6 x2 3x 1 9. 3x 5 2x 3 2x2 1 x 3 6x2 8x 3 4x2 9 5x2 14x 56 232 x 4 10x3 10x2 60x 360 1360 x 6 x2 8x 64 3x3 6x2 12x 24 48 x 2 x 13 23. x2 10x 25 x3 6x2 36x 36 216 x 6 4x2 14x 30 f (x) x 4x2 3x 2 3, f 4 3 f x x 2 15x3 6x 4 34 34 f 2 3 , f x x 2 x2 3 2 x 32 8, f 2 8 f x x 1 34x2 2 43x 2 23, f 1 3 0 3 3 3 45. (a) 1 47. (a) 97 (b) 4 (b) 5 3 (c) 4 (d) 1954 (c) 17 (d) 199 333200_02_AN.qxd 12/9/05 1:27 PM Page A109 49. 51. 53. 55. 2, 3, 1 1 2, 5, 2 Zeros: Zeros: x 2x 3x 1; 2x 1x 5x 2; x 3 x 3 x 2; Zeros: x 1x 1 3x 1 3; 1, 1 3, 1 3 Zeros: (b) 2x 1 57. (a) Answers will vary. 3, 3, 2 (c) (e) f x 2x 1x 2x 1 (d) 1 2, 2, 1 7 −1 6 −6 59. (a) Answers will vary. (b) x 1, x 2 (c) (d) (e) f x x 1x 2x 5x 4 1, 2, 5, 4 20 −6 6 −180 61. (a) Answers will vary. x 7 f x x 72x 13x 2 7, 1 (b) 2, 2 3 (c) (d) (e) (c) (e) 320 −9 3 14 −6 6 −6 −40 63. (a) Answers will vary. x 5 f x x 5x 52x 1 (b) (d) ± 5, 1 2 65. (a) Zeros are 2 and x 2 (b) ±2.236. f x x 2x 5x 5 3.732. (c) 2, 0.268, and 67. (a) Zeros are (b) (c) 2x2 x 1, x 3 2 x 2 h t t 2t 2 3t 2 3 71. x2 3x, x 2, 1 69. 73. (a) and (b) 1800 3 1200 13 M 0.242x3 12.43x 2 173.4x 2118 Answers to Odd-Numbered Exercises and Tests A109 (c) t 3 4 5 6 7 8 M(t) 1703 1608 1531 1473 1430 1402 t 9 10 11 12 13 M(t) 1388 1385 1392 1409 1433 Answers will vary. (d) 1614 thousand. No, because the model will approach negative infinity quickly. is a zero of f. 4 7 75. False. 77. True. The degree of the numerator is greater than the 85. 0; degree of the denominator. x2n 6xn 9 c 210 ± 5 3 f x x3 7x2 12x f x x3 x2 7x 3 7 5 91. 89. , 2 x 3 81. The remainder is 0. is a factor of f. 3 ± 3 2 79. 83. 87. 93. 95. Section 2.4 (page 167) Vocabulary Check (page 167) (b) i 1. (a) iii 3. complex numbers; 5. complex conjugates (c) ii a bi 2. 1; 1 4. principal square 1. 7. 13. 21. 27. 35. 41. 47. 53. 59. 63. 67. 73. 5. 4 3i a 6, b 5 11. 8 11 i 25. 15. 9. 0.3i 3. 53 i 17. 14 20i a 10, b 6 2 33 i 1 6i 3 32 i 5 i 10 25i, 8 10 41i 41 120 27 1681 23 21 52 75 310i 23. 12 30i 6 3i, 45 8, 8 43. 3 4 5i 5 1 2 29. 37. 20 55. 10 51. 5 2i 1681i 61. 49. 31. 24 39. 19. 4 7 6i 9 40i 1 6 33. 1 5 i, 6 5i 45. 5 6i 57. 62 949 297 949i 65. 1 ± i 2 ± i 1 2 ± 515 7 5 7 5i 69. 5 2 , 3 2 75. 1 6i 71. 2 ± 2i 77. 83. (a) (b) 81. 20 10i i 79. 3753i 9 16i, z2 z1 z 11,240 877 (b) 16 i 4630 877 (c) 16 85. (a) 16 87. False. If the complex number is real, the number equals its (d) 16 conjugate. C H A P T E R 2 333200_02_AN.qxd 12/9/05 1:27 PM Page A110 Answers to Odd-Numbered Exercises and Tests A110 89. False. 33. (a) i44 i150 i74 i109 i61 1 1 1 i i 1 2 x 2 3x2 23 93. 91. Proof 97. 31 99. 103. 1 liter x2 3x 12 27 2 101. a 95. 3Vb 2b Section 2.5 (page 179) Vocabulary Check (page 179) 1. Fundamental Theorem of Algebra 2. Linear Factorization Theorem 4. conjugate 6. Descartes’ Rule of Signs 5. irreducible over the reals 7. lower; upper 3. Rational Zero 3. 2, 4 0, 6 ±1, ±3, ±5, ±9, ±15, ±45, ± 1 5. 1. 9. 11. 1, 2, 3 19. 25. (a) (b) 2, 3, ± 2 3 ±1, ±2, ±4 y 13. 1, 1, 4 1, 2 21. 6, ±i 7. 2, ± 3 2, ± 9 ± 5 2, 1, 10 15. 6, 1 23. 2, 1 ±1, ±3 2, ± 15 2 , ± 45 2 17. 1 2, 1 (c) 2, 1, 2 (c) 1 4, 11, ±3, ± 1 2, ± 3 2, ± 1 4 10 x − 4 − 6 ±1, ±2, ±4, ±8, ± 1 2 16 −8 −4 8 27. (a) (b) 29. (a) (b) 31. (a) ±1, ±1.414 f x x 1x 1x 2x 2 0, 3, 4, ±1.414 hx xx 3x 4x 2x 2 39. (b) 35. (a) (b) x3 x2 25x 25 3x4 17x3 25x2 23x 22 x3 4x 2 31x 174 37. 41. 43. (a) (b) (c) 45. (a) (b) (c) x2 9x2 3 x2 9x 3x 3 x 3i x 3i x 3x 3 x2 2x 2x2 2x 3 x 1 3x 1 3x2 2x 3 x 1 3x 1 3x 1 2 i x 1 2 i 2, ±5i ±2i, 1, 1 2 ±5i; 55. 3 ± i , 1 51. 4 x 5i x 5i x 2 3x 2 3 x 3x 3x 3ix 3i z 1 iz 1 i x 2x 2 i x 2 i 3 47. 49. 2, 3 ± 2 i, 1 53. 2 ± 3; 57. ±3, ±3i; 59. 1 ± i; 61. 2, 2 ± i; 63. 2, 1 ± 2 i; 65. 1 5, 1 ± 5 i; 67. 2, ±2i; 69. ±i, ±3i; 71. 10, 7 ± 5i 73. 2, 1 2, ±i 77. 81. No real zeros 85. One or three positive zeros 87. Answers will vary. 3 91. 4 101. b 99. d 103. (a) 93. 100. a 15 1, 1 2 x 22x 2ix 2i x ix i x 3ix 3i 3 75. 4, 1 ± 1 2i 79. No real zeros 83. One positive zero 89. Answers will vary. 95. 97. ±1, 1 4 ±2, ± 3 2 102. c x 2x 1 2 ix 1 2 i 5x 1x 1 5 ix 1 5 i (c) 1 2, 1, 2, 4 (b) (c) 9 x x 9 − 2x x 5 − 2 1 x V x9 2x15 2x Domain: 0 < x < 9 2 V 125 100 75 50 25 1, ±3, ± 1 2, ± 3 2, ± 1 4, ± 3 4, ± 1 8, ± 3 (b) 6 8, ± 1 (c) 16, ± 3 1, 3 16, ± 1 4, 1 8 32, ± 3 32 x 5 4 2 3 1 Length of sides of squares removed −1 3 −2 105. 1 1.82 centimeters 5.36 centimeters 11.36 centimeters 2, 7 (d) 2, 8; x 38.4, 8 is not in the domain of V. or $384,000 333200_02_AN.qxd 12/9/05 1:27 PM Page A111 Answers to Odd-Numbered Exercises and Tests A111 107. (a) V x3 9x2 26x 24 120 1. (a) (b) 4 feet by 5 feet by 6 feet x 40, 109. or 4000 units 111. No. Setting p 9,000,000 and solving the resulting equation yields imaginary roots. 113. False. The most complex zeros it can have is two, and the Linear Factorization Theorem guarantees that there are three linear factors, so one zero must be real. r1, r2, r3 115. 119. The zeros cannot be determined. k 4 121. (a) k > 4 123. Answers will vary. There are infinitely many possible 5 r1, 5 r2, 5 r3 0 < k < 4 k < 0 117. (d) (b) (c) functions for f. Sample equation and graph: f x 2x3 3x2 11x 6 y 8 4 (−2, 03, 0) 4 8 12 x y 10 8 6 (0, 4) (4, 8) (2, 4) (− 2, 0) −2 2 4 6 8 x 125. Answers will vary. x2 b 127. (a) (b) 11 9i 131. 129. 133. y x2 2ax a2 b2 20 40i 135. (6, 4) 4 3 2 1 − 1 − 2 137. (2, 2) (4, 2) (0, 0) 2 3 4 5 6 x (2, 4) y y 4 3 (0, 2) (1, 2) (−1, 0) − 2 − 1 xx 1 2 Section 2.6 (page 193) Vocabulary Check (page 193) 1. rational functions 3. horizontal asymptote 2. vertical asymptote 4. slant asymptote x 0.5 0.9 f x 2 10 0.99 100 0.999 1000 x 1.5 1.1 f x 2 10 x 5 10 f x 0.25 0.1 1.01 100 100 0.01 1.001 1000 1000 0.001 (b) Vertical asymptote: x 1 Horizontal asymptote: y 0 (c) Domain: all real numbers x except x 1 3. (a) x 0.5 0.9 f x 1 12.79 x 1.5 1.1 f x 5.4 x 5 f x 3.125 17.29 10 3.03 0.99 147.8 1.01 152.3 100 3.0003 0.999 1498 1.001 1502 1000 3 (b) Vertical asymptotes: Horizontal asymptote: x ±1 y 3 (c) Domain: all real numbers x except x ±1 5. Domain: all real numbers x except x 0 Vertical asymptote: Horizontal asymptote: x 0 y 0 7. Domain: all real numbers x except x 2 Vertical asymptote: Horizontal asymptote: x 2 y 1 9. Domain: all real numbers x except x ±1 Vertical asymptotes: x ±1 11. Domain: all real numbers x Horizontal asymptote: 15. c 13. d 17. 1 21. Domain: all real numbers x except 14. a 19. 6 y 3 16. b Vertical asymptote: x 4; 23. Domain: all real numbers x except Vertical asymptote: x 3; 25. Domain: all real numb
ers x except x ±4; horizontal asymptote: x 1, horizontal asymptote: 3; x 1, 1 2; y 1 y 1 2 x 2 Vertical asymptote: horizontal asymptote: x 1 2; 27. (a) Domain: all real numbers x except 0, 1 (b) y-intercept: (c) Vertical asymptote: x 2 Horizontal asymptote 333200_02_AN.qxd 12/9/05 1:27 PM Page A112 A112 (d) Answers to Odd-Numbered Exercises and Tests y 2 1 −1 −2 ( 0, ( 1 2 x − 3 −1 29. (a) Domain: all real numbers x except 0, 1 (b) y-intercept: 2 (c) Vertical asymptote: x 2 Horizontal asymptote: y (d, − 1 2 −1 − 1 −2 x 31. (a) Domain: all real numbers x except 2, 0 5 0, 5 (b) x-intercept: y-intercept: x 1 (c) Vertical asymptote: Horizontal asymptote: x 1 y 2 (d) y 6 (0, 5 33. (a) Domain: all real numbers x 0, 0 (b) Intercept: (c) Horizontal asymptote: (d) y y 1 3 2 (0, 0 35. (a) Domain: all real numbers s (b) Intercept: (d) 0, 0 y (c) Horizontal asymptote: y 0 2 1 (0, 0) −1 −2 s 1 2 37. (a) Domain: all real numbers x except x ±2 (b) x-intercepts: y-intercept: 1, 0 0, 1 (c) Vertical asymptotes: Horizontal asymptote: y (d) and 4, 0 x ±2 y 1 6 4 2 −6 −4 (1, 0) (4, 0) 6 x 39. (a) Domain: all real numbers x except x ±1, 2 (b) x-intercept: 3, 0, 1 0, 3 (c) Vertical asymptotes: y-intercept: 2 2, 0 Horizontal asymptote: y (d) x ±1 x 2, y 0 9 6 3 − 2( , 01 ( −4 −3 ( 0, − 3 2 ( (3, 0) 3 4 x 41. (a) Domain: all real numbers x except 0, 0 (b) Intercept: (c) Vertical asymptote: x 2 x 2, 3 Horizontal asymptote: y y 1 (d) 6 4 2 −6 −4 −2 (0, 0) −4 −6 x 4 6 333200_02_AN.qxd 12/9/05 1:27 PM Page A113 43. (a) Domain: all real numbers x except 2, 0 1 0, 1 (b) x-intercept: y-intercept: 3 (c) Vertical asymptote: Horizontal asymptote: x 3 2 y 1 x 3 2, 2 (d 45. (a) Domain: all real numbers t except 1, 0 0, 1 (b) t-intercept: y-intercept: t 1 (c) Vertical asymptote: None Horizontal asymptote: None (d1, 0) 2 3 1 (0, −1) t 47. (a) Domain of f: all real numbers x except Domain of g: all real numbers x x 1; Vertical asymptotes: none (b) (c) x 1 1 0.5 0 Undef. 2 1.5 1.5 1 1 1 0 0 x f x gx 3 4 4 (d) −4 1.5 2.5 2.5 2 2 3 3 1 −3 (e) Because there are only a finite number of pixels, the graphing utility may not attempt to evaluate the function where it does not exist. 49. (a) Domain of f: all real numbers x except Domain of g: all real numbers x except x 0, 2 x 0 Answers to Odd-Numbered Exercises and Tests A113 1 x ; (b) (c) x f x gx (d) −3 Vertical asymptote: x 0 0.5 0 0.5 2 2 Undef. Undef. 2 2 1 1 1 1.5 2 2 3 2 3 Undef2 3 (e) Because there are only a finite number of pixels, the graphing utility may not attempt to evaluate the function where it does not exist. 51. (a) Domain: all real numbers x except x 0 (b) x-intercepts: (c) Vertical asymptote: Slant asymptote: 2, 0, 2, 0 x 0 y x (d) y y = x x (2, 0) 6 (− 2, 0) −6 −4 2 −2 −4 −6 53. (a) Domain: all real numbers x except b) No intercepts (c) Vertical asymptote: Slant asymptote: x 0 y 2x (d) y 6 4 2 y = 2x −6 −4 −2 2 4 6 x −6 55. (a) Domain: all real numbers x except x 0 (b) No intercepts (c) Vertical asymptote: x 0 Slant asymptote: y x 333200_02_AN.qxd 12/9/05 1:27 PM Page A114 A114 (d) Answers to Odd-Numbered Exercises and Tests 57. (a) Domain: all real numbers t except t 5 0, 0.2 y- (b) intercept: (c) Vertical asymptote: Slant asymptote: y (d) t 5 y t 5 25 20 15 y = 5 − t (0, −0.2) 5 − 20 − 15 − 10 −5 t 10 (c) Vertical asymptote: Slant asymptote: y (d) x 2 y 2x 7 18 12 6 −6 −5 −4 −3 −1 −12 −18 −24 −30 −36 8 −8 (0, 0.5) (1, 0) x 3 (0.5, 0) y = 2x − 7 10 65. −14 Domain: all real numbers x except Vertical asymptote: Slant asymptote1 67. −12 12 −4 12 59. (a) Domain: all real numbers x except 0, 0 (b) Intercept: (c) Vertical asymptotes: Slant asymptote: y x x ±1 (d) y y = x 2 (0, 0 61. (a) Domain: all real numbers x except x 1 0, 1 (b) y-intercept: (c) Vertical asymptote: Slant asymptote: y (d0, −1 Domain: all real numbers x except Vertical asymptote: Slant asymptote: y x 3 1, 0 (b) 1, 0, 1, 0 1 ±1 (b) 69. (a) 71. (a) 73. (a) 2,000 0 0 100 (b) $28.33 million; $170 million; $765 million p 100. (c) No. The function is undefined at (b) 1500 deer 75. (a) 333 deer, 500 deer, 800 deer 77. (a) Answers will vary. (b) 4, (c) 200 63. (a) Domain: all real numbers x except x 1, 2 (b) y-intercept: x-intercepts: 0, 0.5 0.5, 0, 1, 0 4 0 40 11.75 inches 5.87 inches 333200_02_AN.qxd 12/9/05 1:27 PM Page A115 79. (a) Answers will vary. (b) Vertical asymptote: x 25 Horizontal asymptote: y 25 Answers to Odd-Numbered Exercises and Tests A115 17. 3, 1 −3 −2 −1 0 1 x 19. , 4 21 4 21, −4 + 21 −4 − 21 (c) 200 (d) 25 0 x y 65 − 10 − 8 − 6 − 4 − 2 0 2 21. 1, 1 3, 30 35 40 45 150 87.5 66.7 56.3 50 50 55 60 45.8 42.9 23. 1 0 − 1 3, 2 3, 2 3 4 x x x (e) Yes. You would expect the average speed for the round trip to be the average of the average speeds for the two parts of the trip. (f) No. At 20 miles per hour you would use more time in one direction than is required for the round trip at an average speed of 50 miles per hour. 81. False. Polynomials do not have vertical asymptotes. 83. Answers will vary. Sample answer: f x 2x2 x2 1 85. 87. 89. x 7x 8 x 5x 2ix 2i x ≥ 10 3 10 3 x 0 1 2 3 4 5 6 93. Answers will vary. Section 2.7 (page 204) 91. 3 < x < 7 − Vocabulary Check (page 204) 1. critical; test intervals 3. P R C 2. zeros; undefined values 1. (a) No 3. (a) Yes 2, 3 5. 2 3, 3 9. (b) Yes (b) No 7 7. 2, 5 (c) Yes (c) No (d) No (d) Yes −3 −2 −1 76 −4 −8 −2 , 5 1, 0 2 4 11. 13. −6 −5 −4 3, 2 15. −3 −2 −1 0 1 2 −3 −2 − 25. 27. (, 0 0, 3 2 1 2 29. 33. 0 −1 −2 2, 0 2, 1 6 −2 −5 (a) (b) x ≤ 1, x ≥ 3 0 ≤ x ≤ 2 37. , 1 0, 1 x 2 31. 2, 35. 8 −12 12 −8 2 ≤ x ≤ 0, (a) (b) 392 −1 0 1 2 −2 −1 0 1 2 3 4 5 41. 5, 15 5 3 6 9 12 15 18 45. 3 4, 3 6, − 3 4 3 x x 43. 5, 3 2 1, − 3 2 −5 47. −3 −1 −4 −2 3, 2 0, 3 0 −3 −2 −1 0 1 2 3 −4 −2 0 2 4 6 8 49. , 1 2 3, 1 3 51. 8 x 53. 6 −6 12 −6 −a) (b) −2 x ≥ 2 (a) (b) < x < x x 6 333200_02_AN.qxd 12/12/05 11:17 AM Page A116 A116 Answers to Odd-Numbered Exercises and Tests 2, 2 55. 57. 5, 0 7, 59. 0.13, 25.13 63. t 10 67. (a) 69. 13.8 meters 71. 73. (a) 80 seconds ≤ L ≤ , 3 4, 61. 3.51, 3.51 65. 2.26, 2.39 (b) 4 seconds < t < 6 seconds 36.2 meters 40,000 ≤ x ≤ 50,000; 50.00 ≤ p ≤ 55.00 (b) (c) (d) 0 0 23 t 24 26 28 30 32 34 C 70.5 71.6 72.9 74.6 76.8 79.6 2011 t 31 t 36 37 38 39 C 83.2 85.4 87.8 90.5 t 40 41 42 43 C 93.5 96.8 100.4 104.4 2016 to 2021 37 ≤ t ≤ 41 ≥ 2 75. ohms 77. True. The test intervals are (e) R1 79. 4, . , 4 4, a > 0 and 83. (a) If c ≤ 0, b (f) Answers will vary. , 3, 3, 1, 1, 4, and 81. , 230 230, a > 0 can be any real number. If and c > 0, b < 2ac or b > 2ac. (b) 0 2x 52 85. 87. x 3x 2x 2 89. 2x2 x Review Exercises (page 208) 1. (a Vertical stretch (b4 −3 −2 −1 1 2 3 4 x x −3 −4 Vertical stretch and reflection in the x-axis (d) 1 2 3 4 x −4 −3 −2 (c) −4 −3 −2 y 4 3 1 −1 −1 −2 −3 −4 y 4 1 −1 −1 −2 −3 −4 1 2 3 4 x Vertical shift gx x 12 1 3. Horizontal shift f x x 42 6 5, 1 − 1 − 2 Vertex: x 1 Axis of symmetry: 0, 0, 2, 0 x-intercepts: f t 2t 12 , 3 Vertex: t 1 Axis of symmetry: , 0 1 ± 6 2 2 41 t-intercepts: hx x 5 2 4 7. 11. y 2 −2 −4 x 2 − 8 − 4 −6 4, 6 Vertex: Axis of symmetry: x-intercepts: hx 4x 1 x 4 4 ± 6, 0 2 12 2 y 20 15 10 5 9. x t − 3 −2 −1 1 2 3 x 1 Vertex: Axis of symmetry: 2, 12 x 1 2 No x-intercept x 5 f x 1 3 2 13 − 10 5 Vertex: 2, 41 Axis of symmetry: 4 x-intercepts: x 5 2 , 0 ±41 5 2 2 41 12 Vertex: − 6 2, 41 x 5 Axis of symmetry: 2 , 0 x-intercepts: ±41 5 12 2 333200_02_AN.qxd 12/9/05 1:27 PM Page A117 f x 1 2 15. 19. (a) x 42 1 17. y f x x 12 4 y 100 x (b) A 100x x2 x 50, y 50 (c) x 21. 1091 units 23. y 25. y 3 1 −1 −2 −3 −2 275 29. Falls to the left, falls to the right 31. Rises to the left, rises to the right 33. 35. 37. 0, even multiplicity; 39. (a) Rises to the left, falls to the right 7, 3 2, 0, ± 3, odd multiplicity; turning point: 1 odd multiplicity; turning points: 2 5 3, (b) 1 odd multiplicity; turning points: 2 (c) Answers will vary. (d) y 4 3 2 1 (−1, 0 41. (a) Rises to the right, rises to the left (b) 3, 0, 1 (c) Answers will vary. (d) y (−3, 0) 3 − 4 − 2 − 1 (1, 0) 3 4 x 2 1 (0, 0) − 15 − 18 − 21 Answers to Odd-Numbered Exercises and Tests A117 43. (a) 45. (a) 1, 0 (b) 1, 0, 1, 2 0.900 (b) 0.200, 1.772 47. 51. 53. 8x 5 2 3x 2 49. 5x 2 x2 3x 2 1 x2 2 6x3 8x2 11x 4 8 2x2 11x 6 x 2 55. 57. (a) Yes 9 59. (a) 61. (a) Answers will vary. (b) Yes (b) 421 (c) Yes (d) No (b) x 7, x 1 7, (d) 1, 4 1 2 3 x (c) (e) f x x 7x 1x 4 −8 80 −60 5 63. (a) Answers will vary. (b) x 1, x 4 f x x 1x 4x 2x 3 2, 1, 3, 4 40 (c) (d) (e) −3 5 65. 71. 77. 6 2i 40 65i 21 13 1 13i −10 67. 1 3i 73. 79. ± 4 46i 69. 3 7i 23 75. 17 10 17i 81. 1 ± 3i 4, ± 5 4, ± 15 4 x 2 3i 8, 1 101. 4, ± 3 3, 1 2, 2 ± i 87. 2, ± 3 1, 8 4, 6, ±2i 83. 85. 0, 2 ±1, ±3, ±5, ±15, ± 1 2, ± 15 2 , ± 1 2, ± 5 89. 4, 3 1, 3, 6 95. 91. 93. 3x4 14x3 17x2 42x 24 97. 4, ±i 99. 0, 1, 5; f (x x x 1x 5 103. 4, 2 ± 3i; gx x 42x 2 3i 105. 107. Two or no positive zeros, one negative zero 109. Answers will vary. 111. Domain: all real numbers x except 113. Domain: all real numbers x except 115. Vertical asymptote: x 12 x 6, 4 Horizontal asymptote: 117. Vertical asymptote: Horizontal asymptote: x 3 y 0 x 3 y 0 119. (a) Domain: all real numbers x except x 0 (b) No intercepts (c) Vertical asymptote: x 0 Horizontal asymptote: y 0 333200_02_AN.qxd 12/9/05 1:27 PM Page A118 A118 Answers to Odd-Numbered Exercises and Tests (dd) y 4 2 (0, 0 121. (a) Domain: all real numbers x except x 1 (b) x-intercept: y-intercept: 2, 0 0, 2 (c) Vertical asymptote: x 1 Horizontal asymptote: y y 1 (d) − 8 129. (a) Domain: all real numbers x except 1.5, 0 (b) x-intercept: (c) Vertical asymptote: x 0 Horizontal asymptote: y 2 (d) y x 0, 1 3 6 4 (0, 2) (−2, 0 123. (a) Domain: all real numbers (c) Horizontal asymptote: (d) x y 1 y −8 −6 −4 2 −2 −2 −4 −6 −( (b) Intercept: 0, 0 131. (a) Domain: all real numbers x (b) Intercept: (d) 0, 0 y (c) Slant asymptote: y 2x 0, 0) 2 3 x − 2 125. (a) Domain: all real numbers x y 0 (c) Horizontal asymptote: (d) y (b) Intercept: 0, 0 2 1 (0, 0) − 1 − 2 x 1 2 127. (a) Domain: all real numbers x (c) Horizontal asymptote:
y 6 (b) Intercept: 0, 0 3 2 1 (0, 0) −3 − 2 −1 1 2 3 −2 −3 x 133. (a) Domain: all real numbers x except (b) y-intercept: x-intercepts: (c) Vertical asymptote: Slant asymptote: y (d) 0, 0.5 2 3, 0, 1( ( 0, − 1 2 ( −2 −1 −2 (1, 0) 2 3 4 x 135. $0.50 is the horizontal asymptote of the function. 333200_02_AN.qxd 12/9/05 1:27 PM Page A119 137. (a) 2 in. 2 in. 2 in. x y 2 in. (b) x 4 y 4 30 y 4x 14 x 4 Area x4x 14 x 4 2x2x 7 x 4 4 < x < 200 (c) (d) 4 0 32 2 9.48 inches 3, 1 141. 9.48 inches 4, 0 4, 145. 4 5, 1 1, 4, 3 0, 139. 143. 147. 4.9% 149. False. A fourth-degree polynomial can have at most four zeros, and complex zeros occur in conjugate pairs. 151. Find the vertex of the quadratic function and write the function in standard form. If the leading coefficient is positive, the vertex is a minimum. If the leading coefficient is negative, the vertex is a maximum. 153. An asymptote of a graph is a line to which the graph increases or decreases x becomes arbitrarily close as without bound. Chapter Test (page 212) 1. (a) Reflection in the x-axis followed by a vertical translation (b) Horizontal translation y x 3 2 6 2. 3. (a) 50 feet (b) 5. Yes, changing the constant term results in a vertical translation of the graph and therefore changes the maximum height. 4. Rises to the left, falls to the right Answers to Odd-Numbered Exercises and Tests A119 5. 7. 6. 3x x 1 x2 1 4x 1x 3x 3; 1 4, ± 3 Solutions: 2x3 4x 2 3x 6 9 x 2 8. (a) 10. 11. 12. 14. 3 5i (b) 7 9. 2 i 13. f x x4 9x3 28x2 30x f x x4 6x3 16x2 24x 16 2, ± 5i 2, 4, 1 ± 2 i x -intercepts: No -intercept Vertical asymptote: Horizontal asymptote: y 2, 0, 2 (−2, 0) (2, 0) −2 −1 1 2 −2 x 15. 1.5, 0 x -intercept: 0, 0.75 y -intercept: Vertical asymptote: Horizontal asymptote2 −4 (0, 0.75) x 2 4 (− 1.5, 0) −8 −6 −4 16. No -intercept x 0, 2 y -intercept: Vertical asymptote: Slant asymptote: x 1 y x 1 y 10 8 6 4 2 −4 −6 −8 −6 −4 6 8 2 4 (0, −2) 17. x < 4 or 18. x < 6 or 0 < x < 4 −8 −6 −4 −2 0 2 4 6 x 333200_03_AN.qxd 12/9/05 1:30 PM Page A120 A120 Answers to Odd-Numbered Exercises and Tests Problem Solving (page 215) 1. Answers will vary. 3. 2 inches 2 inches 5 inches y x2 5x 4 5. (a) and (b) 7. (a) (b) a bi a bi a2 abi abi b2i2 a2 b2 f x x 2x 2 5 x3 2x 2 5 f x x 3x 2 1 x3 3x 2 1 9. 11. (a) As a a < 0, b b > 0, (b) As For b < 0, lated to the left. increases, the graph stretches vertically. For the graph is reflected in the x-axis. increases, the vertical asymptote is translated. the graph is translated to the right. For the graph is reflected in the x-axis and is trans- Chapter 3 Section 3.1 (page 226) Vocabulary Check (page 226) 1. algebraic 2. transcendental 3. natural exponential; natural 4. 5. A Pert A P1 r n nt 1. 946.852 7. d 11. 8. c x 2 1 3. 0.006 9. a 5. 1767.767 10. b 0 1 1 2 0.5 0.25 2 − 1 1 2 3 − 1 13. x 2 1 f x 36 6 0 1 x 1 2 0.167 0.028 15. x f x 2 1 0 0.125 0.25 0. 17. Shift the graph of f four units to the right. 19. Shift the graph of f five units upward. 21. Reflect the graph of f in the x-axis and y-axis and shift six units to the right. 23. −3 3 −1 25. 3 −1 4 0 27. 0.472 33. x f x 29. 3.857 1022 31. 7166.647 2 1 0.135 0.368 0 1 1 2 2.718 7.389 5 y 5 4 3 2 1 −3 − 2 −1 1 2 3 x − 1 35.055 0.149 0.406 1.104 333200_03_AN.qxd 12/9/05 1:30 PM Page A121 37. x f x 2 1 0 1 4.037 4.100 4.271 4.736 7654321 x Answers to Odd-Numbered Exercises and Tests A121 59. t A t A 10 20 30 $22,986.49 $44,031.56 $84,344.25 40 50 $161,564.86 $309,484.08 61. $222,822.57 65. (a) (c) V1 10,000.298 V2 1,000,059.6 63. $35.45 (b) V1.5 100,004.47 67. (a) 25 grams (b) 16.21 grams (c) 30 41. − 10 22 0 23 69. (a) 0 0 110 39. −7 43. 7 −1 4 5 3 −3 x 2 0 47. x 3 49. x 1 3 51. x 3, 1 45. 53. 55. 573200.21 $3205.09 $3207.57 12 365 Continuous $3209.23 $3210.06 $3210.06 1 2 4 $4515.28 $4535.05 $4545.11 12 365 Continuous $4551.89 $4555.18 $4555.30 10 20 30 $17,901.90 $26,706.49 $39,841.40 40 50 $59,436.39 $88,668.67 C H A P T E R 3 5000 120 0 25 50 75 100 0 0 x (b) Model 12.5 44.5 81.82 96.19 99.3 Actual 12 44 81 96 99 (c) 63.14% (d) 38 masses x → , f x → 2 f x gx hx 75. x < 0 (a) 71. True. As 73. 77. f x hx y but never reaches 2. (b) x > 0 y = 3x y = 4x 3 2 1 −2 −1 1 2 x − → gx. f x → gx. 79. −3 As As 81. y ± 25 x2 333200_03_AN.qxd 12/9/05 1:30 PM Page A122 A122 83. Answers to Odd-Numbered Exercises and Tests y 12 9 6 3 −18 −15 x 3 − 6 −3 −3 −6 −9 85. Answers will vary. Section 3.2 (page 236) Vocabulary Check (page 236) 1. logarithmic aloga x x 4. 2. 10 5. x y 3. natural; e 43 64 3612 6 1 log6 36 1. 7. 13. 21. 2 31. 2 23. y 0.097 3. 9. 72 1 5. 49 log5 125 3 15. 2 log7 1 0 25. 1.097 3225 4 11. 4 19. 0 29. 1 log81 3 1 17. 4 27. 4 0, 1, 0 Domain: x-intercept: Vertical asymptote: x 0 71, 5, 0 Domain: x-intercept: Vertical asymptote: x 0 4 6 8 x 37. y 4 2 −2 −4 42. e e1.386. . . 4 e0 1 55. 39. c 45. 49. 53. 57. 63. 69. 40. f 41. d 47. 51. e0.693 . . . 1 2 e5.521 . . . 250 ln 20.0855 . . . 3 ln 0.6065 . . . 0.5 0.575 y 65. 3 67. 43. b 44. a 61. 2.913 ln 4 x ln 1.6487 . . . 1 2 59. 2 3 Domain: x-intercept: Vertical asymptote: 1, 2, 0 x 1 3 2 1 −1 1 2 3 4 5 x −1 −2 −3 y 2 1 , 0 Domain: 1, 0 x-intercept: Vertical asymptote: x 0 33. y 6 4 2 −2 − 4 − 6 35. −3 −2 −1 x 1 −2 x 0 73. 2 75. 3 0, 9, 0 Domain: x-intercept: Vertical asymptote: 2 4 6 8 10 12 x −2 77. 5 −3 2, Domain: 1, 0 x-intercept: Vertical asymptote: x 2 0 9 −1 x 3 x 7 79. 87. (a) 30 years; 20 years 81. x 4 83. (b) $396,234; $301,123.20 85. x 5, 5 (c) $246,234; $151,123.20 (d) x 1000; $1000. The monthly payment must be greater than 333200_03_AN.qxd 12/12/05 11:18 AM Page A123 89. (a) 100 (b) 80 (c) 68.1 (d) 62.3 0 12 0 91. False. Reflecting f x. the graph of gx about the line y x will determine 93. 952 − 1 1 2 x −2 −1 1 2 x − 1 − 2 −1 −2 Answers to Odd-Numbered Exercises and Tests A123 7. (a) 13. 21. log x log 2.6 0.417 6 ln 5 (b) ln x ln 2.6 15. 2.633 23. 2 is not in the domain of log3 x. 37. 2 43. 1 log5 x a 1 2 log2 3 2 ln y 5 ln z ln 3x 63. log4 z y 69. ln x x 13 ln 3xx 32 x2 1 75. 5. (a) 3 log 10 log x (b) 3 ln 10 ln x 2.000 11. 3 log5 2 9. 1.771 3 17. 2 3 25. 4 31. 4.5 39. 45. 49. 53. 57. 19. 27. 2.4 33. 29. 9 35. 7 41. 47. 4 log8 x ln x ln y 2 ln z 1 51. 2 log2 4 ln x 1 1 2 log4 5 log4 x 1 2 ln z ln z 2 lnz 1 1 3 ln x 1 2 log5 x 2 log5 y 3 log5 z 3 ln x 1 4 4 lnx2 3 3 ln y 55. 61. 59. 65. log2 x 4 2 67. log3 45x 4 ln 77. 73. x x2 44 xz3 y2 3y y 42 y 1 log2 32 log2 4; 60 dB log8 log2 32 10log I 12; y 256.24 20.8 ln x ln 1 0 u v 2 f x log x log 2 ln x ln 2 79. 81. 85. 87. False. 91. False. 95. Property 2 83. 3 lnx 2 ln x ln 2 89. False. 93. Answers will vary. f x log x log 1 2 97. ln x ln 3 3 −3 6 −3 3 −3 The functions f and g are inverses. 40 97. (a) The functions f and g are inverses. 71. log g f 0 0 1000 The natural log function grows at a slower rate gx; than the square root function. 15 (b) g f 0 0 20,000 The natural log function grows at a slower rate gx; than the fourth root function. (c) True (b) True (d) False 99. (a) False 101. (a) 4 (b) Increasing: Decreasing: 1, 0, 1 99. f x log x log 11.8 ln x ln 11.8 −1 −2 8 (c) Relative minimum: 103. 15 105. 4300 1, 0 107. 1028 −1 2 −2 5 Section 3.3 (page 243) 101. f x hx; Property 2 Vocabulary Check (page 243) 1. change-of-base 2. 3. c 4. a 5. b log x log a ln x ln a 1. (a) log x log 5 (b) ln x ln 5 3. (a) log x 1 log 5 (b) ln x ln 1 5 y 2 1 −1 − 333200_03_AN.qxd 12/9/05 1:30 PM Page A124 A124 Answers to Odd-Numbered Exercises and Tests 103. 107. 3x4 2y 3, x 0 1 1, 3 109. 105. 1, x 0, y 0 1 ± 97 6 Section 3.4 (page 253) Vocabulary Check (page 253) 1. solve x y 2. (a) 3. extraneous (b) x y (c) x (d) x (b) No (b) Yes 1. (a) Yes 3. (a) No 5. (a) Yes, approximate 7. (a) No 9. 2 17. 25. 13. 2 19. 64 1.618, (c) Yes, approximate (c) Yes (c) Yes, approximate (b) No ln 2 0.693 3, 8 23. 15. 21. 0.618 9, 2 1.465 1.994 (b) Yes 5 11. e 1 0.368 2, 1 27. ln 5 ln 3 ln 80 2 ln 3 3 ln 565 ln 2 1 ln 7 ln 5 3 5 ln 5 1.609 2.209 0.511 ln 29. 35. 41. 45. 49. 55. 63. 65. 67. 71. 6.142 43. 1 3 log3 2 0.059 47. ln 12 3 0.828 51. 0 53. ln 8 3 3 ln 2 1 3 0.805 57. ln 4 1.386 ln 1498 3.656 61. 1 2 21.330 59. 2 ln 75 8.635 ln 4 365 ln1 0.065 ln 2 12 ln1 0.10 365 12 6.960 6 −6 69. 300 15 −6 9 −30 0.427 8 −4 −20 12.207 −1200 3.847 73. −40 40 16.636 40 2 − 10 75. e3 0.050 77. e2.4 2 5.512 79. 1,000,000 81. e103 5 5.606 83. e2 2 5.389 e23 0.513 85. 89. No solution 93. No solution 87. 23116 14.988 1 1 e 2.928 1 17 2 97. 91. 95. 7 1.562 725 12533 8 180.384 105. 5 99. 2 101. 103. −8 10 −2 2.807 107. (a) 8.2 years 109. (a) 1426 units 111. (a) 10 10 −5 30 −1 20.086 (b) 12.9 years (b) 1498 units The yield (b) V 6.7; will approach 6.7 million cubic feet per acre. (c) 29.3 years 113. 2001 115. (a) y 100 100%. and y 0; The range falls between 0% and (b) Males: 69.71 inches Females: 64.51 inches 117. (a) x y (b) 200 0.2 0.4 0.6 0.8 1.0 162.6 78.5 52.5 40.5 33.9 The model appears to fit the data well. 0 0 1.2 (c) 1.2 meters (d) No. According to the model, when the number of ’s is between 2.276 meters and 4.404 is less than 23, meters, which isn’t realistic in most vehicles. g x 119. 121. logb uv logb u logb v True by Property 1 in Section 5.3. u v logb u logb v logb False. 1.95 log100 10 log 100 log 10 1 123. Yes. See Exercise 93. 125. Yes. Time to double: t ln 2 r Time to quadruple: t ln 4 r ; 2ln 2 r 31. ln 5 1.609 33. ln 28 3.332 37. 2 39. 4 0 0 1500 333200_03_AN.qxd 12/9/05 1:30 PM Page A125 Answers to Odd-Numbered Exercises and Tests A125 4xy23y y 127. 131. 129. 5 33 133. 14 12 4 −3 −2 −3 135. 1.226 137. 5.595 Section 3.5 (page 264) Vocabulary Check y aebx y aebx; 1. y a b ln x; 2. 3. normally distributed 5. sigmoidal (page 264) y a b log x 4. bell; average value y e 0.7675x 31. 35. (a) Decreasing due to the negative exponent. (b) 2000: population of 2430 thousand y 5e0.4024x 33. 2003: population of 2408.95 thousand (c) 2018 k 0.2988; 5,309,734 hits (b) 12,180 years old V 6394t 30,788 37. 41. (a) 43. (a) (c) 32,000 39. 3.15 hours 4797 years old (b) V 30,788e0.268t The exponential model depreciates faster. 0 0 t (d) 4 1 3 V 6394t
30,788 24,394 11,606 V 30,788e0.268t 23,550 13,779 (e) Answers will vary. St 1001 e0.1625t 45. (a) (b) 5. d Time to Double 19.8 yr 7.75 yr 6.3 yr 15.4 yr 6. f Amount After 10 years $1419.07 $1834.36 $1505.00 $10,000.00 4. a 3. b Annual % Rate 3.5% 8.9438% 11.0% 4.5 ( 120 90 60 30 5 10 20 15 25 Time (in years) (c) 55,625 t 30 C H A P T E R 3 1. c 2. e Initial Investment 7. $1000 9. $750 11. $500 13. $6376.28 15. $112,087.09 17. (a) 6.642 years (c) 6.302 years (b) 6.330 years (d) 6.301 years 19. 21. r t r t 2% 4% 6% 8% 10% 12% 54.93 27.47 18.31 13.73 10.99 9.16 2% 4% 6% 8% 10% 12% 55.48 28.01 18.85 14.27 11.53 9.69 23. A Continuous compounding 2.00 1.75 1.50 1.25 1.00 ) = 0.07t A = 1 + 0.075 [[ [[ t 2 4 6 8 10 t Half-life (years) 25. 1599 27. 5715 29. 24,100 Initial Quantity 10 g 2.26 g 2.16 g Amount After 1000 Years 6.48 g 2 g 2.1 g 47. (a) 0.04 (b) 100 70 115 0 49. (a) 203 animals 1200 (c) (b) 13 years Horizontal asymptotes: y 0, y 1000. The population size will approach 1000 as time increases. (b) 108.3 199,526,231 0 0 40 51. (a) (c) 107.9 79,432,823 104.2 15,849 53. (a) 20 decibels (c) 40 decibels (b) 70 decibels (d) 120 decibels 55. 95% 105.1 61. 65. (a) 150,000 57. 4.64 63. 3:00 A.M. 59. 1.58 106 moles per liter (b) 21 years; Yes 0 0 24 333200_03_AN.qxd 12/9/05 1:30 PM Page A126 A126 Answers to Odd-Numbered Exercises and Tests 67. False. The domain can be the set of real numbers for a 87. logistic growth function. f x 69. False. The graph of is the graph of gx shifted upward five units. 71. (a) Logarithmic (d) Linear 73. (a) (c) Exponential (f) Exponential (b) Logistic (e) None of the above y (b) (c) (d) 3 (0, 5) 5 10 1 2, 7 2 (−1, 2) 3 2 1 − 3 −2 − 1 1 2 3 − 1 75. (a3, 3) 2 4 6 8 10 14 (14, −2) 77. (a) y 1 1 2 3 4b) (c) (d) 146 17 2 , 1 5 11 2 (b) (c) 1 8 5 8, 1 8 (d) 1 79. y 81. y 2 10 8 6 4 2 − 2 −2 83. 2 6 8 10 12 y 7 6 5 4 3 2 1 −4 −2 85. x x y 3 1 −3 −2 −1 1 2 −1 −2 −3 y 14 12 10 8 6 4 2 89. y 14 12 10 10 x 93. Answers will vary. − 8 − 6 − 4 − 2 91. 2 y 5 4 3 2 1 −6 − 5 −4 − 3 −2 − 1 32 4 x −2 −3 −5 Review Exercises (page 271) 8. d 5. 1456.529 10. b 3. 0.337 1. 76.699 9. a 7. c 11. Shift the graph of f one unit to the right. 13. Reflect f in the x-axis and shift two units to the left. 15.25 4.063 4.016 17. x 2 f x 0.377 .65 7.023 18.61 − 3 − 6 − 9 − 12 − 15 333200_03_AN.qxd 12/9/05 1:30 PM Page A127 19. x f x 1 0 1 4.008 4.04 4. 21. x f x 2 1 3.25 3. 23. 31. − 4 − 2 x 4 2 4 25. x 22 5 x hx 2 1 2.72 1.65 x 0 1 27. 2980.958 29. 0.183 1 2 0.61 0.37 33. x 3 2 1 0 1 2.72 7.39 20.09 f x 0.37 Answers to Odd-Numbered Exercises and Tests A127 35. n A n A 1 2 4 12 $6569.98 $6635.43 $6669.46 $6692.64 365 Continuous $6704.00 $6704.39 (b) 0.487 43. 37. (a) 0.154 39. (a) $1,069,047.14 log4 64 3 41. 45. 3 47. 53. Domain: 3 0, 1, 0 x-intercept: Vertical asymptote: y 4 3 2 1 − 57. Domain: 5, (c) 0.811 (b) 7.9 years ln 2.2255 . . . 0.8 49. x 7 51. 55. Domain: x 5 0, 3, 0 x 0 x-intercept: Vertical asymptote-intercept: Vertical asymptote: 9995 59. 3.118 65. Domain: 12 61. 0, 63. 2.034 e3, 0 x-intercept: Vertical asymptote1 1 2 3 4 5 x 67. Domain: , 0, 0, ±1, 0 x-intercept: Vertical asymptote 69. 53.4 inches 75. log 2 2 log 3 1.255 71. 1.585 73. 2.322 77. 2 ln 2 ln 5 2.996 C H A P T E R 3 333200_03_AN.qxd 12/9/05 1:30 PM Page A128 A128 Answers to Odd-Numbered Exercises and Tests 79. 83. 87. 1 2 log5 x 81. 2 ln x 2 ln y ln z x 4y log2 5x 89. ln 1 log3 2 1 85. 3 log3 x lnx 3 ln x ln y 91. log8 y7 3x 4 2x 1 x 12 0 ≤ h < 18,000 93. ln 95. (a) (b) 100 0 0 Vertical asymptote: h 18,000 20,000 (c) The plane is climbing at a slower rate, so the time required increases. (d) 5.46 minutes 99. ln 3 1.099 97. 3 103. 109. 113. 115. e4 54.598 ln 22 ln 2 ln 2 0.693, 4.459 101. 16 ln 12 2.485 105. 111. ln 17 ln 5 1.760 ln 5 1.609 2 117. 20 −4 11 107. x 1, 3 −4 8 −12 2.447 4e 7.5 452.011 1 121. 125. e 4 1 53.598 − 8 7.480; 0.392 1 3e8.2 1213.650 3e 2 22.167 119. 123. 127. No solution 131. 1 −4 − 7 129. 0.900 133. −5 8 1 −9 1.643 135. 15.2 years 140. d 145. 2008 149. (a) 0.05 141. a 137. e No solution 139. f y 2e0.1014x 147. (a) 13.8629% (b) $11,486.98 138. b 143. 142. c (b) 71 40 0 100 103.5 watt per square centimeter 151. 153. True by the inverse properties 155. b and d are negative. a and c are positive. Answers will vary. Chapter Test (page 275) 1. 1123.690 5. x f x 2. 687.291 3. 0.497 4. 22.198 1 10 1 2 3.162 0 1 1 2 1 0.316 0..005 0.028 0.167 1 6 x f x y 1 −.865 0.632 y 1 2 1.718 1 6.389 0 0 x 6. 7. 10 − 4 −3 − 2 −1 1 2 3 4 −. (a) 9. 0.89 (b) 9.2 1 2 5.699 1 6 3 2 6.176 2 4 6.301 6.602 Vertical asymptote 333200_03_AN.qxd 12/9/05 1:30 PM Page A129 10. 5 0 x f x y 7 9 11 13 1.099 1.609 1.946 2.197 Vertical asymptote: x 4 4 2 −2 −4 2 6 8 x 11.099 2.609 2.792 2.946 Answers to Odd-Numbered Exercises and Tests A129 (page 276) 2 4 6 x Cumulative Test for Chapters 1–3 1, 3 Distance: 41 ; 2 1. (a) Midpoint: 2. y 3. −6 −4 x 4 8 y 2 −2 −4 −10 5. y 2x 2 16 12 8 −4 −8 −12 −8 −4 4. y 6 4 Vertical asymptote: x 6 −4 −2 2 4 6 x −2 −4 y 5 4 2 1 − 12. 1.945 15. 17. 14. 1.328 13. 0.115 log2 3 4 log2 a ln 5 1 log 7 2 log x log y 3 log z 16. 2 ln x ln 6 18. log3 13y 21. x 2 19. 22. ln x 4 y4 x ln 44 5 20. ln x2x 5 y3 0.757 23. ln 197 4 1.321 24. e12 1.649 e114 0.0639 y 2745e0.1570x 25. 27. 29. (a) 1.597 800 26. 501 28. 55 58.720 75.332 86.828 103.43 110.59 117.38 6. For some values of x (b) Division by 0 is undefined. y. there correspond two values of s 2 s (c) 7. (a) 3 2 8. (a) Vertical shrink by 1 2 (b) Vertical shift of two units upward (c) Horizontal shift of two units to the left 3x 4 (c) 4x2 11x 3 Domain: all real numbers x except x 1 4 (b) x 1 x2 . (a) ; (d) (b) 5x 2 x 3 4x 1 x 1 x2 1 x2x 1 x 1 x 1 x2 1 2x 12 Domain of Domain of x 2 Domain of f g: g f: (b) (d) ; 10. (a) (c) 11. (a) 12. (a) h1(x) 1 5 x 82 5 13. Yes; 15. 16. y 3 4 −8 −6 −2 2 4 6 x Domain: all real numbers x such that x ≥ 1 2x2 6 (b) all real numbers x such that all real numbers x 2 f g and g f: x 2 all real numbers 14. 2438.65 kilowatts x ≥ 6 y 3 2 1 −2 − 120 110 100 90 80 70 60 50 40 1 (b) 103 centimeters; 103.43 centimeters y 6 17. x 6 −4 −6 −8 −2 −3 4 2 5 3 Age (in years) 333200_03_AN.qxd 12/9/05 1:30 PM Page A130 A130 18. Answers to Odd-Numbered Exercises and Tests y 12 10 6 4 2 −10 −8 −6 −4 −2 − 2 s 2 4 x 42x 1x 1 3ix 1 3i 2, ±2i; x 2x 2ix 2i 7, 0, 3; xxx 3x 7 4, 1 2, 1 ± 3i; 3x 2 3x 2 2x2 1 2x3 x2 2x 10 25 x 2 4 −3 3 19. 20. 21. 22. 23. 24. Interval: 25. Intercept: −6 1, 2; 1.20 0, 0 Vertical asymptotes: Horizontal asymptote: y x ±3 y 0 (0, 0) 1 654 x −10 , 1 1, 0 26. y-intercept: x-intercept: Horizontal asymptote: Vertical asymptote1, 0) x 1 2 (0, −1 27. y-intercept: x-intercepts: Slant asymptote: Vertical asymptote: 0, 6 2, 0, 3, 0 y x 4 x 1 y (0, 6) (−3, 0) 2 −8 −6 (−2, 0) 2 4 6 8 28. x ≤ 2 or 0 ≤ x ≤ 2 −3 −2 −1 0 1 2 3 x x 29. All real numbers such that x x < 5 or x > 1 −6 −5 −4 −3 −2 −1 0 1 2 x 30. Reflect f in the x-axis and y-axis, and shift three units to the right. −10 7 f −7 11 g 31. Reflect f in the x-axis, and shift four units upward. 6 f g 8 −6 33. 0.067 34. 1.717 35. 0.281 x > 4 x ln 12 2 38. 1.242 3 ln 2 2.079 32. 1.991 36. lnx 4 lnx 4 4 ln x, , ln x > 0 x2 x 5 ln 3 1.099 e6 2 401.429 or 37. 39. 40. 41. (a) 50 7 20 13 (b) S 0.274t2 4.08t 50.6 333200_04_AN.qxd 12/9/05 1:32 PM Page A131 (c) 50 (c) 2,900,000 Answers to Odd-Numbered Exercises and Tests A131 7 20 13 The model is a good fit for the data. (d) 65.9 Yes, this is a reasonable answer. Problem Solving (page 279) 1. a = 0..5 x and y 1.2x 0 ≤ a ≤ 1.44 a = 2 a = 1.2 1 2 3 4 x − → , the graph of 3. As the graph of xn. ex increases at a greater rate than 21. 30 y2 y1 0 200,000 85 (d) The exponential model is a better fit. No, because the model is rapidly approaching infinity. 17. 19. 1, e2 y4 x 1 1 2 x 12 1 3 x 13 1 4 x 14 −3 4 −4 y = ln x y4 9 The pattern implies that ln x x 1 1 2 x 12 1 3 x 13 . . . . 5. Answers will vary. 7. (a) 6 y = ex (c) −6 −6 9. −2 6 y = ex y3 −b) y1 6 y = ex y2 6 −6 6 −2 100 0 1500 17.7 cubic feet per minute 23. (a) 9 25. (ab)–(e) Answers will vary. (b)–(e) Answers will vary. Chapter 4 Section 4.1 (page 290) Vocabulary Check (page 290) f1x lnx x2 4 2 ln c2 ln c1 ln 1 1 k2 2 252,6061.0310t 400.88t2 1464.6t 291,782 1 k1 t 13. y1 y2 11. c 15. (a) (b) 2. angle 5. acute; obtuse 1. Trigonometry 4. radian 6. complementary; supplementary 8. linear 9. angular 10. 3. coterminal 7. degree 2r2 A 1 1. 2 radians 3. 7. (a) Quadrant I 9. (a) Quadrant IV 11. (a) Quadrant III 3 radians (b) Quadrant III 5. 1 radian (b) Quadrant III (b) Quadrant II 333200_04_AN.qxd 12/9/05 1:32 PM Page A132 A132 13. (a) 15. (a) Answers to Odd-Numbered Exercises and Tests (b) 37. (a) (b) π 5 4 y y π 11 b) x x y y 405° 480° x x 405, 315 39. (a) 600, 120 41. (a) 43. (a) Complement: 324, 396 180, 540 Supplement: (b) (b) 72; (b) Complement: none; Supplement: 11 (b) Complement: none; Supplement: 45. (a) Complement: ; Supplement: 162 65 101 30 −3 47. (a) 51. (a) 55. 2.007 63. 69. 73. (a) 75. (a) 77. (a) 6 79. 5 50 85. 29 25.714 114.592 85.308 240 36 2 30 radians radians 6 270 (b) 5 6 (b) 210 3.776 57. 65. 49. (a) (b) 9 53. (a) 420 59. 9.285 67. 4 3 (b) 61. 66 0.014 337.500 71. (a) 54.75 756.000 128.5 (b) (b) 81. (b) (b) 330.007 145 48 3 34 48 32 83. radians 7 15 inches 47.12 inches 87. 8 3 radian 2 9 89. 3 meters 91. square inches 8.38 square inches 2 1.14 0.071 radian 4.04 93. 12.27 square feet 97. 101. (a) 728.3 revolutions per minute (b) 4576 radians per minute 95. 591.3 miles 5 12 radian 99. 103. (a) 10,400 radians per minute 32,672.56 radians per minute 17. (a) 19. (a) 13 6 8 3 , 11 6 , 4 3 (b) , 7 17 6 6 , 23 25 12 12 (b) 21. (a) Complement: ; 6 Supplement: 2 3 (b) Complement: none; Supplement: 4 23. (a) Complement: Supplement: 1 0.57; 2 1 2.14 (b) Complement: none; Supplement: 210 60 165 29. 27. 25. 31. (a) Quadrant II 33. (a) Quadrant III 35. (a) (b) Quadrant IV (b) Quadrant I (b) y y 30° x 150° x 94253 400, 1000 2400, 6000 (b) 105. (a) (b) 107. feet per minute 9869.84 feet per minute radians per minute centimeters per mi
nute 140° 35 109. False. A measurement of A 476.39 square meters 1496.62 square meters 4 radians corresponds to two complete revolutions from the initial to the terminal side of an angle. 333200_04_AN.qxd 12/9/05 1:32 PM Page A133 Answers to Odd-Numbered Exercises and Tests A133 111. False. The terminal side of the angle lies on the -axis. 113. Increases. The linear velocity is proportional to the radius. 115. The arc length is increasing. If is constant, the length of x 117. 121. −2 the arc is proportional to the radius 2 2 210 119. s r . y y = x5 x − 2)5 123. −3 −2 y 6 5 4 3 1 −1 −2 −3 y = x5 1 2 3 x y = 2 − x5 Section 4.2 (page 299) Vocabulary Check (page 299) 1. unit circle 3. period 2. periodic 4. odd; even csc 17 15 sec 17 8 cot 8 15 csc 13 5 sec 13 12 cot 12 5 3 , 1 2 2 7. 9. 1 2 , 3 2 15. 19. 1 sin 2 6 cos 3 2 6 tan 3 6 3 sin 11 1 6 2 11 3 6 2 11 3 6 3 cos tan sin 15 17 cos 8 17 tan 15 8 sin 5 13 cos 12 13 tan 5 12 2 2 2 2 0, 1 , sin cos 4 4 2 2 2 2 1. 3. 5. 11. 13. 17. 21. 2 2 2 2 1 tan 4 sin7 4 cos7 4 1 tan7 4 1 sin3 2 0 cos3 2 tan3 2 is undefined. 23. 25. 27. sin cos tan 4 1 sin 2 cos 0 2 tan 2 sin4 3 2 3 1 cos4 2 3 3 tan4 3 is undefined. sec csc 2 2 3 4 3 4 3 1 cot 4 1 csc 2 sec is undefined. 2 cot 0 2 23 csc4 3 3 2 sec4 3 cot4 3 3 3 8 3 2 3 cos cos 1 2 31. 0 35. 33. 29. sin 5 sin 0 cos cos15 2 2 sin sin9 7 4 4 1 3 37. (a) (b) 3 4 41. (a) 5 0.1288 47. 1 53. (a) 55. (a) 0.25 or 2.89 57. (a) (b) (b) 4 5 49. 1.3940 0.4 2 2 1 5 39. (a) 43. 0.7071 51. (b) 5 45. 1.0378 1.4486 (b) 1.82 or 4.46 t y 0 1 4 0.25 0.0138 1 2 0.1501 3 4 0.0249 1 0.0883 t 5.5 (b) 59. False. sint sin t (c) The displacement decreases. means that the function is odd, not that the sine of a negative angle is a negative number. (b) cos t1 65. y -axis symmetry cos t1 61. (a) (c) f 1x 2 3 sin t1 x 1 sin t1 f 1x x2 4, x ≥ 0 69. 63. 67. y 8 6 4 2 −6 −4 −2 −2 −4 −6 −8 2 4 6 8 10 x −6 −5 −2 y 4 3 2 1 −1 −1 −2 −3 − 333200_04_AN.qxd 12/9/05 1:32 PM Page A134 A134 Answers to Odd-Numbered Exercises and Tests Section 4.3 (page 308) Vocabulary Check (page 308) (b) iv 1. (a) v 2. opposite; adjacent; hypotenuse 3. elevation; depression (c) vi (d) iii (e) i (f) ii 1. sin cos tan 3. sin cos tan 5. sin cos tan 3 5 4 5 3 4 9 41 40 41 9 40 1 3 22 3 2 4 csc sec cot csc sec cot 5 3 5 4 4 3 41 9 41 40 40 9 csc 3 sec 32 4 cot 22 The triangles are similar, and corresponding sides are proportional. 3 5 4 5 3 4 7. sin cos tan The triangles are similar, and corresponding sides are proportional. 5 3 5 4 4 3 csc sec cot 9. 4 3 θ 7 cos 7 4 tan 37 7 csc 4 3 sec 47 7 7 3 cot 11. sin 3 2 1 2 cos tan 3 csc cot 23 3 3 3 sin 213 13 2 cos 313 13 csc sec 13 2 13 3 tan 2 3 60; 3 2 3 3 23. 30; (d) 3 2 13 2 3 (c) (d) 1 2 2 3 (c) (d) 2 4 43. (a) 0.1736 1 3 (b) 0.1736 13 3 15. 17. 25. θ ; 6 1 2 45; 19. 60; 3 21. 27. (a) 3 (b) 4 213 13 29. (a) 31. (a) 3 (b) 313 13 (c) (b) 22 3 33– 41. Answers will vary. 45. (a) 0.2815 47. (a) 1.3499 49. (a) 5.0273 51. (a) 1.8527 (b) 3.5523 (b) 1.3432 (b) 0.1989 (b) 0.9817 53. (a) 30 55. (a) 60 57. (a) 60 6 3 3 (b) 30 (b) 45 (b) 45 6 4 4 59. 303 61. 323 3 63. 443.2 meters; 323.3 meters 65. 30 67. (a) 371.1 feet (b) 341.6 feet 6 (c) Moving down line at 61.8 feet per second Dropping vertically at 24.2 feet per second x1, y1 x2, y2 283, 28 28, 283 69. 71. (a) (b) sin 85 h 20 (c) 19.9 meters 20 h 85° 2 3 θ 1 10 3 θ 1 13. sin 310 10 10 10 10 3 cos csc sec 10 cot 1 3 (d) The side of the triangle labeled will become shorter. (e) h Angle, 80 70 60 50 Height 19.7 18.8 17.3 15.3 Angle, 40 30 Height 12.9 10.0 20 6.8 10 3.5 333200_04_AN.qxd 12/9/05 1:32 PM Page A135 Answers to Odd-Numbered Exercises and Tests A135 (f) As → 0, h → 0. 7. sin h 20 θ 75. False, 2 2 2 2 1. 9. 73. True, csc x 1 sin x 1.7321 0.0349. . 77. False, 79. Corresponding sides of similar triangles are proportional. 81. (a) sin 0.1 0.2 0.3 0.4 0.5 0.0998 0.1987 0.2955 0.3894 0.4794 (b) x x 2 83. (c) As approaches 0, sin approaches 0. , x ±6 85. 2x2 5x 10 x 2x 22 Section 4.4 (page 318) 19. sin 529 29 229 29 cos tan 5 2 sin 685849 cos 355849 5849 5849 29 5 29 2 csc sec cot 2 5 csc sec 5849 68 5849 35 tan 68 35 11. Quadrant III 15. sin cos tan 17. sin cos tan 3 5 4 5 3 4 15 17 8 17 15 8 10 10 310 10 1 3 3 2 1 2 cos tan cos 21. sin cot 35 68 13. Quadrant II 5 3 5 4 4 3 17 15 17 8 8 15 csc sec cot csc sec cot csc 10 sec 10 3 cot 3 csc 23 3 sec 2 C H A P T E R 4 Vocabulary Check (page 318) y r cot 1. 6. 2. csc 3. y x 4. r x 5. cos 7. reference 1. (a) sin cos tan csc sec cot 3. (a) sin cos b) sin cos tan csc sec cot (b) sin cos 15 17 8 17 15 8 17 15 17 8 8 15 17 17 417 17 tan csc sec 3 3 2 23 3 3 cot 24 25 7 25 24 7 5. sin cos tan csc sec cot 25 24 25 7 7 24 tan 1 4 csc 17 sec 17 4 cot 4 tan 3 cot 3 3 23. sin 0 cos 1 tan 0 25. sin 2 2 cos 2 2 tan 27. sin 1 25 5 5 5 cos tan 2 is undefined. csc sec 1 cot is undefined. csc 2 sec 2 cot 1 csc 5 2 sec 5 cot 1 2 333200_04_AN.qxd 12/9/05 1:32 PM Page A136 A136 Answers to Odd-Numbered Exercises and Tests 29. 0 37. 23 31. Undefined 33. 1 39. 65 y 35. Undefined 79. 0.4142 203° ′θ −245° 43. 3.5 y 3.5 ′θ 47. sin 51. sin 750 750 750 cos tan cos tan 3 4 3 4 3 4 3 11 4 11 4 11 4 55. sin cos tan 2 2 2 2 1 81. (a) 83. (a) 85. (a) 87. (a) (b) (b) 30 60 210 7 6 135 3 4 150 5 6 , 150 5 6 6 , 120 2 3 3 , 225 5 4 N 22.099 sin0.522t 2.219 55.008 F 36.641 sin0.502t 1.831 25.610 N 34.6, F 1.4 45 (b) 4 , 330 11 6 , 315 7 4 , 330 11 6 (b) February: March: May: June: August: September: November: N 41.6, F 13.9 N 63.4, F 48.6 N 72.5, F 59.5 N 75.5, F 55.6 N 68.6, F 41.7 N 46.8, F 6.5 (c) Answers will vary. 89. (a) 2 centimeters 1.98 (c) centimeters (b) 0.14 centimeter x x 95. As 91. 0.79 ampere 93. False. In each of the four quadrants, the signs of the secant function and cosine function will be the same, because these functions are reciprocals of each other. 0 y 0 cm and sin y12 decreases from 1 to 0. Thus, 90 without bound. When x y decreases from 12 cm to increases from 0 cm to 12 cm. Therefore, cos x12 increases from 0 to 1 and tan yx and increases , the tangent is undefined. -intercepts: increases from 90 , to x 1, 0, 4, 0 y -intercept: Domain: all real numbers 0, 4 x 2, 0 x -intercept: 0, 8 y -intercept: Domain: all real numbers x 97. 99. (1, 0) 2 4 6 8 x (0, −4) (−4, 0) −6 −8 8 6 4 2 −2 −2 −4 −8 y 12 10 (0, 8) (−2, 0) x −8 −6 −4 2 4 6 8 −4 x x y 3 y ′θ 41. π 2 3 ′θ 45. sin 225 cos 225 2 2 2 2 tan 225 1 49. sin150 1 2 3 2 cos150 3 tan150 3 1 sin 2 6 cos 3 2 6 tan 3 3 6 1 sin3 2 0 cos3 2 tan3 2 4 5 0.3420 13 2 69. 61. 53. 57. 59. is undefined. 67. 73. 4.6373 75. 0.3640 77. 63. 8 5 1.4826 65. 0.1736 71. 3.2361 0.6052 333200_04_AN.qxd 12/12/05 11:19 AM Page A137 Answers to Odd-Numbered Exercises and Tests A137 29. y 101 (− 0, ( (7, 0) 6 8 x 7, 0 x -intercept: 0, 7 y -intercept: 4 Vertical asymptote: x 2 Horizontal asymptote: y 0 27. g Domain: all real numbers x 2 except x 31. 35. 39. 43. 103. y 5 4 3 2 1 ) 2) 0 105. y 12 9 6 (−1, 0) (1, 0) − 12 − 9 − 6 − 3 3 6 9 12 x x 0, 1 y -intercept: Horizontal asymptote: 2 y 0 Domain: all real numbers x ±1, 0 x -intercepts: x 0 Vertical asymptote: Domain: all real numbers except x 0 x Section 4.5 (page 328) Vocabulary Check (page 328) 1. cycle 2. amplitude 3. 4. phase shift 5. vertical shift 2 b 1. Period: Amplitude: 3 7. Period: 2 3. Period: 4 Amplitude: 9. Period: 5 5. Period: 6 Amplitude: 1 2 5 2 Amplitude: 3 Amplitude: 3 11. Period: 3 Amplitude: g g 1 2 f is a shift of is a reflection of 13. Period: 1 1 Amplitude: 4 15. units to the right. 17. xin the axis. 19. The period of f is twice the period of 21. f is a shift of three units upward. 23. The graph of has twice the amplitude of the graph of g 25. The graph of g is a horizontal shift of the graph of g. f g f f. units to the right1 y 4 3 2 1 x π3 2 g f π3 33. 37. 41. 45. x x x x − π3 2 − π 2 π 2 π3 2 −4 y 2 π− 2 π2 π4 − 1 − 2 y 3 2 −1 2 3 −2 −1 y 3 −3 y 4 3 1 2 3 x π2 1 4 − 3 y 2 1 −− π −2 −3 333200_04_AN.qxd 12/9/05 1:32 PM Page A138 A138 47. 51. − 4 − 6 y 2.2 1.8 Answers to Odd-Numbered Exercises and Tests 493 –2 –1 −1 1 2 3 x 53. x π π 2 y 4 2 −8 4 −0.1 0 y 55. x 0.1 0.2 57 59. − 3 −6 6 π x π4 −4 61. 0.12 −20 20 3 −0.12 63. a 2, d 1 65. a 4, d 4 67. a 3, b 2, c 0 69. a 2, b 1, c 4 71. −2 2 − 11 6 73. (a) 6 seconds (b) 10 cycles per minute (c) v 1.00 0.75 0.50 0.25 −0.25 −1.00 2 4 8 10 t 75. (a) Ct 56.55 26.95 cos 6 t 3.67 (b) 100 0 0 12 The model is a good fit. 100 (c) 0 0 12 The model is a good fit. 77.90; (d) Tallahassee: Chicago: The constant term gives the annual average temperature. 56.55 (e) 12; yes; one full period is one year. (f) Chicago; amplitude; the greater the amplitude, the greater the variability in temperature. 1 440 (b) 440 cycles per second 77. (a) second 79. (a) 365; answers will vary. (b) 30.3 gallons; the constant term (c) 60 124 < t < 252 81. False. The graph of f x sin x graph of the two graphs look identical. cos x sinx 83. True. Because f x sinx 2 translates the exactly one period to the left so that reflection in the -axis of y sinx x , 2 is a y cos x . 2 2 0 0 365 333200_04_AN.qxd 12/9/05 1:32 PM Page A139 Answers to Odd-Numbered Exercises and Tests A139 7. 11. 15. 19. 23. 85. y 2 1 f = g − π3 2 π 2 π3 2 x −2 Conjecture: sin x cosx 2 87. (a) −2 2 − 2 2 The graphs appear to coincide from 2 to . 2 (b) −2 2 − 2 2 The graphs appear to coincide from 2 to . 2 (c) x7 7! , x 6 6! −2 2 − 2 2 −2 2 −2 2 The interval of accuracy increased. 91. x 2 89. 1 2 log10 3 ln t lnt 1 3x y4 93. log10 xy 95. ln 97. Answers will vary. Section 4.6 (page 339) Vocabulary Check (page 339) 1. vertical 5. 4. 2 7. 2. reciprocal 6. x n 3. damping , 1 1, 1. e, 5. f, 4 2 2. c, 6. b, 4 3. a, 1 4. d. 13. 17. 212 −1 1 2 x x −3 − 2π x − π x π 25. y 4 3 2 1 −4 4 x − π3 2 π 2 x y y y −3 −2 −1 −1 π−2 3 2 1 6 4 2 333200_04_AN.qxd 12/9/05 1:32 PM Page A140 Answers to Odd-Numbered Exercises and Tests 29. y 2 1 4 −4 3 −3 x π2 2 2 5 4 41. 7 4 , 3 , 4 4 , A140 27. y 4 3 2 1 −π −1 π 2π 3π x 31. 35. 39. 43. 47. −.6 −6 6 − 0. 33. 5 − 2 37. 3 2 − 2 45 49. Even (b π3 4 x π 51. (a) y 3 2 1 −1 53. (c)
−3 approaches 0 and f approaches cosecant is the reciprocal of the sine. g because the 2 − 2 3 The expressions are equivalent except that when y1 is undefined. sin x 0, 55. −2 4 −4 2 The expressions are equivalent. f → 0 as x → 0. g → 0 as x → 0. 58. a, 60. c, f → 0 as x → 0. g → 0 as x → 0. 57. d, 59. b, 61. y 3 2 1 −1 −2 −3 63. y 3 2 1 2 3 x π− π –1 x −3 −2 −1 The functions are equal. The functions are equal. 65. 1 67. 6 −8 8 −9 9 −1 x → , gx → 0. As 6 69. −6 x → , f x → 0. As 71. 2 0 −2 As x → 0, y → . 8 −6 6 −1 x → 0, gx → 1. As 73. − 2 −2 75. x → 0, f x As d 7 cot x d oscillates between 1 and 1 14 10 6 2 −2 −6 −10 −14 π 4 π 2 π 3 4 x π Angle of elevation 333200_04_AN.qxd 12/9/05 1:32 PM Page A141 Answers to Odd-Numbered Exercises and Tests A141 77. (a) 50,000 R C 0 0 100 17. −1.5 1 −1 f g the coordinate of csc x is the 43. 0.3 45. 0.1 47. 0 49. 3 5 51. 5 5 (b) As the predator population increases, the number of prey decreases. When the number of prey is small, the number of predators decreases. C : 24 months H: 24 months; 12 months; 12 months R : L: (c) 79. (a) (c) 1 month (b) Summer; winter 81. True. For a given value of ysin x. reciprocal of the coordinate of f from the left, y2 from the right, 83. As approaches 2 approaches x, x f approaches . approaches x. As 85. (a) 2 −2 −3 0.7391 3 (b) 1, 0.5403, 0.8576, 0.6543, 0.7935, 0.7014, 0.7640, 0.7221, 0.7504, 0.7314, . . . ; 0.7391 87. − 3 2 6 − 6 3 2 The graphs appear to coincide on the interval 1.1 ≤ x ≤ 1.1. ln 54 2 2 e73 3 ln 2 0.693 1.684 1031 1.994 91. 89. 93. 95. ± e3.2 1 ±4.851 97. 2 Section 4.7 (page 349) Vocabulary Check (page 349) 1. 2. 3. y sin1 x; 1 ≤ x ≤ 1 y arccos x; 0 ≤ y ≤ y tan1 x. 11. 6 2 3 3. 3 13. 3 5. 6 7. 5 6 9. 3 15. 0 1.5 1.25 23. 31. 0.85 37. arctan 25. 0.32 33. 1.29 x 4 0.85 19. 1.29 27. 1.99 35. , 3 21. 29. 0.74 3 3 , 1 39. arcsin x 2 5 41. arccos x 3 2x 53. 12 13 55. 34 5 57. 5 3 59. 61. 1 4x 2 63. 1 x 2 65 67. 69. −3 2 −2 3 71. 73. 75. y ±1 , x > 0; Asymptotes: 9 x2 81 x 1 x2 2x 10 9 x2 81 , x < 0 y π 77. −1 1 2 3 x y π2 π − π −2 −1 1 2 g is a The graph of horizontal shift one unit to the right of f. 79. y π 81. y π −4 −2 2 4 x − 333200_04_AN.qxd 12/9/05 1:32 PM Page A142 A142 Answers to Odd-Numbered Exercises and Tests 83. 2 85. 103. Domain: Range: , 1 1, 2, 0 0, 2 −2 4 − y π 2 −2 −1 1 2 x − π 2 87. −1 −4 0 4 −2 32 sin2t 89. 4 −2 6 −6 1 5 2 The graph implies that the identity is true. 91. (a) arcsin 5 s 93. (a) 1.5 (b) 0.13, 0.25 0 − 0.5 6 (b) 2 feet (c) 95. (a) 26.0 As 0; x (b) 24.4 feet increases, 97. (a) arctan x 20 (b) 14.0, 31.0 approaches 0. is not in the range of the arctangent. 99. False. 5 4 101. Domain: , Range: 0 105. (a) 107. (a) (b) 4 f f 1 2 2 (c) 1.25 (d) 2.03 f 1 f − − −2 2 −2 (b) The domains and ranges of the functions are restrictdiffer because of f f 1 f 1 f ed. The graphs of the domains and ranges of and 111. 117.391 and f f 1. 109. 1279.284 113. 7 115. sec 47 7 7 3 cot 7 cos 4 tan 37 7 csc 4 3 3 4 θ 11 6 11 5 sec 6 5 cot 511 11 sin tan csc 611 11 6 5 θ 11 117. Eight people 119. (a) $21,253.63 (c) $21,285.66 (b) $21,275.17 (d) $21,286.01 Section 4.8 (page 359) Vocabulary Check (page 359) 1. elevation; depression 3. harmonic motion 2. bearing 333200_04_AN.qxd 12/9/05 1:32 PM Page A143 1. 7. a 3.64 c 10.64 B 70 a 49.48 A 72.08 B 17.92 13. 19.99 inches 19. (a) 5. c 11.66 A 30.96 B 59.04 11. 2.56 inches 3. 9. a 8.26 c 25.38 A 19 a 91.34 b 420.70 B 7745 15. 107.2 feet 17. 19.7 feet h x y 47° 40′ 50 ft 35° h 50tan 4740 tan 35 (b) (c) 19.9 feet (b) tan 121 2 171 3 (c) 35.8 21. 2236.8 feet 23. (a) 1 12 ft 2 θ 1 17 ft 3 2.06 27. 0.73 mile 25. 29. 554 miles north; 709 miles east 31. (a) 58.18 nautical miles west; 104.95 nautical miles south distance 130.9 nautical miles (b) 68.82 meters 37. 1933.3 feet W; (b) S 36.7 N 58 E N 56.31 W 3.23 miles or 17,054 feet 78.7 y 3 r 35.3 49. 33. (a) 35. 39. 41. 47. 43. a 12.2, b 7 45. 29.4 inches 51. d 4 sint 53. 59. (c) 528 32 d 3 cos4t 3 (d) 1 16 (d) 1 120 (b) 4 (c) 4 (b) 60 (c) 0 (b) 8 π 8 π 4 π3 8 π 2 t 55. (a) 4 1 16 57. (a) 61. (a) y 1 − 1 Answers to Odd-Numbered Exercises and Tests A143 63. (a) Base 1 Base 2 Altitude Area 8 8 8 8 8 8 8 8 16 cos 30 8 sin 30 8 16 cos 40 8 sin 40 8 16 cos 50 8 sin 50 8 16 cos 60 8 sin 60 8 16 cos 70 8 sin 70 8 16 cos 80 8 sin 80 8 16 cos 90 8 sin 90 59.7 72.7 80.5 83.1 80.7 74.0 64.0 (b) Base 1 Base 2 Altitude Area 8 8 8 8 8 8 8 16 cos 56 8 sin 56 8 16 cos 58 8 sin 58 8 16 cos 59 8 sin 59 82.73 83.04 83.11 8 16 cos 60 8 sin 60 83.14 8 16 cos 61 8 sin 61 8 16 cos 62 8 sin 62 83.11 83.04 (c) (d) 83.14 square feet A 641 cos sin 100 0 0 90 83.1 square feet when The answers are the same. 60 65. False. The tower is leaning, so it is not perfectly vertical and does not form a right angle with the ground. C H A P T E R 4 67. No. 69. N 24 E y 4x 6 y 7 6 5 3 2 1 means 24 degrees east of north. 71. y 4 5x 22 5 y 7 6 4 3 2 1 −4 −3 −2 −1 −1 1 2 3 4 x −2 −1 −1 1 2 3 4 5 x 333200_04_AN.qxd_pg A144 1/9/06 8:57 AM Page A144 A144 Answers to Odd-Numbered Exercises and Tests Review Exercises (page 365) 1. 0.5 radian 3. (a) y 5. (a) y 41. sin 441 41 cos 541 41 43. sin 3 2 cos 1 2 π 11 4 (b) Quadrant II , 5 4 3 4 (c) 7. (a) y 70° x x x π − 4 3 (b) Quadrant II , 10 3 2 3 (c) 9. (a) y x 59. −110° (b) Quadrant III 250, 470 (c) 128.571 , , 13. 15. 27. 25. 23. 29. sin csc cos 2 0.589 19. 478.17 inches 430, 290 200.535 662 3 400 3 2 1 2 3 2 (b) Quadrant I (c) 11. 8.378 17. 21. (a) radians per minute (b) inches per minute Area 339.28 square inches 1 1 2 2 7 6 7 6 7 tan 6 sin2 3 cos2 3 tan2 3 11 sin 4 sin17 6 75.3130 3 3 3 2 1 2 3 3 sin 4 sin 7 6 39. 3.2361 csc2 3 sec2 3 cot2 3 sec cot 37. 31. 35. 33. 23 3 23 3 2 3 3 15 15 55. 71.3 meters csc tan 4 5 41 4 41 5 sec tan 3 csc 23 3 sec 2 cot 5 4 cot (b) (b) 22 3 15 4 51. 0.5621 (c) (c) (d) 32 4 415 15 53. 3.6722 45. (a) 3 47. (a) 1 4 49. 0.6494 57. 3 3 2 4 (d) csc 5 4 sec 5 3 cot 3 4 sin 4 5 cos 3 5 tan 4 3 sin 15241 241 cos 4241 241 csc sec 241 15 241 4 cot 4 15 82 9 csc sec 82 cot 1 9 17 4 csc sec 17 cot 1 4 61. 63. tan 15 4 sin 982 82 82 82 cos tan 9 sin 417 17 17 17 cos tan 4 65. sin 11 6 67. cos 5 6 11 5 tan csc 611 11 cot 511 11 55 8 cos tan 355 55 csc 8 3 sec 855 55 55 3 cot 333200_04_AN.qxd 12/9/05 1:33 PM Page A145 69. sin 21 5 21 tan 2 csc 521 21 sec 5 2 cot 221 21 71. 84 y 73. 5 y 264° ′θ ′θ x x − π 6 5 1 3 ; tan 2 3 1 cos7 2 3 ; 75. 77. 79. sin 3 sin7 3 tan7 3 3 ; cos 2 3 3 ; 2 3 2 2 ; sin 495 cos 495 2 2 ; tan 495 1 81. sin240 3 2 ; cos240 1 2 ; tan240 3 0.7568 85. 0.0584 83. 89. y 2 1 − π3 2 π 2 x − 2 87. 3.2361 91. y 6 4 2 −2 −6 x π6 Answers to Odd-Numbered Exercises and Tests A145 93. 951 −2 −3 −4 π t 97. (a) 99. y 2 sin 528x (b) 264 cycles per second 101− π 105− −1 −2 −3 −4 π 6 x − π3 2 π 2 −3 −4 −9 9 −6 x → , f x → As 111. 0.41 113. 6 0.46 115. 6 −1.5 119. 1.24 121. 0.98 125. 1.5 −4 − 2 − 2 103. 107. 109. 117. 123. 127. 4 5 129. 13 5 131. 4 x2 x 133. 66. 333200_04_AN.qxd 12/12/05 11:20 AM Page A146 A146 Answers to Odd-Numbered Exercises and Tests 135. 1221 miles, 137. False. The sine or cosine function is often useful for 85.6 6. 70 y modeling simple harmonic motion. 139. False. For each there corresponds exactly one value of 2 141. d; The period is 143. b; The period is 2 and the amplitude is 2. 145. The function is undefined because 147. The ranges of the other four trigonometric functions are and the amplitude is 3. sec 1cos . y. , , 1 1, . 149. (a) or A 0.4r2, r > 0; s 0.8r, r > 0 (b) A 50, > 0; s 10, > 0 30 The area function increases more rapidly. Chapter Test (page 369) 1. (a) y π 5 4 x , 3 4 13 4 225 (b) (c) 4. sin 2. 3000 radians per minute 310 10 10 10 cos csc sec square feet 3. 709.04 10 3 10 tan 3 5. For 0 ≤ < : 2 313 13 213 13 13 3 13 2 sin cos csc sec cot 1 3 For ≤ < 3 : 2 sin 313 13 csc cos 213 13 13 3 13 2 sec cot 2 3 cot 2 3 290° x ′θ 7. Quadrant III sin 4 10. 5 tan 4 3 csc 5 4 sec 5 3 cot 3 4 y 12. 8. 11. 150, 210 sin 15 17 cos 8 17 tan 15 8 csc 17 15 cot 8 15 13. 9. 1.33, 1.81 y 4 3 2 1 4 3 1 −1 −2 −3 −4 4 −4 14. −6 x π2 −π − π 2 α π π 2 6 15. 6 0 −2 32 16. 18. Period: 2 a 2, b 1 2 , c 4 Not periodic 5 2 17. y π −2 x 1 2 − π 19. 310.1 20. d 6 cos t Problem Solving 11 2 1. (a) radians or (page 371) 990 (b) 816.42 feet 333200_05_AN.qxd 12/9/05 1:50 PM Page A147 Answers to Odd-Numbered Exercises and Tests A147 (b) 3705 feet (c) 3. (a) 4767 feet w 2183 tan 63 w 3705 feet, 3000 5. (a) (b) 3 −1 −2 2 −2 3 −1 2 Even Even h 51 50 sin8t 7. 2 9. (a) 2 E P I 7300 7380 −2 2 (b) 7348 I 7377 E P −2 (c) P7369 0.631 E7369 0.901 I7369 0.945 (b) 11. (a) 3.35, 7.35 0.65 (c) Yes. There is a difference of nine periods between the values. 40.5 (b) 1.75 feet x 1.71 feet; y 3.46 feet 13. (a) (c) (d) As you move closer to the rock, must get smaller and and will decrease along with d 2 smaller. The angles d the distance y, 1 so will decrease. Chapter 5 Section 5.1 (page 379) Vocabulary Check (page 379) 1. 5. 9. tan u cot2 u cos u 2. 6. 10. cos u sec2 u tan u 3. cot u 7. cos u 4. csc u 8. csc u 1. 5. 9. sin x 3 2 cos x 1 2 tan x 3 csc x 23 3 sec x 2 cot x 3 3 sin x 5 13 cos x 12 13 tan x 5 12 sec x 13 12 csc x 13 5 cot x 12 5 sin x 1 3 cos x 22 3 2 4 tan x csc x 3 sec x 32 4 cot x 22 3. sin 2 2 2 2 tan 1 cos sec 2 csc 2 cot 1 7. sin cos 2 3 tan 5 3 5 2 sec 3 2 csc 35 5 cot 25 5 25 5 5 5 cos 11. sin tan 2 csc 5 2 sec 5 cot 1 2 13. sin 1 cos 0 tan cot 0 csc 1 sec is undefined. csc is undefined. 16. a 22. c 17. b 23. f cos2 29. tan x 37. cos u sin u sec x 1 51. cot2 xcsc x 1 4 cot2 x 61. 1 cos y 39. 45. sec4 x 57. 2 csc2 x 15. d 21. b 27. 35. 1 43. 49. 55. 59. 65. 18. f 24. a 31. cos x 1 sin y 19. e 25. e 33. 41. sin2 x sec sin2 x tan2 x 47. sin2 x cos2 x 53. sin2 x 1 2 sin x cos x 63. 2 sec x 67. 3sec x tan x C H A P T E R 5 20. c 26. d 333200_05_AN.qxd 12/9/05 1:50 PM Page A148 A148 69. 1 0 0 x y1 y2 x y1 y2 y1 71. x y1 y2 x y1 y2 12 Answers to Odd-Numbered Exercises and Tests 109. Not an identity because tan k sin k cos k 1 sin 115. x 25 119. 5x2 8x 28 x2 4x 4 123. 2 0.2 0.4 0.6 0.8 1.0 113. Answers will vary. x2 6
x 8 x 5x 8 117. 111. An identity because sin 1 0.1987 0.3894 0.5646 0.7174 0.8415 121. 0.1987 0.3894 0.5646 0.7174 0.8415 1.2 1.4 0.9320 0.9854 0.9320 0.9854 y2 y 2 1 −1 −2 −3 −4 x π2 π3 2 0.2 0.4 0.6 0.8 1.0 Section 5.2 (page 387) 1.2230 1.5085 1.8958 2.4650 3.4082 1.2230 1.5085 1.8958 2.4650 3.4082 1.2 1.4 5.3319 11.6814 5.3319 11.6814 Vocabulary Check (page 387) 1. identity 4. cot u 8. sec u 2. conditional equation cos2 u 6. sin u 7. tan u 3. csc u 5. 1–37. Answers will vary. 39. (a) (b) 0 1 y2 y1 csc x 5 sec 73. 81. 75. 2 tan x 83. 77. 3 sin 3 tan 3 cos 3; sin 0; cos 1 79. 85. 4 sin 22; sin 2 2 ; cos 2 2 87. 0 ≤ ≤ lncot x 91. 95. (a) 0 ≤ < 89. lncsc t sec t 93. 2 , 3 2 < < 2 (b) 97. (a) 1.6360 0.6360 1 csc2 132 cot 2 132 1.8107 0.8107 1 cot2 2 csc2 2 7 7 cos90 80 sin 80 0.9848 cos 0.8 sin 0.8 0.7174 2 tan 99. 101. True. For example, , 0 103. 1, 1 107. Not an identity because cos ± 1 sin2 sinx sin x. 105. (b) − 5 5 −5 5 (c) Answers will vary. 41. (a) −2 5 −1 y2 y1 2 Identity (b) (c) Answers will vary. 43. (a) (b) Not an identity 5 −1 −2 2 Identity (c) Answers will vary. 333200_05_AN.qxd 12/9/05 1:50 PM Page A149 45. (a) (b) y2 y1 −2 3 −3 2 (c) Answers will vary. Not an identity 47 and 49. Answers will vary. 55. Answers will vary. 57. False. An identity is an equation that is true for all real 51. 1 53. 2 values of . 59. The equation is not an identity because sin ±1 cos2 . 7 4 Possible answer: 2 3 26i 3 ± 21 61. 65. 8 4i 63. 1 ± 5 67. Section 5.3 (page 396) Vocabulary Check (page 396) 1. general 2. quadratic 3. extraneous 1–5. Answers will vary. 7. 2 n, 2n 2 3 6 4 3 5 6 11. n, n 9. 3 2n, 2 3 2n 13. 17. 21. 25. 31. 37. 23. n 2 8 0, n, 2n 12 2 6n, 2 6n 11 6 3 , , 6 , 39. 15. 3 n, n 2 3 19. 0 11 6 29. , , 3 27. No solution 5 3 5 6 2 4n, 4n 7 2 45. 2.678, 5.820 41. 1 4n 49. 0.860, 3.426 43. 47. 1.047, 5.236 51. 0, 2.678, 3.142, 5.820 53. 0.983, 1.768, 4.124, 4.910 55. 0.3398, 0.8481, 2.2935, 2.8018 57. 1.9357, 2.7767, 5.0773, 5.9183 59. , 4 5 4 , arctan 5, arctan 5 61. , 3 5 3 Answers to Odd-Numbered Exercises and Tests A149 4 5 4 0.7854 3.9270 63. (a) 3 (b) 2 0 −3 Maximum: Minimum: 0.7854, 1.4142 3.9270, 1.4142 65. 1 67. (a) All real numbers except x x 0 y -axis symmetry; Horizontal asymptote: y 1 (d) Infinitely many solutions (b) (c) Oscillates (e) Yes, 0.6366 69. 0.04 second, 0.43 second, 0.83 second 71. February, March, and April 75. (a) Between (b) 5 times: t 8 seconds and t 16, 48, 80, 112, 144 73. t 24 36.9, 53.1 seconds seconds 0.6 < x < 1.1 77. (a) 2 (b) 0 2 −2 A 1.12 79. True. The first equation has a smaller period than the second equation, so it will have more solutions in the interval 0, 2. 81. 1 83. C 24 a 54.8 b 50.1 87. 85. cos 390 sin 390 1 2 3 2 3 3 tan 390 sin1845 cos1845 2 2 2 2 tan1845 1 C H A P T E R 5 Vocabulary Check (page 404) 1. 2. 4. 5. sin u cos v cos u sin v cos u cos v sin u sin v sin u cos v cos u sin v cos u cos v sin u sin v 3. 6. tan u tan v 1 tan u tan v tan u tan v 1 tan u tan v 1. (a) 2 6 4 (b) 1 2 2 33. 2 35. 6 n, n 89. 1.36 91. Answers will vary. Section 5.4 (page 404) 333200_05_AN.qxd 12/9/05 1:50 PM Page A150 A150 Answers to Odd-Numbered Exercises and Tests 3. (a) 5. (a) 2 6 4 (b) 2 1 2 1 2 (b) 3 1 2 7. sin 105 cos 105 2 4 2 4 3 1 1 3 tan 105 2 3 9. sin 195 2 4 1 3 cos 195 2 4 3 1 tan 195 2 3 11. sin cos tan 13. sin cos 3 1 2 4 2 1 3 4 11 12 11 12 11 12 17 12 17 12 17 12 tan 2 3 31. 39. 49. 16 65 5 3 3 2 33. 3 2 35. 1 37. 63 65 63 16 41. 51. 1 43. 53. 0 65 56 45. 3 5 47. 44 117 55–63. Answers will vary. 65. sin x 67. cos 69. 2 71. 5 , 4 7 4 sin2t 0.6435 (c) 1 cycle per second feet sinu ± v sin u cos v ± cos u sin v cos x cos 2 sin x sin 2 sin x 2 81– 83. Answers will vary. 2 sin 13 sin 3 0.3948 85. (a) 4 (b) 2 cos (b) 4 13 cos3 1.1760 73. , 4 75. (a) 7 4 y 5 12 (b) 5 12 77. False. 79. False. cosx 87. (a) 89. 95. 2 cos −2 91. Proof 93. 15 3 −3 2 3 1 sin 285 2 4 2 3 1 cos 285 4 tan 285 2 3 sin165 3 1 cos165 1 3 2 4 2 4 tan165 2 3 2 4 1 3 sin cos 13 12 13 12 13 tan 12 sin13 12 cos13 12 tan13 12 cos 40 tan 239 25. 3 1 27. sin 1.8 29. tan 3x 15. 17. 19. 21. 23. sin2 sin2 4 f 1x x 15 5 97. 1 4 99. Because 4x 3 101. f is not one-to-one, 6x 3 103. Section 5.5 (page 415) f 1 does not exist. Vocabulary Check (page 415) 1. 3. 4. 6. 2. cos2 u 2 sin u cos u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u ±1 cos u tan2 u 5. 2 1 cos u sin u sin u 1 cos u 1 2 1 2 7. 8. cosu v cosu v sinu v sin u v cosu v 2 sinu v 9. 2 2 sinu v 10. 2 sinu v 2 2 333200_05_AN.qxd 12/9/05 1:50 PM Page A151 Answers to Odd-Numbered Exercises and Tests A151 3 4 cos 2x cos 4x 1. 11. 15. 17 17 , 5 , 12 , 12 , 5 , 6 6 2 3 sin 2x 3. 15 17 17 13 , 12 12 3 7 , , 2 6 21. 11 6 4 cos 2x 9. 0, 5. 8 15 13. 0, 17 8 4 3 7. 2 , 3 17. , , 3 2 0, 2 19. 23. sin cos tan 27. sin 2u 24 25 2u 7 25 2u 24 7 1 8 29. cos 25. sin cos tan 2u 24 25 2u 7 25 2u 24 7 2u 421 25 2u 17 25 2u 421 17 1 cos 4x 1 8 1 cos 2x cos 4x cos 2x cos 4x 1 16 417 17 17 tan 37. 39. 31. 33. 35. 41. 1 4 2 3 2 3 sin 75 1 2 cos 75 1 2 tan 75 2 3 112 30 1 2 112 30 1 2 112 30 1 2 2 2 2 2 43. sin cos tan 45. sin 2 2 cos 2 2 2 2 47. sin cos tan cos tan 178 2 1 89 889 89 889 8 89 5 178 49. 51. sin 1 2 1 2 8 8 2 1 tan sin tan cos tan 2 sin 3x cos 526 26 26 26 5 310 10 10 10 3 53. sin 55. 57. tan 4x 3 , , 5 3 59. 61. 3 , , 5 3 2 0 −2 sin 0 3sin sin 10 sin 2 2 1 2 2 2 0 2 −2 65. 5cos 60 cos 90 69. 5 2 cos 8 cos 2 cos 2y cos 2x 1 2 2 cos 4 sin 77. 79. 2 cos sin 81. sin 2 sin 2 1 2 73. 2 cos 4x cos 2x 2 sin sin 63. 67. 71. 75. 83. 87. 3 1 2 , , 0, 4 2 2 0 −2 89. , 6 5 6 2 0 −2 25 169 −2 91. 111. 115 85 93. 4 13 3 −3 95–109. Answers will vary. 113. 2 −2 3 −3 2 π π 2 x 333200_05_AN.qxd 12/9/05 1:50 PM Page A152 A152 Answers to Odd-Numbered Exercises and Tests 23.85 Review Exercises (page 420) 117. 121. (a) 2x1 x2 119. (b) 0.4482 (c) 760 miles per hour; 3420 miles per hour (d) 2 sin1 1 M u < 0, 123. False. For sin 2u sin2u 2 sinu cosu 2sin u cos u 2 sin u cos u. 125. (a) 4 (b) 0 0 2 , 3 Maximum: 3 cos 4x 1 (b) 4 1 2 sin2 x cos2 x 2 cos4 x 2 cos2 x 1 127. (a) (d) (c) (e) No. There is often more than one way to rewrite a 2 sin2 2 x 1 1 trigonometric expression. 129. (a) (−1, 4) y 6 5 4 3 2 1 (5, 2b) 131. (a) Distance 210 y (c) Midpoint: 2, 1 2 − 1 1 2 x (b) Distance 2 3 133. (a) Complement: 13 35; 2 3, 3 (c) Midpoint: 125 supplement: 2 3. cos x 9. 1. 7. sec x tan x 3 4 csc x 5 3 sec x 5 4 cot x 4 3 5. cos x cot x 2 2 tan x 1 csc x 2 sec x 2 cot x 1 sin2 x 13. 1 sec x 2 sin x 15. 11. 19. 21. 23–31. Answers will vary. cot 2 tan2 17. cot2 x 2n, 2n 35. n n, n 39. 0, 4 3 41. 0 11 45. 0, 9 , 8 11 , 8 13 , 8 15 8 47. 0, arctan4 , arctan4 2, arctan 3, arctan 3 sin 285 cos 285 3 1 2 4 2 3 1 4 tan 285 2 3 33. 37. 43. 49. 51. 53. 55. 61. 3 1 3 1 sin cos 2 4 2 4 25 12 25 12 25 12 sin 15 57. 57 36 1 52 tan 2 3 tan 35 63. 1 52 65–69. Answers will vary. 75. sin 2u 24 25 cos 2u 7 25 tan 2u 24 7 5 47 59. 3 52 57 36 7 , 4 71. 4 77. , 6 11 6 73. 2 −2 2 (b) No complement; supplement: 18 135. (a) Complement: (b) Complement: 4 ; 9 ; 20 supplement: supplement: 17 18 11 20 137. September: $235,000; October: $272,600 139. 127 feet 79. 83. −2 3 4 cos 2x cos 4x 41 cos 2x 1 cos 4x 81. 1 cos 4x 2 3 sin75 1 2 2 3 cos75 1 2 tan75 2 3 333200_05_AN.qxd 12/9/05 1:50 PM Page A153 Answers to Odd-Numbered Exercises and Tests A153 < < 2 7–12. Answers will vary. 5. 0, 2 6. −2 3 2 2 < ≤ , 3 −3 y1 1 16 y2 10 15 cos 2x 6 cos 4x cos 6x 1 cos 2x 7 2 2 cos 2sin 6 sin 2 16. sin 2 14. tan 2 2 3 1 2 2 3 2 3 87. sin 10 u 10 2 310 u 10 2 u 2 1 3 cos tan 89. sin 91. cos 5x 85. sin 1 2 19 12 19 12 19 12 314 u 14 2 70 u 2 14 35 u 2 5 cos tan cos tan 93. 1 2 sin 3 95. 1 2 99. 2 sin x sin 6 101. 103. 2 2 0 −2 cos 2 cos 8 97. 2 cos 3 sin 13. 15. 17. 0, 15 or 12 10 105. 1 2 feet 19 11 , 6 6 , 6 18. , 6 20, cos 2u 3 22. 6 5 107. False. If cos2 109. True. then 2 < < , cos2 > 0. depends on the quadrant in which 4 sinx cosx 4sin x cos x 4 sin x cos x 22 sin x cos x 2 sin 2x The sign of lies. 2 111. Reciprocal identities: sin 1 cos 1 csc , sec , tan 1 cot , csc 1 sin , Quotient identities: sec 1 cos , tan sin cot 1 cos , cot cos tan sin sin2 cos2 1, Pythagorean identities: 1 tan2 sec2 , 1 ≤ sin x ≤ 1 for all 1.8431, 2.1758, 3.9903, 8.8935, 9.8820 1 cot2 csc2 y2 115. y1 x 1 (b) 113. 117. Chapter Test (page 423) 2. 1 3. 1 4. csc sec 1. sin 313 13 cos 213 13 13 3 13 2 csc sec cot 2 3 21. 2.938, 2.663, 1.170 5 , tan 2u 4 sin 2u 4 23. 24. Day 123 to day 223 t 0.26 minute 25. 0.58 minute 0.89 minute 1.20 minutes 1.52 minutes 1.83 minutes Problem Solving (page 427) 1. (a) cos ± 1 sin2 tan ± cot ± sec ± sin 1 sin2 1 sin2 sin 1 1 sin2 csc 1 sin sin ±1 cos2 1 cos2 cos 1 1 cos2 csc ± tan ± sec 1 cos cot ± cos 1 cos2 3. Answers will vary. 5 333200_06_AN.qxd 12/9/05 1:51 PM Page A154 A154 Answers to Odd-Numbered Exercises and Tests 7. sin 2 2 2 2 1 cos 1 cos sin 2 1 cos cos tan 9. (a) 20 0 0 365 t 91, t 274; (b) (c) Seward; The amplitudes: 6.4 and 1.9 (d) 365.2 days Spring Equinox and Fall Equinox ≤ x ≤ 5 6 (b) 2 3 ≤ x ≤ 4 3 6 11. (a) (c) (d) 13. (a sinu sin u cos v cos w cos u sin v cos w sin u sin v sin w cos u cos v sin w (b) tanu v w tan u tan v tan w tan u tan v tan w 1 tan u tan v tan u tan w tan v tan w 15. (a) 15 (b) 233.3 times per second 0 0 1 Chapter 6 Section 6.1 (page 436) Vocabulary Check (page 436) 1. oblique 2. b sin B 3. 1 2 ac sin B 1. 3. 5. 7. 9. 11. 13. 15. 17. C 105, b 28.28, c 38.64 C 120, b 4.75, c 7.17 B 21.55, C 122.45, c 11.49 B 60.9, b 19.32, c 6.36 B 42 4, a 22.05, b 14.88 A 10 11, C 154 19, c 11.03 A 25.57, B 9.43, a 10.53 B 18 13, C 51 32, c 40.06 C 83, a 0.62, b 0.51 B 48.74, C 21.26, c 48.23 19. 21. No solution 23. Two solutions: B 72.21, C 49.79, c 10.27 B 107.79, C 14.21, c 3.30 25. (a) (c) 27. (a) b ≤ 5, b 5 sin 36 b > 5 sin 36 b ≤ 10.8, b 10.8 sin 10 (b) 5 < b < 5 sin 36 (b) 10.8 < b < 10.8 sin 10 (c) b > 29. 10.4 16.1 37. 41. (a) 10.8 sin 10 31. 1675.2 39. 77 meters 17.5° 18.8° x y 9000 ft 33. 3204.5 35. 15.3 meters (b) 22.6 miles (c) 21.4 miles (d) 7.3 miles z 43. 3.2 miles 45. True. If an angle of a triangle is obtuse grea
ter than 90, then the other two angles must be acute and therefore less than The triangle is oblique. 90. arcsin0.5 sin 47. (a) (b) (c) (d) (e) 1 0 0 Domain: 0 < < Range: 0 < < 6 c 18 sin arcsin0.5 sin sin 27 0 0 c c Domain: Range: 0 < < 9 < c < 27 0.4 0.8 1.2 1.6 0.1960 0.3669 0.4848 0.5234 25.95 23.07 19.19 15.33 2.0 2.4 2.8 0.4720 0.3445 0.1683 12.29 10.31 9.27 increases from 0 to As decreases, and decreases from 27 to 9. c , increases and then 333200_06_AN.qxd 12/9/05 1:51 PM Page A155 49. cos x 51. sin2 x Section 6.2 (page 443) Vocabulary Check 1. Cosines 3. Heron’s Area Formula 2. (page 443) b2 a2 c2 2ac cos B 1. 3. 5. 7. 9. 11. 13. 15. A 23.07, B 34.05, C 122.88 B 23.79, C 126.21, a 18.59 A 31.99, B 42.39, C 105.63 A 92.94, B 43.53, C 43.53 B 13.45, C 31.55, a 12.16 A 14145, C 2740, b 11.87 A 27 10, C 27 10, b 56.94 A 33.80, B 103.20, c 0.54 a c b d 135.1 111.8 102.8 17. 5 19. 10 21. 15 23. 16.25 29. 8 14 16.96 12.07 20 25 25. 10.4 5.69 13.86 20 27. 52.11 45 68.2 77.2 N 37.1 E, S 63.1 E W N S E 3000 m C 1700 m B 3700 m A N 58.4 W 31. 373.3 meters 37. (a) 41. 24.2 miles 43. 45. d (inches) PQ 9.4, QS 5, RS 12.8 33. (b) 72.3 S 81.5 W 35. 43.3 miles 39. 63.7 feet 9 10 12 13 14 Answers to Odd-Numbered Exercises and Tests A155 59. 2 61. 3 63. 3 65. 1 1 4x2 69. cos 1 67. 1 x 2 71. sec 1 csc is undefined. 3 tan 3 sec 23 3 csc 2 73. 2 sin 7 12 sin 4 Section 6.3 (page 456) Vocabulary Check (page 456) 2. initial; terminal 4. vector 1. directed line segment 3. magnitude 5. standard position 7. multiplication; addition 8. resultant 9. linear combination; horizontal; vertical 6. unit vector slopev and have the same magnitude and direction, so they are slopeu 1 4 v 1. 3. 7. 11. 15. u v 17, u equal. v 3, 2; v 13 v 0, 5; v 5 v 8, 6; v 10 y 9. x 5. v 3, 2; v 13 v 16, 7; v 305 13. v 9, 12; v 15 y 17degrees) 60.9 69.5 88.0 98.2 109.6 s (inches) 20.88 20.28 18.99 18.28 17.48 d (inches) 15 16 (degrees) 122.9 139.8 s (inches) 16.55 15.37 47. 46,837.5 square feet 51. False. For s 49. $83,336.37 to be the average of the lengths of the three s sides of the triangle, would be equal to a b c3. v v− 19. y u + 2v 2v 53. False. The three side lengths do not form a triangle. 55. (a) 570.60 57. Answers will vary. (b) 5910 (c) 177 u x 333200_06_AN.qxd 12/9/05 1:51 PM Page A156 A156 Answers to Odd-Numbered Exercises and Tests 21. (a) 3, 4 (b) 1− −+ u 2 3 4 5 (c) 1, 7 y 2 2u − − 10 −3v v− 3u 2 23. (a) 5, 3 (b) 5c) 4i 11j y 12 10 8 v− 3u 2 −3v − 8 − 6 − 4 − 2 − 2 2u 2 4 6 x 27. (a) 2i j (b) 2i j u x 3 − v u v− − 1 y 1 −1 −2 −c) 4i 3j y 1 u v 2u −1 −2 −3 −4 −3v 2u − 2v 2 2 2 , 2 i 25 5 j 39. 10 10 j 310 33. 10 52 2 , i 52 2 43. 7i 4j 45. 3i 8j 29. 1, 0 31. 5 5 37. j 1829 , 29 v 3, 3 2 4529 29 (c) 10 12 10 8 6 4 2 − 12 − 10 − 3v 2 x 25. (a) 3i 2j (b) i 4j 3 −2 −1 x 3 u v+ y 5 4 u v− − 35. 41. 47. y 1 −1 −2 1 2 3 x u 3 2 u 333200_06_AN.qxd 12/9/05 1:51 PM Page A157 v 4, 3 49. y 4 3 2 1 − 1 2w u + 2w 3 4 5 x u 53. v 3; 60 55. 57. v 3 61. v 36 2 , 32 2 y 5 4 3 2 1 150 51. y 2 1 −1 −2 v 7 2, 1 2 w1 2 u 3 2 v 62; 315 v 73 7 4 4 59 63. v 10 5 , 310 1 67. 73. 5, 5 62.7 71.3; 102 50, 102 65. 12.8; 71. 398.32 newtons 75. 228.5 pounds 77. Vertical component: 70 sin 35 40.15 1 2 69. 90 feet per second 70 cos 35 57.34 feet per second 81. 3154.4 pounds 79. Horizontal component: 1758.8 pounds TAC 1305.4 pounds TBC N 21.4 E; 83. 85. 1928.4 foot-pounds 180 89. (a) (b) 0 138.7 kilometers per hour 87. True. See Example 1. (c) No. The magnitude is at most equal to the sum when the angle between the vectors is 0. 1, 3 or 1, 3 91. Answers will vary. 95. 97. 6 sec 93. 8 tan 2 99. n, 2n 101. n, 6 2n, 11 6 2n Answers to Odd-Numbered Exercises and Tests A157 Section 6.4 (page 467) Vocabulary Check 1. dot product 2. (page 467) u v u v 3. orthogonal x 4 1 2 (3u + w) 4. u v v2 v 5. 1. 11. 15. 21. 9 6, 8; 5 1; 541 11 3. vector scalar 23. 6 proj PQ \F PQ \; F PQ \ 9. 8; scalar 12 7. 66, 66; 5. 6 13. 17. 4; scalar 90 25. 27. vector 19. 13 143.13 29. 60.26 31. 90 33. 35. 150 12 37. v y 10 8 6 4 2 −2 −2 −4 u x 2 4 6 −8 −6 −4 −2 2 4 −2 −4 x −8 −6 −4 41. 45. 90 91.33 26.57, 63.43, 90 39. 20 43. 51. Orthogonal 2, 15, 6 45 55. 229 229 5, 3, 5, 3 59. 65. (a) $58,762.50; This value gives the total revenue that can 41.63, 53.13, 85.24 47. Parallel 84, 14, 1 37 57. 0 3 i 1 229.1 1 53. 37 15, 2 10, 60 49. Neither 3 i 1 2 j 2 j, 2 63. 32 61. 2 be earned by selling all of the units. 1.05v Force 30,000 sin d 0 0 6 1 2 3 4 5 523.6 1047.0 1570.1 2092.7 2614.7 7 8 9 10 (b) 67. (a) (b) d Force d Force 3135.9 3656.1 4175.2 4693.0 5209.4 (c) 29,885.8 pounds 69. 735 newton-meters 73. 21,650.64 foot-pounds 75. False. Work is represented by a scalar. 71. 779.4 foot-pounds 77. (a) 2 (b) 0 ≤ < 2 (c) 2 < ≤ 79. Answers will vary. 81. 26 127 85. 0, , , 6 83. 11 6 87. 0, 89. 253 325 91. 204 325 333200_06_AN.qxd 12/9/05 1:51 PM Page A158 A158 Answers to Odd-Numbered Exercises and Tests Section 6.5 (page 478) Vocabulary Check (page 478) 1. absolute value 2. trigonometric form; modulus; argument 4. 3. DeMoivre’s th root n 1. Imaginary axis 3. i Imag nary axis − 4 − 2 2 4 Real axis − Real axis 42 − 2 −4 −6 −8 −7i 2 4 6 8 Real axis 7 5. Imaginary axis − 2 − 4 − 6 − 8 6 − 7i i sin 2 85 3cos 7. 2 11. Imaginary axis 1 2 3 Real axis 13. Imaginary axis 2 1 3 + i − 1 1 2 Real axis 3 − 3i − 1 − 1 − 2 −3 19. − Imaginary axis 21. Imaginary axis 4 2 − 2 − 4 Real axis Real axis 65 cos 2.62 i sin 2.62 7cos 0 i sin 0 23. Imaginary axis 253 −2 3 3+ i −3 − i 1 2 3 4 Real axis Imaginary axis Real axis −1 −2 −3 − 4 23cos 6 i sin 6 10 cos 3.46 i sin 3.46 27. y Imaginar axis 29. 5 4 3 2 1 5 + 2i Imaginary axis Real axis −10 −8 −6 −4 −2 −2 −4 −6 −8 −10 29cos 0.38 i sin 0.38 139cos 3.97 i sin 3.97 31. Imaginar y axis 33. Imaginary axis − 3 2 3 Real axis Real axis 1 1 −1 −2 3 4 33 4 i 32cos 7 4 i sin 7 4 2cos 6 i sin 6 15. Imaginary axis 17. Imaginary axis − 4 −3 −2 −1 Real axis − 4 − 2 2 4 Real axis 35. −2 −3 − 4 −2( 1 + 3i) − 2 −4 −6 −8 −5i 4cos 4 3 i sin 4 3 5cos 3 2 i sin 3 2 3 2 33 2 i − 15 2 8 + 15 2 8 i − 4 −3 −2 −1 152 8 152 8 i Imaginary axis 37. Imaginar y axis 3 2 1 −1 Real axis 8i 2 4 6 8 10 Real axis 10 8 6 4 2 − 2 − 2 8i 9. 10 cos 5.96 i sin 5.96 − 1 − 1 1 2 3 4 5 Real axis − 8 − 5 3i 333200_06_AN.qxd 12/9/05 1:51 PM Page A159 39. Imaginary axis 2 1 − 1 − 2 2.8408 + 0.9643i 1 2 3 4 Real axis 2.8408 0.9643i 4.6985 1.7101i Imaginary axis 41. 45. 43. 2.9044 0.7511i 2 z 2 = i 2 z3 = (−1 + i) 2 2 z = (1 + i) 2 − 2 z 4 = −1 1 Real axis − 1 Answers to Odd-Numbered Exercises and Tests A159 67. Imaginary axis 69. Imaginary axi s 3 1 −1 − 1 − 3 4 2 1 3 Real axis −4 −2 2 4 Real axis − 2 − 4 73. i 71. 77. 1253 2 4 4i 125 2 608.0 144.7i 813 81 2 2 89. (a) 85. 81. i 32i 75. 1283 128i 79. 1 83. 597 122i 87. 32i 5 cos 60 i sin 60 5 cos 240 i sin 240 The absolute value of each is 1. 12cos 10 9 0.27cos 150 i sin 150 i sin 49. 3 3 47. 51. cos 200 i sin 200 (b) Imaginary axis 3 1 cos 30 i sin 30 −3 −1 1 3 Real axis 53. 55. cos 59. (a) (b) 61. (a) (b) (c) 57. 7 4 i sin i sin 2 3 2 4cos 302 i sin 302 i sin 3 2cos 22cos 4 4 4 cos 0 i sin 0 4 (c) 4 3 3 2cos 2cos 4 2 2 7 2 2i 22cos 4 2i 2i 2 2i 2 2 2i i sin i sin i sin 7 4 4 7 4 63. (a) 5cos 0.93 i sin 0.93 2cos 5 3 i sin 5 3 cos 1.97 i sin 1.97 0.982 2.299i (c) 91. (a) 15 2 i − 3 i, i sin i sin 5 2 2cos 2cos 2cos 15 2 2 9 8 9 14 9 5 2 2 9 8 9 14 9 i sin (b) Imaginary axis 3 1 5 2 0.982 2.299i 5cos 0 i sin 0 (b) (c) 65. (a) (b) (c) 5 13 10 13 15 13 13 cos 0.98 i sin 0.98 − 3 − 1 − 1 1 3 Real axis cos 5.30 i sin 5.30 0.769 1.154i i 0.769 1.154i − 3 (c) 1.5321 1.2856i, 1.8794 0.6840i, 0.3473 1.9696i C H A P T E R 6 333200_06_AN.qxd 12/9/05 1:51 PM Page A160 Answers to Odd-Numbered Exercises and Tests (b) Imaginary axis 3 4 7 4 i sin i sin 3 5cos 4 7 5cos 4 52 52 2 2 52 52 2 2 i i −6 −2 6 4 2 −2 −4 −6 2 4 6 Real axis A160 93. (a) (c) 95. (a) 5cos 5cos 5cos 4 9 10 9 16 9 i sin i sin i sin 4 9 10 9 16 9 (c) 101. (a) 99. (a) cos 0 i sin 0 cos cos cos cos 2 5 4 5 6 5 8 5 i sin i sin i sin i sin 2 5 4 5 6 5 8 5 −2 1, 0.3090 0.9511i, 0.8090 0.5878i, 0.8090 0.5878i, 0.3090 0.9511i 5cos 3 5cos i sin 5 5cos 3 5 3 i sin i sin 3 (b) Imaginary axis 2 −2 Real axis 2 (b) Imaginary axis Real axis 4 6 (c) 97. (a) 0.8682 4.9240i, 4.6985 1.7101i, 3.8302 3.2140i 2cos 0 i sin 0 2cos 2 2cos i sin 2cos 3 2 i sin i sin 2 3 2 (b) Imaginary axis 3 1 (b) Imaginary axis 6 4 2 −6 −2 2 4 6 Real axis − 4 − 6 (c) 103. (a) i, 5, 5 2 53 5 2 2 22cos 22cos 22cos 22cos 22cos 3 20 11 20 19 20 27 20 7 4 i sin i sin i sin i sin i 53 2 3 20 11 20 19 20 27 20 7 4 i sin − 3 − 1 1 3 Real axis (b) Imaginary axis − 1 − 3 (c) 2, 2i, 2, 2i 1 − 2 − 1 1 2 Real axis − 2 (c) 2.5201 1.2841i, 0.4425 2.7936i, 2.7936 0.4425i, 1.2841 2.5201i, 2 2i 333200_06_AN.qxd 12/9/05 1:51 PM Page A161 105. 107. 109. 111. cos cos cos cos i sin i sin i sin i sin i sin i sin 11 11 8 8 15 15 8 8 3cos 5 3 3cos 5 3cos i sin 7 3cos 5 9 3cos 5 2cos 3 8 2cos 7 8 2cos 11 8 2cos 15 8 7 5 9 5 3 8 7 8 11 8 15 8 i sin i sin i sin i sin i sin i sin 62cos 62cos 62cos 7 12 5 4 23 12 i sin i sin i sin 7 12 5 4 23 12 Imaginary axis 1 2 − 1 2 Imaginary axis 4 Real axis − 4 − 2 2 4 Real axis − 4 Imaginary axis − 3 − 1 3 1 − 3 Imaginary axis 2 Real axis 3 −2 Real axis 2 − 2 113. True, by the definition of the absolute value of a complex number. cos 1 and/or 2 0. r2 i sin 1 2 0 if cos 2 i sin 2 115. True. r1r2 z1z2 0 r1 and only if 117. Answers will vary. 119. (a) 121. Answers will vary. 123. (a) (b) r 2 2cos 30 i sin 30 2cos 150 i sin 150 2cos 270 i sin 270 B 68, b 19.80, c 21.36 B 60, a 65.01, c 130.02 B 47 45, a 7.53, b 8.29 16; 4 1 135. 133. 5 125. 127. 129. 131. 16; 2 Answers to Odd-Numbered Exercises and Tests A161 Review Exercises (page 482) 15. 33.5 19. 31.01 feet C 74, b 13.19, c 13.41 A 26, a 24.89, c 56.23 C 66, a 2.53, b 9.11 B 108, a 11.76, c 21.49 A 20.41, C 9.59, a 20.92 B 39.48, C 65.52, c 48.24 1. 3. 5. 7. 9. 11. 13. 7.9 17. 31.1 meters A 29.69, B 52.41, C 97.90 21. A 29.92, B 86.18, C 63.90 23. A 35, C 35, b 6.55 25. A 45.76, B 91.24, c 21.42 27. 4.3 29. 31. 61
5.1 meters 37. 39. 45. (a) (d) 47. (a) (d) 49. (a) (d) 51. (a) 53. feet 33. 9.80 u v 61, slopeu 7, 5 7, 7 41. 4, 3 11, 3 1, 6 17, 18 7i 2j 20i j 3i 6j slopev 43. 9, 2 3i 4j 22, 7 2, 9 5i 6j 12.6 feet, (b) (b) (b) (b) (c) (c) 35. 8.36 5 6 4, 43 3, 9 (c) 15, 6 6i 3j (d) 18i 12j 12i 30, 9 (c) 55. y y v − 5 2 −2 −4 −6 −8 −10 −12 10 20 25 30 2u 2 +u v x 20 10 v − 10 3v x 10 20 30 C H A P T E R 6 6i 4j 3i 4j 59. 102cos 135 i sin 135j v 7; 60 v 32; 225 57. 61. 63. 67. 69. The resultant force is 133.92 pounds and 65. v 41; 38.7 5.6 from the 85-pound force. 71. 422.30 miles per hour; 77. 50; scalar 75. 130.4 79. 73. 45 6, 8; vector (b) 8i 81. 83. 160.5 2 11 12 85. Orthogonal 4, 1, 16 89. 17 93. 48 13 17 87. Neither 91. 1, 4 5 2 1, 1, 9 2 1, 1 95. 72,000 foot-pounds 3sin 11 sin 5 333200_06_AN.qxd 12/9/05 1:51 PM Page A162 A162 97. Answers to Odd-Numbered Exercises and Tests Imaginary axis 99. Imaginar y axis 5 3+ i 1 2 3 4 5 Real axis 5 4 3 2 1 −1 −1 34 11 6 3 2 10 3 i sin i sin 10 3 i sin 3 i sin 2035 828i 7i 10 Real axis 7 101. 103. 105. (a) 107. 111. (a) 7 4 z2 z1 7 4 52cos i sin 6cos 5 5 i sin 6 6 11 4cos 6 3 10cos 2 40cos cos z1 2 z2 5 6253 2 625 2 z1z2 (b) 3 i i sin 3cos 3cos 3cos 3cos 3cos 3cos 109. 4 7 12 11 12 5 4 19 12 23 12 i sin i sin i sin 4 7 12 11 12 5 4 19 12 23 12 Imaginary axis i sin i sin (b Real axis 4 (c) 113. (a) (b) i, 32 2 i, 0.7765 2.898i, 32 2 32 2.898 0.7765i, 32 2 2 0.7765 2.898i, 2.898 0.7765i 2cos 0 i sin 0 2cos 2 3 2cos 4 3 Imaginary axis 2 3 4 3 i sin i sin 3 −3 −1 1 3 Real axis −3 115. 117. i i sin i sin (c) 3cos 3cos 3cos 3cos 2, 1 3 i, 1 3 i 32 2 4 3 32 4 2 5 32 4 2 7 32 4 2 4 3 4 5 4 7 4 32 2 32 2 32 2 32 2 i sin i sin i i i Imaginary axis 4 2 −4 −2 2 4 Real axis −2 −4 i sin 2cos 2cos 2cos 2 7 6 11 6 2i 2 7 3 i 6 11 3 i 6 i sin i sin Imaginary axis 3 1 −1 −3 −3 Real axis 3 333200_06_AN.qxd 12/12/05 11:21 AM Page A163 119. True. sin 90 is defined in the Law of Sines. so u v v, x2 8i 0 v vu . are x 2 2i and b2 a2 c2 2ac cos B, the direction is the same and the magnitude is k the result is a vector in the opposite direction times as great. (b) 64 121. True. By definition, 123. False. The solutions to 125. x 2 2i. a2 b2 c2 2bc cos A, c2 a2 b2 2ab cos C 127. A 129. If C and k > 0, times as great. k < 0, If and the magnitude is k 4cos 60 i sin 60 4cos 180 i sin 180 4cos 300 i sin 300 131. (a) 133. z1z2 4; z1 z2 cos2 i sin2 cos 2 i sin 2 Chapter Test (page 486) C 88, b 27.81, c 29.98 A 43, b 25.75, c 14.45 1. 2. 3. Two solutions: B 29.12, C 126.88, c 22.03 B 150.88, C 5.12, c 2.46 5. A 23.43, B 33.57, c 86.46 4. No solution 6. 7. 2052.5 square meters A 39.96, C 40.04, c 15.02 9. 14, 23 10. 11. 4, 6 y 8 4 2 u u v 36, 22 13. y 42 36 30 24 18 12 6 v− 3u 5 5u − 6 6 −3v 24 30 36 42 29.1 8. 606.3 miles; , 3034 17 10, 4 1834 17 12. y 12 10 8 6 4 2 u u v− −2 −2 2 − v 8 10 12 x 14. 4 5, 3 5 x x Answers to Odd-Numbered Exercises and Tests A163 15. 18. 20. 22. 24. 25. 16. 135 104 pounds 17. No 21. 3 33 i 5832i 19. 7 4 23. 12 7 12 13 12 19 12 i i sin i sin i sin 14.9; 37 26 65613 2 250.15 pounds 5, 1; 29 1, 5 26 52cos 7 4 6561 2 4 42cos 4 42cos 4 42cos 4 42cos 3cos 3cos 3cos 12 7 12 13 12 19 12 i sin sin i sin i sin i sin Imaginar y axis 4 2 1 −4 −2 −1 1 2 4 Real axis −2 −4 C H A P T E R 6 Cumulative Test for Chapters 4–6 (page 487) 1. (a) y (b) (c) (d) 240 2 3 60 x −120° (e) sin120 3 2 cos120 1 2 csc120 23 3 sec120 2 tan120 3 cot120 3 3 2. 134.6 3. 3 5 333200_06_AN.qxd 12/9/05 1:51 PM Page A164 Answers to Odd-Numbered Exercises and Tests 5. y 3 −1 −2 −3 π 2 π3 2 x 7. a 3, b , c 0 38. 39. 40. 43. 5 3 5 i sin i sin 3cos 5 3cos 3 5 3cos i sin 3cos 7 5 3cos 9 5 395.8 Area 63.67 t d 4 cos 4 i sin 7 5 9 5 radians per minute; i sin square yards 41. 5 feet 8312.6 inches per minute 22.6 42. 44. 32.6; 543.9 kilometers per hour 45. 425 foot-pounds Problem Solving (page 493) 1. 2.01 feet 3. (a) A 75 mi 135° 15° y B 75° 30° x 60° Lost party 9. 6.7 10. 3 4 A164 4. 8 π2 −b) Station A: 27.45 miles; Station B: 53.03 miles (c) 11.03 miles; 5. (a) (i) 2 (iv) 1 (b) (i) 1 (iv) 1 (c) (i) 5 2 (iv) 1 (d) (i) 25 (iv) 1 5 S 21.7 E (ii) (v) 1 (ii) (v) 1 32 (ii) 13 (v) 1 (ii) (v) 1 52 w 1 2 7. 9. (a) u v F1 F2 P Q The amount of work done by of work done by F2. F1 (iii) 1 (vi) 1 (iii) (vi) 1 13 (iii) 85 2 (vi) 1 (iii) (vi) 1r 52 is equal to the amount (b) F1 F2 60° 30° P Q is The amount of work done by as the amount of work done by F1. F2 3 times as great 2 3 2 tan , , 3 , 2 16 20. 63 sin 5 3 21. 4 3 11. 1 4x2 12. 1 13. 14–16. Answers will vary. 17. , 5 2 23. , 18. 22. 11 6 6 5 5 19. sin 25 5 2 cos 6x cos 2x B 26.39, C 123.61, c 15.0 B 52.48, C 97.52, a 5.04 B 60, a 5.77, c 11.55 A 26.38, B 62.72, C 90.90 24. 25. 26. 27. 28. 29. 36.4 square inches 31. 3i 5j 32. 2 2 , 30. 85.2 square inches 2 2 5 33. i sin 3 4 36. 123 12i 34. 35. 37. 5, 1 21 13 3 4 1 1, 5; 13 22cos cos 0 i sin sin i sin cos cos 333200_07_AN.qxd 12/12/05 11:23 AM Page A165 Chapter 7 Section 7.1 (page 503) Vocabulary Check (page 503) 1. system of equations 3. solving 5. point of intersection 4. substitution 2. solution 6. break-even (d) Yes (d) No 0, 0, 2, 4 (c) No (c) No (b) No (b) Yes 7. 1. (a) No 3. (a) No 2, 2 2, 6, 1, 3 5. 0, 5, 4, 3 11. 9. 0, 1, 1, 1, 3, 1 13. 20 1, 1 3 , 40 21. 19. 3 2, 4, 0, 0 27. No solution 25. 5 2, 3 33. 35. 31. 4, 1 37. 43. 2, 2, 4, 0 39. No solution 5, 5 45. 2 2 6 2, 3 1 15. 17. 23. No solution 29. 1, 4, 4, 7 4, 3 41. 4, 3, 4, 3 5 −6 6 − 2 10 0, 1 47. −2 16 −3 4, 2 −24 24 −16 0, 13, ±12, 5 1, 2 51. 0.287, 1.751 1 2, 2, 4, 1 49. 55. 59. 63. (a) 781 units 65. (a) 8 weeks 4 2, 0, 29 57. 10, 21 1, 0, 0, 1, 1, 0 10 53. No solution 61. 192 units (b) 3708 units (b) 1 2 3 4 360 24x 336 312 228 264 24 18x 42 60 78 96 5 6 7 8 360 24x 240 216 192 168 24 18x 114 132 150 168 67. More than $11,666.67 Answers to Odd-Numbered Exercises and Tests A165 69. (a) (b) x 0.06x 27,000 y 0.085y 25,000 2,000 0 12,000 10,000 Decreases; Interest is fixed. (c) $5000 71. (a) Solar: Wind: 150 (b) 0.1429t2 4.46t 96.8 16.371t 102.7 8 0 13 10.3, 66.01. (c) Point of intersection: Consumption of solar and wind energy are equal at this point in time in the year 2000. t 10.3, 135.47 (d) (e) The results are the same, but due to the given parameters, t 135.47 is not of significance. (f) Answers will vary. 6 meters 9 meters 75. 8 kilometers 12 kilometers 73. 77. 79. False. To solve a system of equations by substitution, you can solve for either variable in one of the two equations and then back-substitute. 9 inches 12 inches 81. 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression found in Step 1 into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step 2. 4. Back-substitute the value obtained in Step 3 into the expression obtained in Step 1 to find the value of the other variable. 5. Check that the solution satisfies each of the original C H A P T E R 7 equations. y 2x y 0 (b) 2x 7y 45 0 30x 17y 18 0 83. (a) 85. 89. 91. Domain: All real numbers except (c) y 3 0 87. y x 2 Horizontal asymptote: Vertical asymptote4 93. Domain: All real numbers except x y 1 Horizontal asymptote: Vertical asymptotes: x ±4 333200_07_AN.qxd 12/12/05 11:24 AM Page A166 A166 Answers to Odd-Numbered Exercises and Tests Section 7.2 (page 515) Vocabulary Check (page 515) 51. (a) x 0.2x y 10 0.5y 3 (b) 12 1. elimination 3. consistent; inconsistent 2. equivalent 4. equilibrium point 1. 2, 1 3. 1, 1 (c) 20% solution: 50% solution: 62 3 liters 31 3 liters y 4 3 2 3x + 2y = 1 x + y = 0 −4 −3 −2 −1 2 3 4 −2 −3 −4 7. a, 3 2a 5 2 3x − 2y = 5 y 4 3 2 1 −3 −2 −1 2 3 4 5 −2 −6x + 4y = −10 −6 18 −4 Decreases 55. 400 adult, 1035 student y 0.97x 2.1 y 2x 4 53. $6000 57. 61. 63. (a) 65. False. Two lines that coincide have infinitely many points (b) 41.4 bushels per acre y 0.32x 4.1 y 14x 19 59. of intersection. 69. 67. No. Two lines will intersect only once or will coincide, and if they coincide the system will have infinitely many solutions. 39,600, 398. It is necessary to change the scale on the axes to see the point of intersection. k 4 x ≤ 22 3 x ≤ 19 16 71. 73. 75. − 22 3 x 19 16 − 9 − 8 −7 −6 − 5 −1 0 1 2 3 77. 2 < x < 18 −2 − 3 0 3 6 9 12 15 18 x 795 − 4 −3 − 2 − 81. ln 6x 83. log9 87. Answers will vary. 12 x 85. No solution Section 7.3 (page 527) Vocabulary Check (page 527) 1. row-echelon 3. Gaussian 5. nonsquare 2. ordered triple 4. row operation 6. position 2 −1 1 2 4 5 6 2x + y = 5 −3 −4 5. No solution −2x + 2y = 5 y 4 1 −4 −2 − 1 2 3 4 x x 9. 1 3, 2 3 −2 −4 y 4 3 −4 −3 −2 −1 −2 −3 −4 x − y = 2 3x − 6y = 5 x 2 3 4 9x + 3y = 1 12 7 , 18 7 31 17. 13. 27. 5 6 2 29. 4, 1 3, 4 21. 90 31, 67 a, 1 6a 5, 2 5 2, 3 15. 11. 4 18 5 , 3 19. No solution 5 23. Infinitely many solutions: 35, 43 25. 35 31. b; one solution; consistent 32. a; infinitely many solutions; consistent 33. c; one solution; consistent 34. d; no solutions; inconsistent 6, 3 35. 37. 43. 550 miles per hour, 50 miles per hour 45. 49. Cheeseburger: 310 calories; fries: 230 calories 2,000,000, 100 80, 10 2, 1 4, 1 41. 39. 47. (c) No (c) Yes 3, 10, 2 1. (a) No 3. (a) No 5. 11. 1, 2, 4 2y y x 2x (b) No (b) No 7. 3z 5 2z 9 3z 0 (d) Yes (d) No 2, 2, 2 1 9. 43 6 , 25 6 15. First step in putting the system in row-echelon form 1, 2, 3 13. 19. No solution 23. 4, 8, 5 1 21. 3a 10, 5a 7, a 2a, 21a 2, 8a 2 25. 29. 3 2a 1 a 3, a 1, a 3a 1, a 5, 2, 0 2, 1, 3 2, 2 17. 27. 333200_07_AN.qxd 12/9/05 1:53 PM Page A167 Answers to Odd-Numbered Exercises and Tests A167 75. x 5 y 5 5 77. x ± 2 2 y 1 2 1 or x 0 y 0 0 79. False. Equation 2 does not have a leading coefficient of 1. 81. No. Answers will vary. 83. x 2y z 0 x y 3z 1 3x y z 9 x 2y 4z x 4y 8z x 6y 4z 5 13 7 x 2z 0 2y z 0 x y z 5 x 2y 4z 9 y 2z 3 x 4z 4 12 85. 91. 11 i 89. 80,000 7 2i 95. 7 2 22 3i 87. 6.375 93. 97. (a) (b) 4, 0, 3 y 25 20 15 C H A P T E R 7 − − 10 − 15 − 20 4, 3 2, 3 99. (a) (b) y 30 20 10 − 5 − 3 −2 1 2 4 x x −30 −40 −50 −60 0 2 4 5 4.996 4.938 4 1 101. x y 2 5 y 12 10 31. 37. 41. 43. 1, 1, 1, 1 33. No solution 9
a, 35a, 67a 39. s 16t 2 32t 500 y 1 2x 2 2x 45. 35. 0, 0, 0 s 16t 2 144 5 −4 −3 x2 y 2 4x 0 47. −3 3 −3 8 6 y x2 6x 8 10 −2 −6 49. x2 y 2 6x 8y 0 10 −2 6 −12 51. 6 touchdowns, 6 extra-point kicks, and 1 field goal 53. $300,000 at 8% $400,000 at 9% $75,000 at 10% 250,000 1 2s 125,000 1 2s 125,000 s s in growth stocks in certificates of deposit in municipal bonds in blue-chip stocks 55. 59. Vanilla 2 lb Hazelnut 4 lb French Roast 4 lb X 4 lb 57. Brand Y 9 lb Brand Z 9 lb Brand Television 30 ads Radio 10 ads Newspaper 20 ads 61. 63. (a) Not possible (b) No gallons of 10%, 6 gallons of 15%, 6 gallons of 25% (c) 4 gallons of 10%, No gallons of 15%, 8 gallons of 25% 1, I1 y x2 x I3 69. y 0.0075x2 1.3x 20 24 x2 3 10x 41 y 5 2, 1 I2 6 65. 67. 71. (a) (b) 100 (c) 75 0 x y 175 100 120 140 75 68 55 The values are the same. (d) 24.25% (e) 156 females Touchdowns 8; Two-point conversions 1; Field goals 2; Extra-point kicks 5 73. 333200_07_AN.qxd 12/9/05 1:53 PM Page A168 A168 103. Answers to Odd-Numbered Exercises and Tests x y 2 1 5.793 4.671 0 4 1 2 3.598 3.358 53. (a) (b) 2 3 x 4 x y x 12 xx 6 −4 2 8 10 x − 6 2 8 10 x −c) The vertical asymptotes are the same. 55. (a) 5 3 x 3 x 3 y 24x 3 x2 b4 4 6 8 x −4 2 4 6 8 x −4 −6 −8 y = 5 x + 3 −4 −6 −8 y = 3 x − 3 (c) The vertical asymptotes are the same. 2000 7 4x (b) Ymax Ymin (c) 1000 2000 11 7x 7 4x 2000 2000 11 7x , 0 < x ≤ 1 (d) Maximum: Minimum: 400F 266.7F Ymax 0 −100 Ymin 1 59. False. The partial fraction decomposition is 40, 40 105. 107. Answers will vary. Section 7.4 (page 539) Vocabulary Check (page 539) 1. partial fraction decomposition 3. linear; quadratic; irreducible 2. improper 4. basic equation 1. b A x 5. 2. c B x 14 3. d 4. a B x2 A x 7. B x 52 C x 53 A x 5 A x 1 x Bx C x2 1 1 x 1 Dx E x2 12 1 x 19. 2 C x 10 A 11. x 1 x 1 1 2 Bx C x2 10 1 x 1 15. 2x x2 2 1 2x 1 x2 x 1 39. 31. 1 1 x 1 x 2 4x 4 x2 1 23. x 1 x2 x2 x 2x 1 x 2 x2 2 1 2x 1 1 x 1 2x 7 17 2x 1 2 x2 2x x2 22 3 2x 1 1 x2 2 x 1 4 x 12 2 x 47. 51. 9. 13. 17. 21. 25. 29. 33. 35. 37. 41. 43. 45. 49. 1 x 13 2 1 x2 2x 1 2 x 1 3 x 4 A x 10 1 1 a x 2a B x 10 1 a x C x 102. 1 1 a y 63. 1 a y 61. 1 x 2 27. 3 x 3 9 x 32 57. (a) 333200_07_AN.qxd 12/9/05 1:53 PM Page A169 65. y 67. 8 6 4 2 −2 − 2 − 4 2 4 8 10 x −3 −2 y 5 4 3 −1 −1 −2 −3 1 2 4 5 x 69. y 5 −20 − 15 − 10 5 10 15 20 x Section 7.5 (page 548) Vocabulary Check (page 548) 1. solution 4. solution 2. graph 5. consumer surplus 3. linear 33 1 3 4 5 7. 5. 92 −1 1 2 3 4 − 2 −2 y 115 −4 −2 6 4 3 2 1 −2 No solution x 2 3 Answers to Odd-Numbered Exercises and Tests A169 15. 2 0 − 2 −6 −9 19. 23. 6 6 9 4 −4 3 −9 27. y ≤ 1 2 x 2 13. 17. 21. 25. y 3 2 −3 −2 −1 1 2 3 x −2 −3 2 −2 6 0 4 −2 1 3 4 −8 −3 −5 (b) No (b) No (c) Yes (c) Yes (d) Yes (d) Yes (0, 1) (1, 0) 1 2 x 37. (−1, 4) (−1, 0) − 4 − 3 41. (− 2, 0 29. 31. (a) No 33. (a) Yes 35. y 3 2 −1 y (− 1, 0) −2 39. 4 1 −1 −2 −2 −1 2 3 4 x −3 −1 1 3 4 −2 −3 ( 5, 0 ( x 1 2 3 4 ( 10 333200_07_AN.qxd 12/9/05 1:53 PM Page A170 A170 43. y 3 2 1 Answers to Odd-Numbered Exercises and Tests 45. y 4 2 (4, 2) −1 1 2 3 4 5 x −4 −2 x 2 4 (1, −1) − 2 − 3 47. y 4 3 2 1 (4, 4) (− 1, −1) 1 2 3 4 5 51. 55. 59. − 3 − 4 −6 5 − x2 y2 ≤ 16 x ≥ 0 y ≥ 0 −2 −4 7 − 1 49. −5 53. 5 −2 −1 x 6 7 7 57. 61 63b) Consumer surplus: $1600 Producer surplus: $400 (b) Consumer surplus: $40,000,000 Producer surplus: $20,000,000 65. (a) p 50 40 30 20 10 67. (a) p 160 140 120 100 80 Consumer Surplus Producer Surplus p = 50 − 0.5x p = 0.125x (80, 10) x 10 20 30 40 50 60 70 80 Consumer Surplus Producer Surplus p = 140 − 0.00002x (2,000,000, 100) p = 80 + 0.00001x 1,000,000 2,000,000 x 69 ≤ 12 ≤ 15 ≥ 0 ≥ 0 71. x x y y y ≤ ≥ ≥ ≥ 20,000 2x 5,000 5,000 73. 55x x 70y ≤ ≥ y ≥ 7500 50 40 y 12 10 6 4 2 2 y 15,000 10,000 y 120 100 80 60 40 20 4 6 8 10 x 10,000 15,000 x 20 40 60 80 100 120 x 75. (a) 10y 10y 20y 20x 15x 10x x y ≥ 300 ≥ 150 ≥ 200 ≥ 0 ≥ 0 (b) y 30 x 30 (c) Answers will vary. y 19.17t 46.61 77. (a) (b) 225 8 0 14 (c) Total retail sales h 2 a b $821.3 billion 79. True. The figure is a rectangle with a length of 9 units and a width of 11 units. 81. The graph is a half-line on the real number line; on the rectangular coordinate system, the graph is a half-plane. 333200_07_AN.qxd 12/9/05 1:53 PM Page A171 Answers to Odd-Numbered Exercises and Tests A171 83. (a) y2 x2 ≥ y > x > 10 x 0 (b) −6 17. y 10 (0, 8) 6 4 −4 19. y 10 (0, 8) (c) The line is an asymptote to the boundary. The larger the circles, the closer the radii can be and the constraint will still be satisfied. 87. c 88. a 28x 17y 13 0 91. 86. b 5x 3y 8 0 x y 1.8 0 85. d 89. 93. 95. (a) 2.17t 22.5 0.241t2 7.23t 3.4 271.05t y1 y2 y3 (b) 60 y3 y1 y2 5 30 18 (c) The quadratic model is the best fit for the data. (d) $48.66 Section 7.6 (page 558) Vocabulary Check (page 558) 1. optimization 3. objective 5. vertex 2. linear programming 4. constraints; feasible solutions 1. Minimum at Maximum at 5. Minimum at Maximum at 9. Minimum at Maximum at 11. Minimum at 0, 0: 0 5, 0: 20 0, 0: 0 3, 4: 17 0, 0: 0 60, 20: 740 0, 0: 0 3. Minimum at Maximum at 7. Minimum at Maximum at 0 40 0 0, 0: 0, 5: 0, 0: 4, 0: 20 Maximum at any point on the line segment connecting 60, 20 30, 45: 2100 and 13. y 4 3 1 − 1 (0, 2) (0, 0) 2 3 4 5 (5, 0) x 15. y 4 3 1 −1 (0, 2) (0, 0) 2 3 4 5 (5, 0) x Minimum at Maximum at 0, 0: 0 5, 0: 30 Minimum at Maximum at 0, 0: 0 0, 2: 48 4 2 (5, 3) 4 2 (5, 3) 2 4 6 8 x (10, 0) 2 4 6 8 x (10, 0) 5, 3: 35 Minimum at No maximum 21. 15 10, 0: 20 Minimum at No maximum 23. 15 20 −5 50 20 −5 50 24, 8: 104 40, 0: 160 Minimum at Maximum at 25. Maximum at 29. Maximum at 3, 6: 12 0, 5: 25 36, 0: 36 24, 8: 56 Minimum at Maximum at 27. Maximum at 31. Maximum at 0, 10: 10 22 : 271 3 , 19 6 6 33. y (0, 3) ( 20 19 , 45 19 ( 2 1 (0, 0) 1 (2, 0 The maximum, 5, occurs at any point on the line segment . connecting y 20 19, 45 2, 0 and 19 35. (0, 7) 10 6 4 2 (0, 0) 2 4 6 (7, 0) x The constraint x ≤ 10 is extraneous. Maximum at 0, 7: 14 2x y ≤ 4 The constraint is extraneous. Maximum at 0, 1: 4 37. y 3 2 (0, 0) (0, 1) (1, 0) x 3 4 333200_07_AN.qxd 12/12/05 11:28 AM Page A172 A172 Answers to Odd-Numbered Exercises and Tests 39. 750 units of model A 1000 units of model B Optimal profit: $83,750 43. Three bags of brand X Six bags of brand Y Optimal cost: $195 47. $62,500 to type A $187,500 to type B Optimal return: $23,750 41. 216 units of $300 model 0 units of $250 model Optimal profit: $8640 45. 0 tax returns 12 audits Optimal revenue: $30,000 49. 53. 57. 61. 49. True. The objective function has a maximum value at any point on the line segment connecting the two vertices. 51. (a) t ≥ 9 (b) 3 4 ≤ t ≤ 9 53. z x 5y 55. z 4x y 57. 9 2x 3, x 0 x2 2x 13 xx 2 4 ln 38 14.550 4, 3, 7 59. 63. 67. , x ±3 61. ln 3 1.099 65. 65. 1 3e127 1.851 Review Exercises (page 563) 1. 7. 11. 13. 1, 1 3. 0, 0, 2, 8, 2, 8 1.41, 0.66, 0.25, 0.625 9. 1.41, 10.66 5, 4 5. 4, 2 −6 2 −6 6 17. 23. 0, 2 96 meters 144 15. 3847 units meters 8 0, 0 0.5, 0.8 5 a 14 21. 27. d, one solution, consistent 28. c, infinite solutions, consistent 29. b, no solution, inconsistent 30. a, one solution, consistent 25. 5 , a , , 33. 500,000 159 7 7 24 5 22 , 8 5 a 4, a 3, a x2 y2 4x 4y 1 0 y 3x2 14.3x 117.6 41. 37. 5 31. 35. 39. 43. 45. (a) (b) 2, 4, 5 3a 4, 2a 5, a y 2x2 x 5 130 (c) 195.2; yes. 0 6 80 The model is a good fit. 47. $16,000 at 7% $13,000 at 9% $11,000 at 11% 51. A x C B x2 1 25 x 5 B A x x 20 x2 1 55. 8x 5 3x x2 1 9 8x 3 x x2 12 59. 63 10 10 8 6 4 2 − 2 −2 −4 y 100 60 40 20 (0, 80) (40, 60) − 2 (2, 15) (2, 9) (6, 3) 4 y 67. y 16 12 8 4 x 71. (15, 15) ( 15, − 3 2 ( x 12 8 6 4 2 (0, 0) −2 1600 (6, 4) (4, 0) x 2 4 6 8 −400 1600 −400 (0, 0) (60, 0) 20 40 80 100 69. y 6 5 4 3 2 (2, 3) 19. 2, 3 5 (−1, 0) − 4 −3 1 2 3 4 x 73. 20x 12x x − 2 30y ≤ 8y ≤ ≥ y ≥ 24,000 12,400 0 0 75. (a) p 175 150 125 100 75 50 Consumer Surplus Producer Surplus p = 160 − 0.0001x (300,000, 130) p = 70 + 0.0002 x 100,000 200,000 300,000 x (b) Consumer surplus: $4,500,000 Producer surplus: $9,000,000 333200_07_AN.qxd 12/9/05 1:53 PM Page A173 79. y 27 24 21 18 15 12 9 6 3 (0, 25) (5, 15) (15, 0) x 3 6 9 12 15 18 21 24 27 Minimum at No maximum 15, 0: 26.25 77. y 15 12 9 6 3 (0, 10) (5, 8) (0, 0) (7, 0) x 3 6 9 12 15 Minimum at Maximum at y 81. 0, 0: 0 5, 8: 47 6 5 4 3 2 1 (0, 4) (0, 0) (3, 3) (5, 0) x 1 2 3 4 5 6 Minimum at Maximum at 0 0, 0: 3, 3: 48 Answers to Odd-Numbered Exercises and Tests A173 6. y (1, 12) (0.034, 8.619) 16 12 4 x 9. 12. −1 1, 12, 1, 5 1 1 2 3 0.034, 8.619 2, 1 8. 11. 13. 15. y 4 3 2 1 (0, 0) 10. No solution 2, 3, 1 3 2 x2 2 x 2 x 3x x2 2 14. 16. y 6 3 (1, 4) (1, 2) − 12 −9 −6 −3 6 9 12 x − 2 −1 1 3 4 x (− 4, −16) −18 18. Maximum at Minimum at 12, 0: 240 0, 0: 0 ( 1, 15 ( 83. 72 haircuts 85. Three bags of brand X 0 permanents Optimal revenue: $1800 Two bags of brand Y Optimal cost: $105 87. False. To represent a region covered by an isosceles trape- − 2 17. zoid, the last two inequality signs should be 91. ≤. 89. 93. 2 14 95. 3x y 7 6x 3y 1 2x 2y 3z x 2y z x 4y z 97. An inconsistent system of linear equations has no solution. 99. Answers will vary. 7 4 1 −5 −3 − 2 −1 32 5 x −2 −5 ( ( 7, −3 (1, −3) Chapter Test 3, 4 2. 8, 4, 2, 2 1. 3. 4. (page 567) 0, 1, 1, 0, 2, 1 5. y y 8 6 4 2 −2 −4 3, 2 (3, 2) 2 4 6 10 x (−3, 0) − 9 − 6 12 9 6 3 − 3 − 6 (2, 5) x 6 9 19. 8%: $20,000 8.5%: $30,000 20. y 1 2 x2 x 6 21. 0 units of model I, 5300 units of model II Optimal profit: $212,000 Problem Solving (page 569) 1. (− 10, 0) y 12 8 a c (6, 8) b −8 −4 4 8 (10, 0) x −4 −8 −12 3, 0, 2, 5 a 85, b 45, c 20 852 452 202 Therefore, the triangle is a right triangle. 333200_08_AN.qxd 12/12/05 11:30 AM Page A174 A174 Answers to Odd-Numbered Exercises and Tests 5. (a) One (b) Two (c) Four ad bc 3. 7. 10.1 feet high; 11. (a) 13. (a) 5a 16 6 252.7 feet long 2 3, 4 (b) , a 5 5a 16 , a , 6 13a 40 11a 36 14 14 a 3, a 3, a a 0.15a 193a t ≤ ≥ 772t ≥ 32 1.9 11,000 , (b) (c) t 30 25 15. 9. $12.00 , 1 a 1 4a 1 , a (d) Infinitely many 17. (a) x x 0 y < y ≤ 200 ≥ 35 ≤ 130 (b) 20 10 5 −5 −5 y 250 200 150 100 50 5 10 15 2
0 25 30 a (70, 130) (35,130) (c) No, because the total cholesterol is greater than 200 milligrams per deciliter. 50 100 150 250 x (d) LDL: 140 milligrams per deciliter HDL: 50 milligrams per deciliter Total: 190 milligrams per deciliter 50, 120; 170 50 3.4 < 4; (e) answers will vary. Chapter 8 Section 8.1 (page 582) Vocabulary Check (page 582) 2. square 1. matrix 4. row; column 7. row-equivalent 9. Gauss-Jordan elimination 3. main diagonal 5. augmented 6. coefficient 8. reduced row-echelon form 1. 7. 1 2 4 1 11. 15. 7 19 2x 3 3 5 0 3. 5. 3 1 5 12 9. 2 2 1 5 2 1 8 5z 2z 13. 13 10 12 7 2 y 3y 6x 2 0 6 10 3 1 2 4 0 x 2y 7 2x 3y 4 17. 9x 2x x 3x 12y 18y 7y 3z 5z 8z 2z 19. 21 20 1 6 4 2w 0 10 4 10 1 0 0 1 1 3 4 2 5 20 23. Add 5 times Row 2 to Row 1. 25. Interchange Row 1 and Row 2. 1 6 5 4 Add 4 times new Row 1 to Row 3. (b) (d 10 10 2 1 0 3 2 0 27. (a) (c) (e 10 1 3 10 0 0 1 0 1 2 0 The matrix is in reduced row-echelon form. 29. Reduced row-echelon form 31. Not in row-echelon form 33. 37. 41. 45. 47. 53. 59. 65. 67. 71. 75. 81. 1 6 0 1 3 0 35 2y y 2 16 12 5 0 1 39. 43. x y 2z y z z 4 2 2 51. 3, 2 57. Inconsistent 4, 3, 6 69. Inconsistent 63. 61. 49. 55. 7, 3, 4 4, 10, 4 1, 4 8, 0, 2 3, 4 5, 6 4, 3, 2 2a 1, 3a 2, a 4 5b 4a, 2 3b 3a, b, a 0, 2 4a, a 1, 0, 4, 2 2a, a, a, 0 73. 77. Yes; ,, 1, 3 3 1 2 79. No 333200_08_AN.qxd 12/9/05 2:40 PM Page A175 83. 4x2 x 12x 1 85. $150,000 at 7% $750,000 at 8% $600,000 at 10% 1 x 1 3 2 x 12 x 1 y x2 2x 5 87. 89. (a) (b) y 0.004x2 0.367x 5 18 0 0 120 (c) 13 feet, 104 feet (e) The results are similar. (d) 13.418 feet, 103.793 feet (c) (b) 91. (a) 500, 0, x5 s t, 600 s, x4 s, x7 t 600, x4 t, x3 s, x2 500 t, x6 0, x2 0, x7 0, x2 1000, x6 2 4 x1 x5 x1 x6 x1 x5 93. False. It is a 95. False. Gaussian elimination reduces a matrix until a rowechelon form is obtained; Gauss-Jordan elimination reduces a matrix until a reduced row-echelon form is obtained. 0, x3 0 500, x3 0, x7 matrix. 600, x4 500 500, 97. (a) There exists a row with all zeros except for the entry in the last column. (b) There are fewer rows with nonzero entries than there are variables and no rows as in (a). 99. They are the same. 101. 1032 −1 −1 1 2 3 4 x 105 Answers to Odd-Numbered Exercises and Tests A175 Section 8.2 (page 597) Vocabulary Check (page 597) 1. equal 5. (a) iii 6. (a) ii 2. scalars (b) iv (b) iv (c) i (c) i 3. zero; O (d) v (d) iii 4. identity (e) ii 3. 1 3 x 2, y 3 0 9 (c) (b) 3 6 3 3 (b) 5 3 4 5 1 5 (c) 18 6 9 3 12 15 1. x 4, y 22 5. (a) (d) 7. (a) (d) 9. (a) 2 7 1 19 3 9 15 11 16 8 11 3 2 1 4 6 3 4 9 (c) (b) 1 3 6 3 4 5 11. (a), (b), and (d) not possible 0 11 3 6 1 24 1 6 0 0 2 12 (d 11 18 0 12 3 7 1 8 9 9 0 15. 19. 3.739 13.249 0.362 4.252 9.713 (c) 8 15 10 59 1.581 3 3 1 2 13 2 0 11 2 4 16 46 10 26 3 13. 17. 21. 25. 29. 24 12 17.143 11.571 23. 4 32 12 12 2.143 10.286 9 0 10 6 1 17 27. Not possible 31. 3 0 0 0 4 0 0 0 10 Order: 3 2 Order 333200_08_AN.qxd 12/9/05 2:40 PM Page A176 A176 Answers to Odd-Numbered Exercises and Tests 33 35. 41 42 10 7 5 25 7 25 45 65. 37. 41. (a) 43. (a) 45. (a) 47. 51. (a) 53. (a) 55. (a) 57. (a) Order: 151 516 47 0 6 0 10 7 5 4 (c) (c) (b) (b) (b) 13 4 8 8 1 10 14 9 12 14 16 2 48 387 87 (c) Not possible 39. Not possible 6 12 8 6 6 8 2 31 0 10 2 14 10 0 3 3 25 279 20 15 12 10 0 7 8 1 4 8 49. 16 3 x1 4 1 (b) x2 0 1 4 3 x1 36 x2 1 x1 9 2 3 1 3 5 5 x1 20 5 1 2 30 60 84 120 A 125 100 The entries represent the numbers of bushels of each crop that are shipped to each outlet. $6.00 B $3.50 The entries represent the profits per bushel of each crop. BA $1037.50 The entries represent the profits from both crops at each of the three outlets1012.50 2 1 5 8 16 1 2 3 0 6 17 1 2 75 125 100 175 3 2 $1400 x2 x3 x2 x3 (b) (b) (b) 59. 84 42 61. (a) (b) (c) 63. $15,770 $26,500 $21,260 $18,300 $29,250 $24,150 The entries represent the wholesale and retail values of the inventories at the three outlets. 0.314 0.461 0.315 0.435 0.308 0.392 P3 0.300 P4 0.250 P5 0.225 P6 0.213 P7 0.206 P8 0.203 0.308 0.486 0.311 0.477 0.305 0.492 0.175 0.433 0.392 0.188 0.377 0.435 0.194 0.345 0.461 0.197 0.326 0.477 0.198 0.316 0.486 0.199 0.309 0.492 0.175 0.217 0.608 0.188 0.248 0.565 0.194 0.267 0.539 0.197 0.280 0.523 0.198 0.288 0.514 0.199 0.292 0.508 Approaches the matrix 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 0.5 67. (a) Sales $ Profit 115 161 188 447 624.5 731.2 (b) $464 The entries represent the total sales and profits for each type of milk. 69. (a) 2 0.5 3 (b) 120 lb 150 lb 473.5 588.5 The entries represent the total calories burned. 71. True. The sum of two matrices of different orders is undefined. 73. Not possible 79. 2 3 81. 75. Not possible 3 3 AC BC 2 2 77. 2 2 83. AB is a diagonal matrix whose entries are the products of the corresponding entries of A and B. 85. 91. 4 8, 3 7, 1 2 5 ± 37 4 3, 1 87. 0, 93. 89. 4, ± 15 3 i Section 8.3 (page 608) Vocabulary Check (page 608) 1. square 2. inverse 3. nonsingular; singular 4. A1B 333200_08_AN.qxd 12/9/05 2:40 PM Page A177 1–9. AB I 0 1 3 and BA I 3 2 13. 2 1 15. 1 2 1 1 1 2 0 11. 17. Does not exist 1 2 3 1 1 2 19. Does not exist 0 1 4 1 4 1 3 4 7 20 23. 0 0 1 5 21. 25. 29. 33. 37.5 1 4.5 1 0 10 10 .5 3.5 1 1.81 5 2.72 1 3 1 0.90 5 3.63 27. 175 95 14 37 20 3 13 7 1 31. 12 4 8 5 2 4 9 4 6 35. Does not exist 39. 3 19 2 19 2 19 5 19 15 59 70 59 3, 8, 11 55. No solution 7, 3, 2 16 59 4 59 41. Does not exist 43. 45. 51. 57. 61. 5, 0 2, 1, 0, 0 4, 8 5 16, 19 16a 13 5, 0, 2, 3 49. 47. 8, 6 53. 59. 16a 11 2, 2 1, 3, 2 16, a 63. 65. 67. $7000 in AAA-rated bonds $1000 in A-rated bonds $2000 in B-rated bonds 69. $9000 in AAA-rated bonds $1000 in A-rated bonds $2000 in B-rated bonds (b) I1 I2 I3 71. (a) amperes amperes amperes 3 8 5 2 3 5 73. True. If B is the inverse of A, then 75. Answers will vary. 77. x ≥ 5 or x ≤ 9 I1 I2 I3 amperes amperes amperes AB I BA. − 10 − 9 − 8 − 7 −6 −5 −4 x 79. 2 ln 315 ln 3 10.472 81. 26.5 90.510 83. Answers will vary. Answers to Odd-Numbered Exercises and Tests A177 Section 8.4 (page 616) Vocabulary Check (page 616) 1. determinant 3. cofactor 2. minor 4. expanding by cofactors 5. 27 17. 11. 4.842 3 3 3 7. 0 0.002 2, M21 2, C21 2, M21 2, C21 4, M13 9. 6 19. 4, M22 4, C22 1, M22 1, C22 1, M21 4, M32 10, M33 1, C21 4, C13 4, C32 12, M13 7, M31 12, C13 4, C32 10, C33 11, M21 4, M32 11, C21 42, C33 75 33. (a) 96 (b) 1. 5 13. 72 23. (a) (b) 25. (a) (b) 27. (a) M11 C11 M11 C11 M11 M23 C11 C23 M11 M22 C11 C23 75 31. (a) 35. (a) 170 58 43. 51. 412 3. 5 11 15. 6 5, M12 5, C12 4, M12 4, C12 3, M12 4, M31 3, C12 4, C31 30, M12 26, M23 30, C12 7, C31 (b) (b) 170 30 126 45. 53. 29. (a) (b) 61. (a) 3 (b) 2 63. (a) 8 (b) 0 (c) 65. (a) 21 (b) 19 67. (a) 2 (b) 6 (c) 39. 0 168 37. 0 47. 55. 0 2 (c) 0 4 1 57. 0 3 4 1 (c, 4 1 ln x 9 21. 0 2, 2, 3 2, M22 8 2, C22 8 36, 42, M33 36, C22 12 (b) 96 41. 9 12 26, 49. 0 336 (d) 6 59. 410 (d) 0 4 3 9 3 3 0 (d) 399 (d) 12 69–73. Answers will vary. 79. 83. 85. True. If an entire row is zero, then each cofactor in the 8uv 1 75. 81. 77. e5x 1, 4 expansion is multiplied by zero. 87. Answers will vary. 89. A square matrix is a square array of numbers. The deter- minant of a square matrix is a real number. 91. (a) Columns 2 and 3 of A were interchanged. A 115 B A (b) Rows 1 and 3 of were interchanged. A 40 B 93. (a) Multiply Row 1 by 5. (b) Multiply Column 2 by 4 and Column 3 by 3. 95. All real numbers x C H A P T E R 8 333200_08_AN.qxd 12/9/05 2:40 PM Page A178 A178 Answers to Odd-Numbered Exercises and Tests 97. All real numbers such that 99. All real numbers t such that 101. y 12 4 103. − 8 − 4 4 8 12 x 105. Does not exist Section 8.5 (page 628) Vocabulary Check (page 628) 1. Cramer’s Rule 2. collinear x1 3. A ± 1 2 y1 y2 y3 5. uncoded; coded x2 x3 1 1 1 4. cryptogram 67. y 6 4 2 (0, 5) (0, 0) (6, 4) ( 20 3 ( , 0 x 2 4 6 Minimum at Maximum at 0, 0: 0 6, 4: 52 Review Exercises 1. 3 1 3. 1 1 5. (page 632) 10 3 4 5 1 9. 0 0 15 22 3 1 1 2 1 0 13. x 5y 4z 1 y 2z 3 z 4 9 10 3 7z 4x 9x y 2y 4y 5x x 2y 3z 9 y 2z 2 z 0 2z 10 1 17. 2, 2a 1, a 27. 5, 7 23. 2, 3, 1 33. 5, 2, 6 40, 5, 4 19. 1, 0, 4, 3 29. 2, 6, 10, 3 7. 11. 15. 21. 25. 31. 5 2 5. 7 0, 1 32 7 , 30 11. 19. 33 8 2, 1 2 21. 3. Not possible 2, 1, 1 17. 14 y 0 or 9. 15. 7 y 16 5 y 11 2, 2 1, 3, 2 1, 2, 1 1. 7. 13. 23. 28 y 3 27. 31. Collinear y 3 37. 2x 3y 8 0 43. 45. Uncoded: 25. or 29. 250 square miles 33. Not collinear 3x 5y 0 39. 35. Collinear 41. x 3y 5 0 Encoded: 20 18 15, 21 2 12, 5 0 9, 14 0 18, 9 22 5, 18 0 3, 9 20 25 52 49 49 10 27 94 12 22 54 1 1 121 41 55 3 34 7 0 13 27 9 47. 49. 6 35 69 11 20 17 6 16 58 46 79 67 5 41 87 91 207 257 11 5 41 40 80 84 76 177 227 51. HAPPY NEW YEAR 53. CLASS IS CANCELED 55. SEND PLANES 59. False. The denominator is the determinant of the coeffi- 57. MEET ME TONIGHT RON cient matrix. 61. False. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions. 6, 4 1, 0, 3 63. 65. 5, 2, 0 10, 12 2a 3 2, 3, 3 x 12, y 7 1 8 13 15 8 8 20 12 5 20 7 14 42 3 31 (c) 28 44 16 8 8 (c) 35. (a) 37. (a) 39. 17 13 17 2 41. 43. 48 15 18 51 3 33 4 3 10 3 3 100 12 84 2 3 11 3 0 220 4 212 47. 51. 49. 30 51 53. 14 14 36 (b) (d) (b) (d) 54 2 4 45. x 1, y 11 12 5 3 9 7 28 29 39 5 5 11 9 1 10 38 13 38 122 5 71 4 24 32 14 7 17 4 70 2 10 12 4 17 2 8 40 48 333200_08_AN.qxd 12/12/05 11:30 AM Page A179 Answers to Odd-Numbered Exercises and Tests A179 5.5 2 14.5 2.5 3.5 1 9.5 1.5 5. 44 20 14 4 8 22 41 66 55. 61. 65. 22 80 66 57. 24 36 59. 17 36 8 12 76 38 1 12 114 95 133 76 63. 19 42 $274,150 $303,150 The merchandise shipped to warehouse 1 is worth $274,150 and the merchandise shipped to warehouse 2 is worth $303,150. AB I and BA I 73. 6, 1, 1 97. 550 83. 36, 11 89. 42 2 95. 1, M22 1, C22 2 21, 22, M31 5, 67–69. 4 5 71. 2 1 2 1 1 0 75. 79. 85. 91. 99. (a) (b) 101. (a 15 2.5 77. 12 5 6 5 2 13 3 1 7 1 20 3 1 1 6 10 2, 1, 2 2 1 2 5 6 1 3 81. 87. 4 6, 1 3, 1 M11 C11 M11 M21 M32 C11 C21 C31 93. 4, M12 4, C12 30, M12 20, M22 2, M33 30, C12 20, C22 5, C32 105. 279 113. 10 x 2y 4 0 (b) 1, 1, 2 7, M21 7, C21 12, M13 19,
M23 19 12, C13 19, C23 19 2, C33 107. 4, 7 115. Collinear 21, 22, 103. 130 111. 16 117. 121. Uncoded: Encoded: 109. 1, 4, 5 12 15 2 12, 5 21 6 0 20 21 99 119. 15, 15 68 30 11 2x 6y 13 0 15, 0 21 23 0 20 0, 42 60 53 8 45 102 69 123. SEE YOU FRIDAY 125. False. The matrix must be square. 127. The matrix must be square and its determinant nonzero. 129. No. The first two matrices describe a system of equations with one solution. The third matrix describes a system with infinitely many solutions. ±210 3 131. Chapter Test (page 637) 1, 3, 1 2 14 5 8 2 2 4 2. 3. 4. (a 15 12 7 4 4 0 (d) (b) (c) 12 12 14 12 . 2 5 4 13, 22 7. 3, 5 11. 14. Uncoded: Encoded: 3 6 5 8. 12. 196 2, 4, 6 9. 29 10. 43 13. 7 11 14 15, 3 11 0, 15 14 0, 23 15 15, 4 0 0 115 41 59 14 3 11 29 15 14 128 53 60 15. 75 liters of 60% solution 25 liters of 20% solution 1. (a) Problem Solving AT 1 1 AAT 1 1 y (page 639) 2 3 3 2 4 2 2 4 AT 4 3 2 1 T −4 −3 −2 −1 1 2 3 4 x AAT −2 −3 −4 A represents a counterclockwise rotation. (b) AAT is rotated clockwise to obtain AT. AT is then rotated clockwise 90 90 to obtain T. 333200_09_AN.qxd 12/9/05 2:41 PM Page A180 A180 Answers to Odd-Numbered Exercises and Tests 3. (a) Yes 5. (a) Gold Cable Company: 28,750 subscribers (d) No (b) No (c) No 27. 10 29. 18 Galaxy Cable Company: 35,750 subscribers Nonsubscribers: 35,500 Answers will vary. (b) Gold Cable Company: 30,813 subscribers Galaxy Cable Company: 39,675 subscribers Nonsubscribers: 29,513 Answers will vary. (c) Gold Cable Company: 31,947 subscribers Galaxy Cable Company: 42,329 subscribers Nonsubscribers: 25,724 Answers will vary. (d) Cable companies are increasing the number of subscribers, while the nonsubscribers are decreasing. x 6 7. 13. Sulfur: 32 atomic mass units 9–11. Answers will vary. Nitrogen: 14 atomic mass units Fluorine: 19 atomic mass units 2 0 1 5 1 BT 3 BTAT 2 3 AT 1 1 2 ABT 2 4 A1 1 1 0 15. 17. (a) 1 2 1 1 (b) JOHN RETURN TO BASE A 0 19. Chapter 9 Section 9.1 (page 649) Vocabulary Check (page 649) 1. infinite sequence 4. recursively 6. summation notation 8. series 9. 5. factorial n th partial sum 2. terms 3. finite 7. index; upper; lower 3. 2, 4, 8, 16, 32 3, 3 7. 0, 1, 0, 1 1. 4, 7, 10, 13, 16 5. 9. 2, 4, 8, 16, 32 3, 12 47, 15 13, 24 11, 9 11. 37 1 1 1 332, 232, 8 3, 2 3, 2 3, 2 3, 2 1, , 3 2 15. 19. 17. 1 532 21. 0, 0, 6, 24, 60 3, 2, 5 2, 0 1 4 1, 2, 7 5 13. , 1 9 23. , 53 17 5 3, 9 , 27, 1 , 1 25 16 73 (b) 109. (a) (b) (c) 161 81 , 485 243 25. 44 239 31. 0 0 2 0 0 10 10 33. c 34. b 35. d 0 −10 10 an 3n 2 36. a 37. 1nn 1 n 2 39. an 43. an 49. an n 2 1 n 1 2n 1 1 1 n 41. an 45. an 1 n2 47. an 1n1 51. 28, 24, 20, 16, 12 53. 3, 4, 6, 10, 18 55. 6, 8, 10, 12, 14 an 57. 81, 27, 9, 3, 1 243 3n 1 24 67. 90 61. 65. 1 6 1 2 1, , , , 1 30 1 120 1 2n2n 1 9 81. 88 5 71. 79. 89. 95. i1 9 20 i1 2 3 103. 107. (a) an 2n 4 9 9 2 2 27 8 , , 59. 1, 3, 63. 1, 1 2 n 1 69. , 1 24 , 1 720 , 1 40,320 73. 35 75. 40 77. 30 85. 81 87. 47 60 93. 1i13i 6 i1 75 16 99. 101. 3 2 A3 A6 $5306.04, $5630.81, 7 9 i1 i1 97. 91. 8 2i 1 2i1 83. 30 3 2 i 8 5 1 3i 1i1 i 2 105. $5100.00, A2 $5412.16, A5 $5743.43, A8 $11,040.20 60.57n 182 1.61n2 26.8n 9.5 A1 A4 A7 A40 bn cn $5202.00, $5520.40, $5858.30 n an bn cn 8 9 10 11 12 13 311 357 419 481 548 608 303 363 424 484 545 605 308 362 420 480 544 611 The quadratic model is a better fit. (d) The quadratic model; 995 333200_09_AN.qxd 12/12/05 11:31 AM Page A181 Answers to Odd-Numbered Exercises and Tests A181 111. (a) a0 a3 a6 a9 a12 $3102.9, a1 $4425.3, a4 $5091.8, a7 $5550.9, a10 $6251.5, a13 $3644.3, a2 $4698.2, a5 $5245.7, a8 $5735.5, a11 $6616.3 $4079.6, $4914.8, $5393.2, $5963.5, 7000 0 0 14 (b) The federal debt is increasing. 113. True by the Properties of Sums 115. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 21 13, 34 21, 55 34, 8 5, , , , , x, 125. 121. 123. 129. (a) 5 3 1, 2, 2, 3, 117. $500.95 x4 x3 x2 24 6, 2 x4 x2 , x6 , 720 24 2 f 1x x 3 4 8 1 2 6 18 9 7 10 3 7 4 4 2 7 42 23 194 4 1 135. 131. (a) 133. 26 4 1 (c) (c) 89 13 8 , 55 119. Answers will vary. x5 120 x8 40,320 , x10 , x ≥ 0 127. (b) (d) 5 3,628,800 h1x x2 1 26 12 4 24 1 21 2 21 10 16 12 3 25 11 9 31 47 22 10 13 (b) (d) 10 3 8 42 31 25 4 1 3 16 45 48 Section 9.2 (page 659) Vocabulary Check (page 659) 1. arithmetic; common 3. sum of a finite arithmetic sequence an 2. dn c d 2 1. Arithmetic sequence, 3. Not an arithmetic sequence 5. Arithmetic sequence, 7. Not an arithmetic sequence 9. Not an arithmetic sequence 11. 8, 11, 14, 17, 20 d 1 4 13. 7, 3, Arithmetic sequence, 5, Arithmetic sequence, 1, 9 d 3 d 4 15. 17. 3 3 1, 1 3 5 1, 4,1, 1, 1, Not an arithmetic sequence 2,3, Not an arithmetic sequence an an an 25. 21. 3 3n 2 2 xn x 10 3 n 5 8n 108 19. an 5 2n 13 23. an 2 3n 103 29. 27. an 2.6, 3.0, 3.4, 3.8, 4.2 31. 5, 11, 17, 23, 29 33. 2, 2, 6, 10, 14 35. 2, 6, 10, 14, 18 37. d 4; 4n 11 39. 15, 19, 23, 27, 31; 41. 200, 190, 180, 170, 160; 1 1 5 43. 8, 2, 8 ; 45. 59 53. an d 10; 1 8 n 3 4 50. d 1 4, 47. 18.6 10n 210 an 49. b d 1 8; 51. c 55. an 3 8, 14 6 52. a 0 0 10 0 2 10 63. 4000 69. 30,030 71. 355 77. 2725 79. 10,120 83. 2340 seats 61. 265 59. 17.4 67. 1275 75. 520 57. 620 65. 10,000 73. 160,000 81. (a) $40,000 85. 405 bricks 89. (a) 91. $70,500; answers will vary. 93. (a) 25n 225 an (b) $217,500 87. 490 meters (b) $900 Month 1 2 3 4 5 6 Monthly payment Unpaid balance $220 $218 $216 $214 $212 $210 $1800 $1600 $1400 $1200 $1000 $800 C H A P T E R 9 (b) $110 an an 95. (a) (b) (c) 1098n 17,588 1114.9n 17,795; the models are similar. (d) 2004: $32,960 2005: $34,058 32,000 3 20,000 13 d a2 a1 and (e) Answers will vary. a1 a2, and n 1d. 97. True. Given a1 99. Answers will vary. an 333200_09_AN.qxd 12/9/05 2:41 PM Page A182 A182 101. (a) an 33 30 27 24 21 18 15 12 9 6 3 Answers to Odd-Numbered Exercises and Tests (b) y 33 30 27 24 21 18 15 12 10 11 x 7 2 ; an an 27. r 2; 23. 7, 14, 28, 56, 112; an 8 ; r 3 4 , 243 25. 2; 1 128 2 , 81 6, 9, 27 n1 41 2 100exn1; 100e8x 5001.02n1; 1082.372 39. a3 45. b 31. 33. 37. 50,388,480 43. a 44. c 47. 46. d 49. an an 9 29. an 16 41. 15 2n 43 61 3 2 n n1 ; 2 310 35. 45,927 2 a6 − 10 11 n (c) The graph of line. The graph of at the positive integers. y 3x 2 an contains all points on the contains only points 2 3n (d) The slope of the line and the common difference of the arithmetic sequence are equal. 103. 4 105. Slope: y- 1 2; intercept: 0 107. Slope: undefined; y- intercept No y 8 6 4 2 −2 −2 − 4 − 6 −8 2 4 6 8 10 12 14 x 109. x 1, y 5, z 1 111. Answers will vary. Section 9.3 (page 669) Vocabulary Check (page 669) −4 2. an a1r n1 4. geometric series 1. geometric; common 3. Sn 5. S a11 r n 1 r a1 1 r r 3 1. Geometric sequence, 3. Not a geometric sequence r 1 5. Geometric sequence, 2 r 2 7. Geometric sequence, 9. Not a geometric sequence 11. 2, 6, 18, 54, 162 200, 1 20, 1 2, 1 5, 1 15. x4 x2 x3 x 2 128 8 32 21. 64, 32, 16, 8, 4; 1, 1 17. 13. 19. r 1 2; an 2, 2000 , , , 8, 1 4, 1 2, 1 16 1, e, e2, e3, e4 128 1 n 2 51. 0 −16 24 0 0 10 0 10 −15 10 57. 43 63. 592.647 1365 59. 32 65. 2092.596 71. 3.750 7 21 4 n1 85. 5 3 93. 4 11 77. 87. 95. 0.14n1 6 n1 30 7 22 55. 171 53. 511 61. 29,921.311 67. 8 5 69. 6.400 53n1 75. n1 73. 7 79. 2 89. 32 97. n1 16 81. 3 91. Undefined 83. 2 3 20 −15 10 Horizontal asymptote: Corresponds to the sum of the series y 12 99. (a) an 1190.881.006n (b) The population is growing at a rate of 0.6% per year. (c) 1342.2 million. This value is close to the prediction. (d) 2007 101. (a) $3714.87 (d) $3728.32 (b) $3722.16 (e) $3729.52 (c) $3725.85 105. Answers will vary. (b) $26,263.88 (b) $118,788.73 113. $1600 103. $7011.89 107. (a) $26,198.27 109. (a) $118,590.12 111. Answers will vary. $2181.82 115. 119. $3,623,993.23 121. False. A sequence is geometric if the ratios of consecutive 117. 126 square inches terms are the same. 123. Given a real number between r n approaches zero. increases, n r 1 and 1, as the exponent 333200_09_AN.qxd 12/9/05 2:41 PM Page A183 Answers to Odd-Numbered Exercises and Tests A183 139. Answers will vary. −12−10 −8 −6 −4 (d) y 10 8 6 4 2 −4 −6 x 2 4 (0, 0) x2 2x 127. x3x 83x 8 3x2 6x 1 131. 125. 129. 133. 137. , x 3 3x x 3 5x2 9x 30 x 2x 2 3x 12x 5 , x 0, 1 2 2x 1 3 135. Section 9.4 (page 681) Vocabulary Check (page 681) 1. mathematical induction 3. arithmetic 4. second 2. first 1. 5 k 1k 2 3. Sn 37. 5–33. Answers will vary. n 10 10 9 10 45. 979 43. 91 41. 120 51. 0, 3, 6, 9, 12, 15 39. k 12k 22 4 Sn 35. n2n 1 n 2n 1 Sn 47. 70 49. 3402 First differences: 3, 3, 3, 3, 3 Second differences: 0, 0, 0, 0 Linear 53. 3, 1, 2, 6, 11, 17 First differences: Second differences: Quadratic 2, 3, 4, 5, 6 1, 1, 1, 1 55. 2, 4, 16, 256, 65,536, 4,294,967,296 First differences: 2, 12, 240, 65,280, 4,294,901,760 Second differences: 10, 228, 65,040, 4,294,836,480 Neither an 2 n 2 n 3 1 an n2 n 3 57. 59. 61. (a) 2.2, 2.4, 2.2, 2.3, 0.9 (b) A linear model can be used. 2.2n 102.7 2.08n 103.9 142.3; an an (c) (d) Part b: Part c: an These are very similar. an 141.34 P7 63. True. may be false. 65. True. If the second differences are all zero, then the first differences are all the same and the sequence is arithmetic. 4x4 4x2 1 64x3 240x2 300x 125 69. 67. 71. (a) Domain: all real numbers except 0, 0 x x 3 (b) Intercept: (c) Vertical asymptote: Horizontal asymptote: x 3 y 1 73. (a) Domain: all real numbers except t t 0 t -intercept: (b) (c) Vertical asymptote: 7, 0 t 0 Horizontal asymptote: y y 1 (d) 4 2 −8 −6 −4 −2 2 6 8 t (7, 0) −4 −6 −8 Section 9.5 (page 688) Vocabulary Check (page 688) 1. binomial coefficients 2. Binomial Theorem; Pascal’s Triangle 3. ; nCr n r 4. expanding a binomial C H A P T E R 9 3. 1 13. 35 5. 15,504 15. 7. 210 x 4 4x3 6x 2 4x 1 9. 4950 1. 10 11. 56 17. 19. 21. 23. a4 24a3 216a2 864a 1296 y3 12y2 48y 64 x5 5x 4y 10x 3y2 10x 2y 3 5xy 4 y5 r 6 18r 5s 135r 4s2 540r 3s3 1215r 2s4 1458rs5 729s6 25. 243a5 1620a4b 4320a3b2 5760a2b3 3840ab4 1024b5 27. 29. 31. 8x3 12x2y 6xy2 y3 x8 4x6y2 6x 4y 4 4x2y6 y8 1 x 5 2x4 24x3 113x2 246x 207 10y2 x 3 10y3 x 2 5y4 x 5y x 4 y5 33. 35. 32t 5 80t 4s 80t 3s2 40t 2s3 10ts4 s5 33320
0_09_AN.qxd 12/12/05 11:32 AM Page A184 A184 Answers to Odd-Numbered Exercises and Tests 41. 360 x3y2 37. 39. 45. 49. 180 55. 57. x5 10x4y 40x3y2 80x2y3 80xy4 32y5 120x7y3 1,259,712 x2y7 32,476,950,000x4y8 51. 43. 47. 1,732,104 326,592 53. 210 x2 12x32 54x 108x12 81 x 2 3x 43y 13 3x 23y 23 y 59. 3x 2 3xh h 2, h 0 61. 1 x h x h 0 , 4 63. 69. 1.172 73. g − 8 65. 2035 828i 71. 510,568.785 67. 1 4 f − 4 4 93. 954 −2 2 4 6 x −2 −3 −2 −1 1 2 3 −1 x gx x 32 4 5 5 6 97. Section 9.6 (page 698) gx x 2 1 f. Vocabulary Check (page 698) 1. Fundamental Counting Principle 2. permutation is shifted four units to the left of g gx x3 12x2 44x 48 77. 0.171 f t 0.0025t 3 0.015t 2 0.88t 7.7 75. 0.273 79. (a) (b) 24 (c) (d) 0 0 13 gt 0.0025t 3 0.06t 2 1.33t 17.5 60 0 0 g f 13 f t: gt: 33.26 gallons; (e) (f) The trend is for the per capita consumption of bottled water to increase. This may be due to the increasing concern with contaminants in tap water. 33.26 gallons; yes 81. True. The coefficients from the Binomial Theorem can be used to find the numbers in Pascal’s Triangle. 83. False. The coefficient of the x10 -term is 1,732,104 and the coefficient of the x14 -term is 192,456. 85. 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 87. The signs of the terms in the expansion of between positive and negative. 89–91. Answers will vary. 1 10 x yn n! 3. nPr n r! 5. combinations 4. distinguishable permutations 7. 8 5. 3 9. 30 11. 30 (c) 180 (d) 600 27. 120 3. 5 15. 175,760,000 (b) 648 21. (a) 40,320 1. 6 13. 64 17. (a) 900 19. 64,000 25. 336 31. 1,860,480 39. 11,880 45. ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, CABD, CADB, DABC, DACB, BCAD, BDAC, CBAD, CDAB, DBAC, DCAB, BCDA, BDCA, CBDA, CDBA, DBCA, DCBA (b) 384 n 6 or 35. 15,504 33. 970,200 43. 2520 37. 120 41. 420 n 5 23. 24 29. 47. 1,816,214,400 51. AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, 49. 5,586,853,480 DF, EF 53. 324,632 57. (a) 3744 61. 5 65. (a) 146,107,962 63. 20 55. (a) 35 (b) 24 (b) 63 59. 292,600 (c) 203 (b) If the jackpot is won, there is only one winning number. alternate (c) There are 28,989,675 possible winning numbers in the state lottery, which is considerably less than the possible number of winning Powerball numbers. 67. False. It is an example of a combination. 69. They are equal. 71–73. Proof 75. No. For some calculators the number is too great. 77. (a) 35 (c) 83 4 79. (a) (c) 0 (b) 8 (b) 0 81. 8.30 83. 35 333200_09_AN.qxd 12/9/05 2:41 PM Page A185 Section 9.7 (page 709) Vocabulary Check (page 709) 1. experiment; outcomes 3. probability 5. mutually exclusive 7. complement 4. impossible; certain 6. independent (b) i 8. (a) iii 2. sample space (c) iv (d) ii Answers to Odd-Numbered Exercises and Tests A185 71. y 12 10 8 4 2 73. y 2 −8 −6 −4 −2 4 6 8 x −4 −2 2 4 6 8 12 x Review Exercises (page 715) −8 −12 −14 11 17. 12 29. 0.86 1. 8, 5, 4, 4 n an 7. 17. 6050 21n 205 24 15. 7 2, 16 5 3. 72, 36, 12, 3, 3 5 5. an k1 19. 9. 120 20 1 2k $10,067, $10,269, $10,476, $10,687 $22,196.40 A2 A5 A8 25. (a) (b) A1 A4 A7 A10 A120 11. 1 13. 30 21. 5 9 $10,134, $10,338, $10,546, 23. 2 99 A3 A6 A9 $10,201, $10,407, $10,616, 31. 4, 7, 10, 13, 16 d 2 d 1 2 12n 5 35. an 7n 107 39. an 45. 25,250 (b) $192,500 53. 4, 1, 1 4, 1 16, 1 64 9, 6, 4, 8 ; 3.052 105 or n1 3ny 2y 43. 88 r 2 r 2 3, 16 27. Arithmetic sequence, 29. Arithmetic sequence, 33. 25, 28, 31, 34, 37 37. an 41. 80 47. (a) $43,000 49. Geometric sequence, 51. Geometric sequence, 3, 16 8 55. 9, 6, 4, 161 57. an 1001.05n1; 252.695 59. an 15 61. 127 16 71. 8 69. 5486.45 120,0000.7t 77. (a) 79–81. Answers will vary. 1 3 n 85. Sn 91. 5, 10, 15, 20, 25 First differences: 5, 5, 5, 5 Second differences: 0, 0, 0 Linear 83. 87. 465 65. 31 73. 5 2 63. 10 9 at 2 5 67. 24.85 75. 12 (b) $20,168.40 Sn n2n 7 89. 4648 93. 16, 15, 14, 13, 12 First differences: Second differences: 0, 0, 0 Linear 1, 1, 1, 1 95. 15 103. 105. 107. 115. 56 97. 56 99. 35 101. 28 x 4 16x3 96x2 256x 256 a5 15a 4b 90a3b2 270a2b3 405ab4 243b5 41 840i 117. 119. (a) 43% (b) 82% 111. 10,000 109. 11 113. 720 1 9 1 5 1. (b) 27. 15. 3 13 2 5 1 12 3 4 11. 23. H, 1, H, 2, H, 3, H, 4, H, 5, H, 6, T, 1, T, 2, T, 3, T, 4, T, 5, T, 6 ABC, ACB, BAC, BCA, CAB, CBA 3. AB, AC, AD, AE, BC, 5. 3 7. 8 1 19. 3 18 31. 35 35. (a) 243 112 209 7 9. 8 21. 33. (a) 58% (b) 95.6% (c) 0.4% 16 25 274 627 BD, BE, CD, CE, DE 3 13. 26 25. 0.3 (b) PTaylor wins 1 2 PMoore wins PJenkins wins 1 4 49 (c) 323 45. (a) 54 12 (c) 55 55 (b) 0.9998 (c) (c) 41. (a) 43. (a) 47. (a) 51. (a) 0.9702 53. (a) 55. (a) 5 1 (b) 13 2 49. 0.4746 (c) 0.0002 (b) (b) (b) 21 1292 1 120 14 55 225 646 1 24 1 50 97 209 37. (a) (b) (b) 39. (c) (c) (e) 15 16 1 1444 1 8 9 19 1 16 1 38 10 19 729 6859 (c) 4 13 (d) (f) The probabilities are slightly better in European roulette. 57. True. Two events are independent if the occurrence of one has no effect on the occurrence of the other. 59. (a) As you consider successive people with distinct birthdays, the probabilities must decrease to take into account the birth dates already used. Because the birth dates of people are independent events, multiply the respective probabilities of distinct birthdays. 365 363 365 364 365 (c) Answers will vary. is the probability that the birthdays are not distinct, Qn which is equivalent to at least two people having the same birthday. 365 362 (b) (d) 365 (e) n 10 15 20 23 30 40 50 0.88 0.75 0.59 0.49 0.29 0.11 0.03 0.12 0.25 0.41 0.51 0.71 0.89 0.97 Pn Qn (f) 23 61. No real solution 10 69. 67. 11 2 63. 0, 1 ± 13 2 65. 4 333200_09_AN.qxd 12/9/05 2:41 PM Page A186 A186 Answers to Odd-Numbered Exercises and Tests 1 216 121. 125. True. 3 123. 4 n 2n 1n! n 2! n! n! 127. True by Properties of Sums 129. False. When equals 0 or 1, then the results are the same. 131. In the sequence in part (a), the odd-numbered terms are negative, whereas in the sequence in part (b), the evennumbered terms are negative. n 2n 1 r 2 3 1 3 0 4 3 10. 12. 15. 2 1 3 3 2 12 3 1 2 4 13. 2 2 9 9 7 6 0 16. 84 17. 11. 2, 3, 1 14. 175 95 14 6 3 6 2 37 20 3 13 7 1 133. Each term of the sequence is defined in terms of preceding 18. Gym shoes: $198.36 million terms. 135. d 136. a 139. 240, 440, 810, 1490, 2740 137. b 138. c (page 719) , 1 17 4. 6. 86,100 an 2. an n 2 n! 0.8n 1.4 7. 189 , , 1. 1 5 Chapter Test , 1 11 1 1 14 8 3. 50, 61, 72; 140 5. 5, 10, 20, 40, 80 8. 4 10. 12. (a) 72 14. 26,000 18. 25% 9. Answers will vary. x4 8x3y 24x2y2 32xy3 16y4 (b) 328,440 15. 720 13. (a) 330 17. 1 15 16. 108,864 11. (b) 720,720 3.908 1010 Cumulative Test for Chapters 7–9 (page 720) , , 22. 19. 20. , 1 11 3, 4, 2 1 13 Jogging shoes: $358.48 million Walking shoes: $167.17 million 5, 4 1 , 1 1 9 7 5 25. (a) 65.4 24. 920 26. 3, 6, 12, 24, 48 29. 32. 70 36. 720 z4 12z3 54z2 108z 81 34. 453,600 33. 120 37. (b) 13 9 27. 23. an an 1 4 21. 9 n 1! n 3 3.2n 1.4 28. Answers will vary. 30. 210 35. 151,200 31. 600 Problem Solving (page 725) 1. 1, 1.5, 1.416, 1.414215686, 1.414213562, 1.414213562, . . . 2. xn 3. (a) approaches 8 n (b) If n if is odd, is even, an an and 2, 4. y 2 1 −3 −2 −1 1 3 4 6 7 x (c) 0 0 10 n an 1 2 10 101 1000 10,001 4 2 4 2 − 2 −3 − 4 − 5 −6 −8 1. 3. 5. 7. 1, 2, 3 4, 2, 3 2, 3 4 2, 1 2. 1, 2, 1 4. 6 12 10 (0, 5) 8 6 4 2 (4, 4) (6, 0) x 4 8 10 12 2 (0, 0) Maximum at Minimum at 4, 4: z 20 0, 0: z 0 8. $0.75 mixture: 120 pounds; $1.25 mixture: 80 pounds 9. 3 x2 2x 4 y 1 (d) It is not possible to find the value of an as approaches n infinity. 2n 1 5. (a) 3, 5, 7, 9, 11, 13, 15, 17; an (b) To obtain the arithmetic sequence, find the differences of consecutive terms of the sequence of perfect cubes. Then find the differences of consecutive terms of this sequence. (c) 12, 18, 24, 30, 36, 42, 48; (d) To obtain the arithmetic sequence, find the third sequence obtained by taking differences of consecutive terms in consecutive sequences. an 6n 6 (e) 60, 84, 108, 132, 156, 180; an 24n 36 7. sn an n1 1 2 3 4 9. Answers will vary. 11. (a) Answers will vary. $0.71 15. (a) 13. sn 2 1 3 (b) 17,710 (b) 2.53, 24 turns 333200_10a_AN.qxd 12/9/05 2:42 PM Page A187 Chapter 10 Section 10.1 (page 732) Vocabulary Check (page 732) 2. 1. inclination m1 1 m1m2 m2 3. tan Ax1 4. C By1 A2 B2 3. 1 5. 3 7. 3.2236 1. 9. 3 3 3 4 radians, 135 11. 13. 0.6435 radian, 17. 2.1112 radians, 21. 2.1112 radians, 25. 0.1974 radian, 29. 0.9273 radian, 33. 36.9 121.0 121.0 11.3 53.1 radian, 45 4 15. 1.0517 radians, 60.3 19. 1.2490 radians, 23. 1.1071 radians, 71.6 63.4 27. 1.4289 radians, 31. 0.8187 radian, 81.9 46.9 2, 1 ↔ 4, 4: slope 3 2 4, 4 ↔ 6, 2: slope 1 6, 2 ↔ 2, 1: slope 1 4 2, 1: 42.3; 4, 4: 78.7; 6, 2: 59.0 4, 1 ↔ 3, 2: slope 3 7 3, 2 ↔ 1, 0: slope 1 1, 0 ↔ 4, 1: slope 1 5 4, 1: 11.9; 3, 2: 21.8; 1, 0: 146.3 35. 37. 0 39. 7 5 41. 7 45. (a 43. 47. (a) 837 37 1.3152 2 −1 −1 −2 3537 74 53. (c) 35 8 (b) 4 (c) 8 (b) 31.0 51. 0.1003, 1054 feet 22 33.69; 56.31 49. 55. 57. True. The inclination of a line is related to its slope by m tan . , then the angle is in the second quadrant, where the tangent function is negative. If the angle is greater than but less than 2 59. (a) d 4 m2 1 Answers to Odd-Numbered Exercises and Tests A187 (b) d (c) m 0 6 5 2 1 −4 −3 −2 −1 1 2 3 4 −2 (d) The graph has a horizond 0. tal asymptote at As the slope becomes the distance larger, between the origin and y mx 4, the line becomes smaller and approaches 0. xy- intercepts: intercept: 5 ± 5, 0 0, 20 m 63. 61. xy- intercept: intercept: 65. x- intercepts: 7, 0 0, 49 7 ± 53 2 0, 1 , 0 3 yintercept: f x 3x 1 3, 49 Vertex: f x 6x 1 1 12, 289 Vertex: 1 2 49 3 3 2 289 24 12 24 67. 71. 73. 69. 75. f x 5x 17 Vertex: 17 5 2 324 5 5 , 324 5 y 6 5 2 1 −1 −1 − 12 9 6 3 −3 −3 3 6 9 12 x −4 −3 −2 Section 10.2 (page 740) Vocabulary Check (page 740) 1. conic 4. axis 2. locus 5. vertex 3. parabola; directrix; focus 7. tangent 6. focal chord x 1. A circle is formed when a plane intersects the top or bottom half of a double-napped cone and is perpendicular to the axis of the cone. 3. A parabola is formed when a plane intersects the top or
bottom half of a double-napped cone, is parallel to the side of the cone, and does not intersect the vertex. 7. d 8. f 6. b 5. e 11. Vertex: Focus: Directrix: 0, 0 0, 1 2 y 1 2 9. a 13. Vertex: Focus: Directrix: 10. c 0, 0 3 21 −6 −5 −4 −3 −3 −4 333200_10a_AN.qxd 12/9/05 2:42 PM Page A188 A188 Answers to Odd-Numbered Exercises and Tests 15. Vertex: Focus: Directrix: 0, 0 0, 3 2 y 3 2 17. Vertex: Focus: Directrix: 1, 2 13 −2 −1 1 2 3 4 5 x −3 − 19. Vertex: Focus: Directrix: 2, 2 2, 3 y 1 1, 1 1, 2 21. Vertex: Focus: Directrix 321 −2 2 4 x 23. Vertex: Focus: Directrix: 2, 3 4, 3 x 0 − 10 − 8 − 6 − 4 27. Vertex: Focus: Directrix: 2 1 4, 1 0, 1 2 x 1 2 25. Vertex: Focus: Directrix: 2, 1 2, 1 2 x 2 −14 x 10 4 −12 y 2 −2 − 4 −6 −8 4 45. 49. 53. 55. 59. y 22 8x 5 y 22 8x 51. 47. x2 8 y 4 y 6x 1 3 10 −5 25 −10 2, 4 4x y 8 0; 2, 0 15,000 0 0 225 57. 61. 4x y 2 0; 1 y 1 18 x2 2, 0 x 106 units y 1 640 x2 (b) 8 feet 17,5002 miles per hour x2 16,400 y 4100 x2 64 y 75 63. (a) 65. (a) (b) 67. (a) 69. False. If the graph crossed the directrix, there would exist 24,750 miles per hour (b) 69.3 feet points closer to the directrix than the focus. 71. (a) p = 3 p = 2 21 p = 1 p = 4 −18 18 −3 p increases, the graph becomes wider. As 0, 1, 0, 2, 0, 3, 0, 4 (b) (d) Easy way to determine two additional points on the (c) 4, 8, 12, 16; 4p graph 75. ±1, ±2, ±4 m x1 2p ± 1 2, ±1, ±2, ±4, ±8, ±16 f x x3 7x2 17x 15 81. B 23.67, C 121.33, c 14.89 C 89, a 1.93, b 2.33 A 16.39, B 23.77, C 139.84 B 24.62, C 90.38, a 10.88 73. 77. 79. 83. 85. 87. 89. 1 2, 5 3, ±2 −10 2 Section 10.3 (page 750) 29. 35. 41. x2 3 2 y x2 4y x 32 y 1 31. 37. −4 x2 6y y2 8x 43. y2 8x 33. y2 9x 39. y2 4x 4 Vocabulary Check (page 750) 1. ellipse; foci 3. minor axis 2. major axis; center 4. eccentricity 333200_10a_AN.qxd 12/9/05 2:42 PM Page A189 Answers to Odd-Numbered Exercises and Tests A189 2. c 3. d 4. f 1. b 7. Ellipse Center: Vertices: Foci: 0, 0 ±5, 0 ±3, 0 Eccentricity 11. Ellipse Center: Vertices: 0, 0 0, ±3 0, ±2 Foci: Eccentricity: y 2 3 4 2 1 − 15. Circle Center: Radius: 0, 1 2 3 5. a 9. Circle 6. e 0, 0 Center: Radius: 5 y 6 4 2 −6 −2 2 4 6 x −2 −4 −6 3, 5 13. Ellipse Center: Vertices: 3, 10, 3, 0 Foci: Eccentricity: 3, 8, 3, 2 3 5 y 12 8 6 4 2 −2 −1 1 −1 −2 −3 2, 4 17. Ellipse Center: Vertices: 3, 4, 1, 4 4 ± 3 Foci: , 4 2 Eccentricity: 3 2 y −3 −2 −1 x 1 −1 −2 −3 −4 −5 19. Ellipse Center: Vertices: Foci: 2, 3 2, 6, 2, 0 2, 3 ± 5 21. Circle 1, 2 Center: Radius: 6 y Eccentricity: −6 −4 −2 5 3 y 6 4 2 −2 x 2 −8 −6 6 2 −2 −2 −4 −6 −10 2 4 6 8 x 23. Ellipse Center: Vertices: Foci: 3, 1 3, 7, 3, 5 3, 1 ± 26 6 3 Eccentricity: y 8 4 2 −10 −8 −4 −2 2 4 x 25. Ellipse −6 y Center: Vertices: Foci: 3, 5 2 , 3, 5 9, 5 2 2 3 ± 33, 5 2 3 2 Eccentricity: y 3 2 −3 −2 −1 x 1 −1 −4 6 4 2 −2 −6 −8 2 4 6 10 x 29. Ellipse 2, 1 Center: 7 3, 1 3, 1, 5 Vertices: 15, 1 15, 1, 26 34 Foci: Eccentricity 27. Circle Center: Radius: 1, 1 2 3 333200_10a_AN.qxd 12/9/05 2:42 PM Page A190 A190 31. −6 Answers to Odd-Numbered Exercises and Tests 4 −4 33. −4 6 2 −4 ± 5, 1 Center: Vertices: 1 Foci: 2 1 2, 1 1 2 ± 2, 1 x2 y 2 36 11 y 32 9 1 39. 1 1 x2 36 y 2 32 x 22 1 0, 0 0, ± 5 0, ± 2 1 37. y 2 16 y 2 25 Center: Vertices: Foci: x2 4 21x2 400 x 22 16 x 22 4 1 43. y 32 9 y 42 1 1 1 1 53. x2 16 x2 25 y 42 12 y2 16 1 35. 41. 45. 47. 51. 55. 57. (a) y 59. (a) x2 321.84 y2 20.89 1 (b) 14 (0, 10) −21 21 (−25, 0) x (25, 0) −14 (c) Aphelion: (b) x2 625 (c) Yes x2 0.04 61. (a) (b) y2 100 1 y2 2.56 1 y 2 35.29 astronomical units Perihelion: 0.59 astronomical unit (c) The bottom half − 0.8 − 0.4 0.4 0.8 63. − 2 y x 654 ( − 3 5 5 , −2 ) 2− y 4 −− , ) 67. False. The graph of x24 y4 1 is not an ellipse. The 5 degree of y is 4, not 2. 69. (a) A a20 a (b) x2 196 y 2 36 1 (c) a 8 9 10 11 12 13 A 301.6 311.0 314.2 311.0 301.6 285.9 a 10, circle (d) 350 0 0 20 49. x2 48 x 22 4 y 42 64 y 22 1 1 1 The shape of an ellipse with a maximum area is a a 10 circle. The maximum area is found when so the equa(verified in part c) and therefore tion produces a circle. b 10, 71. Geometric 73. Arithmetic 75. 547 77. 340.15 Section 10.4 (page 760) Vocabulary Check (page 760) 1. hyperbola; foci 3. transverse axis; center 5. Ax2 Cy 2 Dx Ey F 0 2. branches 4. asymptotes 3. a 4. d 2. c 1. b 5. Center: 0, 0 ±1, 0 ± 2, 0 Vertices: Foci: Asymptotes: y y ±x 7. Center: 0, 0 0, ±5 0, ± 106 Vertices: Foci: Asymptotes: 2 1 −1 −2 x 2 −8 −6 −2 9. Center: 1, 2 3, 2, 1, 2 Vertices: Foci: Asymptotes: 1 ± 5 86 10 1 2 3 x y 10 8 6 4 2 −2 −4 −6 −10 y 3 2 1 −4 −5 333200_10a_AN.qxd 12/12/05 11:33 AM Page A191 Answers to Odd-Numbered Exercises and Tests A191 11. Center: 2, 6 Vertices: 2, 17 3 Foci: , 2, 19 3 2, 6 ± 6 13 13. Center: Asymptotes: y 6 ± 2 3 2, 3 3, 3, 1, 3 x 2 2 ± 10, 3 Vertices: Foci: Asymptotes: y 3 ± 3x 2 y 2 −2 2 4 6 −6 −10 −12 −14 y 2 −6 −4 −2 2 4 6 8 −4 −6 −8 x x 55. Parabola b to a, the larger the eccentricity of the hyperbola, c2 a2 b2. The larger the ratio of e ca. 53. Ellipse 51. Parabola 57. Ellipse 59. Circle 61. True. For a hyperbola, 63. Answers will vary. y 1 3x 32 2xx 62 4 71. 65. 69. 73. 1 67. xx 4x 4 22x 34x2 6x 9 751 −2 −3 −4 π 3− 2 0, 0 ± 3, 0 Vertices: Foci: ± 5, 0 Asymptotes: y ± 6 3 x Section 10.5 (page 769) Vocabulary Check (page 769) 1. rotation of axes 2. 3. invariant under rotation Ax2 C y2 Dx Ey F 0 4. discriminant 12 1. 3, 0 3. 3 3 2 33 1 2 , 32 2 , 2 2 5. 7. y2 2 x2 2 1 9. y ± x′ y' 2 2 y 2 1 x' x −2 −1 1 2 x −1 −2 y 4 −2 −3 −4 −8 10 y′ −10 −4 −3 − 11. x 322 16 y 2 2 16 1 15. The graph of this 17. Center: equation is two lines intersecting at y 112 8 −8 2 x 1 23. x2 1 1 27. 1 y 2 25 x 42 4 19. Center: 1, 3 1, 3 ± 2 1, 3 ± 25 Vertices: Foci: Asymptotes: y 3 ± 1 3 21. 25. 29. 33. 37. 1 x2 12 17x2 64 y 2 4 17y 2 1024 y 52 16 y 22 4 x 32 9 x 2 1 x 42 9 1 1 x2 4 y 22 4 35. 1 y 2 1 12 4x 22 9 y 22 1 1 31. y 2 9 x 22 1 39. (a) y 2 1693 3300, 2750 1 43. 41. 45. Circle 47. Hyperbola 2.403 (b) feet 125 1, 0 14.83, 0 49. Hyperbola y 8 6 4 −4 y′ −4 x′ 2 4 6 8 x 333200_10a_AN.qxd 12/9/05 2:42 PM Page A192 A192 Answers to Odd-Numbered Exercises and Tests 13. x2 6 y2 3 2 1 y' −3 y 3 2 −3 x' x 2 3 15. y2 x 17. x 12 6 y 1 6 y′ y 2 x′ − 6 − 4 x 2 y′ y 6 4 2 −2 − 4 −4 2 4 x 41. x′ −2 7 37. (a) Hyperbola (b) (c) y 6x ± 36x2 20x2 4x 22 10 −9 6 −6 9 39. (a) Parabola (b) y 4x 1 ± 4x 12 16x2 5x 3 8 (c) 2 −4 y 6 43. y 4 3 1 −6 −4 −2 2 4 6 x − 4 −3 −2 −1 1 3 4 x −6 −2 −3 − 4 2 47. 49. 0, 8, 12, 8 55. No solution 8, 12 1, 3, 1, 3 2, 2, 2, 4 0, 4 53. 0, 3 , 3, 0 45. 51. 57. 59. True. The graph of the equation can be classified by finding the discriminant. For a graph to be a hyperbola, the k ≥ 1 discriminant must be greater than zero. If then the 4, discriminant would be less than or equal to zero. 61. Answers will vary. 63. y 651 −1 −2 −6 −5 −4 −3 −2 x t −4 −3 −2 −1 1 2 3 4 x 1 2 −2 −3 −4 67. y 691 −1 −2 −3 −4 −3 −2 −1 −1 1 2 3 4 5 t Area 45.11 square units 71. 73. Area 48.60 square units 19. −15 45 23. 10 −10 4 −2 21. 15 −9 6 −6 9 26.57 25. 18 −6 6 −9 27 −6 33.69 31. d 32. c 29. b 30. a −4 31.72 28. f 27. e 33. (a) Parabola (b) y 8x 5 ± 8x 52 416x2 10x 2 (c) −4 35. (a) Ellipse 2 1 −3 (b) (c) y 6x ± 36x2 2812x2 45 14 −4 3 −3 5 333200_10a_AN.qxd 12/9/05 2:42 PM Page A193 Answers to Odd-Numbered Exercises and Tests A193 Section 10.6 (page 776) 7. (a) Vocabulary Check (page 776) 1. plane curve; parametric; parameter 2. orientation 3. eliminating the parameter y 4 3 2 1 1. (a) (bc) y 3 x2 The graph of the rectangular equation shows the entire parabola rather than just the right half. The graph of the rectangular equation continues the graph into the second and third quadrants. 5. (a) y 3. (a) y 6 5 4 2 1 −b) y 2 3 x 3 −2 −1 1 2 x −1 (b) y 16x2 9. (a) y 2 1 −1 −3 −2 −1 −2 (b) y x2 4x 4 11. (a) y 14 12 10 8 6 2 −2 2 4 6 8 10 12 14 x y x 2 3 (b) 15. (a) y 4 3 1 −3 −2 −1 1 2 3 −3 −4 (b) y x 1 x 13. (a) y 4 2 1 −4 −2 −1 1 2 4 −2 −4 (b) y2 9 x2 9 1 17. (a) y 3 2 1 −1 x 1 3 4 5 7 −2 −3 −4 −5 (b) x2 16 y2 4 1 19. (a) (b) x 42 4 21. (a) y 12 1 −1 y 4 3 2 1 −2 −1 −1 −2 −3 −4 1 2 3 4 5 6 x (b) y 1 x3 (b) y ln x 333200_10a_AN.qxd 12/9/05 2:42 PM Page A194 A194 Answers to Odd-Numbered Exercises and Tests 23. Each curve represents a portion of the line y 2x 1. (d) 200 Domain , 1, 1 0, 0, (a) (b) (c) (d) Orientation Left to right Depends on Right to left Left to right 27. 25. 29. 33. mx x1 y y1 x 6t y 3t x 4 cos y 7 sin 31. y k2 b 2 x h2 a2 x 3 4 cos y 2 4 sin x 4 sec y 3 tan 35. x t, y 3t 2 x t, y t 2 (b) x t, y t 2 1 x t, y 1 t (b) 37. (a) 39. (a) 41. (a) 43. (a) 45. 34 x t 2, y 3t 4 (b) x t 2, y t 2 4t 4 (b) x t 2, y 1 x t 2, y t 2 4t 5 t 2 47. 6 0 −6 51. 51 6 −6 55. d 18 6 4 −4 , 49. 0 0 −6 53. b 4 −4 2, 2 Domain: Range: 1, 1 Domain: Range: , 57. (a) 100 Maximum height: 90.7 feet Range: 209.6 feet Maximum height: 136.1 feet Range: 544.5 feet 1 0 0 600 59. (a) x 146.67 cos t y 3 146.67 sin t 16t 2 (b) 50 No 0 0 60 (c) 450 Yes 0 0 500 (d) 19.3 61. Answers will vary. x a b sin 63. y a b cos 65. True x t y t 2 1 ⇒ y x2 1 x 3t y 9t 2 1 ⇒ y x2 1 67. Parametric equations are useful when graphing two functions simultaneously on the same coordinate system. For example, they are useful when tracking the path of an object so that the position and the time associated with that position can be determined. 5, 2 1, 2, 1 71. 69. 73. 75 y 75. 3 y Maximum height: 204.2 feet Range: 471.6 feet 105° θ′ x x θ′ − 2π 3 Maximum height: 60.5 feet Range: 242.0 feet 0 0 (b) 220 0 0 (c) 100 250 500 0 0 300 333200_10b_AN.qxd 12/9/05 2:43 PM Page A195 Section 10.7 (page 783) Vocabulary Check (page 783) 1. pole 3. polar 2. directed distance; directed angle x r cos tan y x y r sin r2 x2 y2 4. 1. π 4, 5. π 3. 1 2 3 4 0 π π 2 3π 2 π 2 3π , 4 3 5 6 , 0, 13 6 0, 7. π 2 3π 2 1 2 3 4 0 π π 2 3π 2 1 2 3 4 0 2, 8.64, 2, 0.78 2 2 2 0, 3 11. 2 , 9. 15. 1.1340, 2.2280 22, 10.99, 22, 7.85 2, 2 6, 19. 17. 13. 2, 5 25. 4 7, 0.8571 4 6, 29. r 4 csc 33. r 3 39. r 35. 2 3 cos sin 313, 0.9828 23. 21. 27. 31. 37. 5, 2.2143 13, 5.6952 17 6 , 0.4900 r 10 sec 41. r2 16 sec csc 32 csc 2 4 1 cos or r 4 1 co
s r a 47. 3x y 0 y 4 57. x2 y 22 6x2y 2y 3 4x2 5y 2 36y 36 0 r 2a cos 53. x2 y2 x23 0 49. x2 y2 16 43. 45. 51. 55. 59. 63. x2 y 2 4y 0 61. x2 4y 4 0 Answers to Odd-Numbered Exercises and Tests A195 65. The graph of the polar equation consists of all points that are six units from the pole. x2 y2 36 y 8 4 2 −8 −4 −2 2 4 8 67. The graph of the polar equation consists of all points on the line that make 6 an angle of positive polar axis. 3 x 3y 0 with the −4 −3 −2 69. The graph of the polar equation is not evident by simple inspection, so convert to rectangular form. x 3 0 −4 −8 y 4 3 2 1 −1 −2 −3 −4 −3 −2 −1 1 2 4 −2 −3 −4 x x x r 73. 71. True. Because is a directed distance, the point r, ± 2n. be represented as x h2 y k2 h2 k2 Radius: Center: h2 k2 h, k 75. (a) Answers will vary. r, can b) (c) 2 1 , r2, r1, and the pole are collinear. r1 d r1 2 r2 2 2r1r2 This represents the distance between two points on the line d r1 This is the result of the Pythagorean Theorem. 2. 2 1 r2 2 r2 (d) Answers will vary. For example: 3, 6, 4, 3 Points: Distance: 2.053 Points: Distance: 2.053 3, 76, 4, 43 77. 79. 2 log6 x log6 z log6 3 log6 y x ln x 2 lnx 4 3y 8 2, 3, 3 7, 88 2, 3 log7 81. 83. 35, 8 89. 93. Collinear 5 87. 85. 91. Not collinear ln xx 2 333200_10b_AN.qxd 12/9/05 2:43 PM Page A196 A196 Answers to Odd-Numbered Exercises and Tests Section 10.8 (page 791) Vocabulary Check (page 791) 2. polar axis 3. convex limaçon 5. lemniscate 6. cardioid 1. 2 4. circle 1. Rose curve with 4 petals 5. Rose curve with 4 petals 3. Limaçon with inner loop 7. Polar axis 9. 2 11. , 2 13. Maximum: r 20 when polar axis, pole 3 2 0 1 2 3 31. π 35. π 2 π 2 3π 2 0 2 4 29. π 33. 37. π π π 2 3π 2 π 2 3π 2 π 2 3π 2 0 4 π π 39. 1 2 3 0 1 3 0 3π 2 π 2 0 4 10 5 3π 2 4 −10 5 −5 41. 6 0 2 45. −4 −4 −6 3 43. −11 14 47. 5 −10 3π 2 π 2 3π 2 π 2 −3 2 49. 0 ≤ < 2 51. 3 −3 3 −4 5 2 4 6 8 0 3π 2 −2 0 ≤ < 4 −3 0 ≤ < 0, , , 6 2 3 5 6 19. , 2 3 π 2 0 1 2 Zero: r 0 when 2 15. Maximum: r 4 when Zero: r 0 when 17. 21. 25. π π π π 2 3π 2 π 2 3π 2 π 2 3π 23. 27. π π π 333200_10b_AN.qxd 12/9/05 2:43 PM Page A197 53. −6 4 −4 55. 6 −3 4 −2 5 57. True. For a graph to have polar axis symmetry, replace r, . (b) r, 59. (a) r, or by 3π 2 3π 2 Upper half of circle Lower half of circle (c) π 2 (d 3π 2 3π 2 Full circle Left half of circle 61. Answers will vary. 63. (a) r 2 2 2 sin cos (b) r 2 cos (c) 65. (a) r 2 sin (d) r 2 cos π 2 (b 3π 2 k 0, k 1, k 2, k 3, circle convex limaçon cardioid limaçon with inner loop 3π 2 7 677 8 k = 0 −3 71. 13 5 69. ±3 Answers to Odd-Numbered Exercises and Tests A197 73. x 12 9 y 22 4 1 y 5 3 2 1 −1 −1 −2 −3 −5 −4 −3 −2 x 1 2 3 Section 10.9 (page 797) Vocabulary Check (page 797) 1. conic 4. (a) iii 2. eccentricity; (b) i (c) ii e 3. vertical; right 1. e 1: r parabola 4 1 cos , 2 1 0.5 cos , 6 1 1.5 cos , e 0.5: r e 1.5: r e = 1 7 e = 0.5 −6 e = 1.5 15 ellipse hyperbola 7 3. e 1: r e 0.5: r e 1.5: r parabola 4 1 sin , 2 1 0.5 sin , 6 1 1.5 sin , ellipse hyperbola −16 e = 1 6 e = 0.5 17 e = 1.5 −16 5. f 6. c 7. d 8. e 9. a 10. b 333200_10b_AN.qxd 12/9/05 2:44 PM Page A198 A198 Answers to Odd-Numbered Exercises and Tests 11. Parabola π 2 13. Parabola π 2 31. − 15. Ellipse 3π 2 π 2 3π 2 17. Ellipse 3π 2 3π 2 19. Hyperbola 21. Hyperbola π 2 3π 2 π 0 1 25. −3 1 2 3 5 0 2 −2 2 π 2 3π 2 π 23. Ellipse π −4 27. Ellipse 6 3 −7 33. r 37. r 41. r 45. r 1 1 cos 2 1 2 cos 10 1 cos 20 3 2 cos 35. r 1 2 sin 39. r 43. r 47. r 2 1 sin 10 3 2 cos 9 4 5 sin 49. Answers will vary. r 9.5929 107 51. 1 0.0167 cos 9.4354 107 9.7558 107 Perihelion: Aphelion: r 1.0820 108 1 0.0068 cos miles miles 1.0747 108 1.0894 108 Perihelion: Aphelion: r 1.4039 108 1 0.0934 cos kilometers kilometers 53. 55. Perihelion: Aphelion: 1.2840 108 1.5486 108 0.624 1 0.847 sin 2 ; 57. r miles miles r 0.338 astronomical unit 59. True. The graphs represent the same hyperbola. 61. True. The conic is an ellipse because the eccentricity is less 3 than 1. 63. Answers will vary. 65. r2 24,336 169 25 cos2 67. r2 144 25 cos2 9 69. r2 144 25 sin2 16 0 1 π 2 3π 2 1 −3 Parabola 71. (a) Ellipse (b) The given polar equation, r, the left of the pole. The equation, directrix to the right of the pole, and the equation, has a horizontal directrix below the pole. has a vertical directrix to has a vertical r2, r1, 29. 9 −3 −3 15 (c) r 2 = 4 1 − 0.4 sin θ 10 −12 12 −6 r 1 = 4 1 + 0.4 cos θ r = 4 1 − 0.4 cos θ 333200_10b_AN.qxd 12/12/05 11:35 AM Page A199 Answers to Odd-Numbered Exercises and Tests A199 n 75. n, 3 2 10 2 3 n 81. 72 10 79. n 2 sin 2u 24 25 cos 2u 7 25 6 73. 77. 83. 85. tan 2u 24 7 4 n 1 1 an 4 87. an 9n 89. 220 91. 720 Review Exercises (page 801) 1. 4 radian, 45 3. 1.1071 radians, 63.43 5. 0.4424 radian, 25.35 7. 0.6588 radian, 37.75 9. 22 11. Hyperbola 13. y 2 16x 15. y 22 12 43 5 x −4 −3 −2 −1 −2 − 3 17. 21. y 2x 2; 1, 0 x 22 y2 21 25 y 1 19. 86 meters x 22 4 y 23. y 12 1 35. Center: 3, 5 7, 5, 1, 5 3 ± 25, 5 Vertices: Foci: Asymptotes: y 5 ± 1 2 x 3 37. Center: 1, 1 5, 1, 3, 1 6, 1, 4, 1 Vertices: Foci: Asymptotes: y 1 ± 3 4 y 2 −2 2 4 6 8 −8 −10 y 6 4 2 x 1 −6 −4 4 6 8 −4 −6 −8 39. 72 miles x2 8 45. y2 8 y′ y 4 3 2 41. Hyperbola 1 47. x′ y′ 43. Ellipse x2 3 y2 2 y 2 1 1 x′ −4 −3 −2 x 2 3 4 −2 −1 1 2 x −2 −3 49. (a) Parabola −1 −2 y 24x 40 ± 24x 402 3616x2 30x 18 (b) (c 10 10 x − 6 − 8 − 10 4 3 2 1 −2 −1 −1 −2 −3 1 2 3 4 5 x −3 7 −1 9 25. The foci occur 3 feet from the center of the arch on a line connecting the tops of the pillars. 27. Center: 2, 1 29. Center: 1, 4 1, 0, 1, 8 1, 4 ± 7 7 4 Vertices: Foci: Eccentricity: Vertices: 2, 11, 2, 9 Foci: 2, 1 ± 19 Eccentricity: 19 10 31. y2 x2 8 1 33. 5x 42 16 5y2 64 1 51. (a) Parabola (b) y (c) 2x 22 ± 2x 222 4x2 22x 2 2 7 1 −1 −11 333200_10b_AN.qxd_pg A200 1/9/06 9:01 AM Page A200 A200 Answers to Odd-Numbered Exercises and Tests 53. t x y 3 11 2 8 1 5 0 2 19 15 11 69. 75. 81. 85. 3 2 1 , 2 213, 0.9828 r2 10 csc 2 x2 y2 3x 71. , 32 2 32 2 r 7 79. 77. x2 y 2 25 83. x2 y2 y23 87. 73. 2, 2 r 6 sin y 20 16 12 4 x 8 12 − 12 − 8 − 4 − 4 − 8 55. (a 57. (a 2x (b) 59. (ab) (b) y 4x x2 y 2 36 61. 65 cos y 4 6 sin 63. x 3 tan y 4 sec π 67 3π 2 2, , 2, 9 4 5 4 7, 1.05, 7, 10.47 89. Symmetry: , 2 Maximum value of r No zeros of π 2 polar axis, pole r: r 4 for all values of π 0 2 3π 2 91. Symmetry: , 2 polar axis, pole Maximum value of r: r 4 when , Zeros of r: r 0 when 0, , 2 , 3 2 π π 2 3π 2 0 4 93. Symmetry: polar axis r: Maximum value of r: when Zeros of r 4 r 0 π 2 when 0 2 4 6 8 0 π 0 2 3π 2 333200_10b_AN.qxd 12/9/05 2:44 PM Page A201 Answers to Odd-Numbered Exercises and Tests A201 95. Symmetry: 2 Maximum value of r: Zeros of r: r 0 π 2 when when r 8 2 3.4814, 5.9433 π 2 4 6 0 3π 2 97. Symmetry: , 2 polar axis, pole Maximum value of r: r 3 , 4 2 , , when 0, 3 , 4 5 , 4 7 4 when Zeros of r: r 0 π π 2 3π 2 0 4 99. Limaçon 101. Rose curve −16 8 −8 8 −6 4 −4 103. Hyperbola 105. Ellipse π 2 1 3π 2 π π 0 π 2 3π 2 1 3 4 0 107. r 4 1 cos 109. r 5 3 2 cos 111. r 7978.81 1 0.937 cos ; 11,011.87 miles 113. False. When classifying an equation of the form Ax2 Bxy Cy2 Dx Ey F 0, its graph can be determined by its discriminant. For a graph to be a parabola, its discriminant, must equal zero. So, if C 115. False. The following are two sets of parametric equations B2 4AC, A then or equals 0. B 0, for the line. x t, y 3 2t x 3t, y 3 6t 117. 5. The ellipse becomes more circular and approaches a circle of radius 5. 119. (a) The speed would double. (b) The elliptical orbit would be flatter; the length of the major axis would be greater. 121. (a) The graphs are the same. (b) The graphs are the same. 3 2 Chapter Test 1. 0.2783 radian, (page 805) 15.9 2. 0.8330 radian, 47.7 3. 72 2 4. Parabola: Vertex: Focus: y 1, 0 2, 0 y2 4x 2 −1 2 3 4 5 6 x −2 −3 −4 6 5. Hyperbola: x 22 4 y2 1 2, 0 0, 0, 4, 0 2 ± 5, 0 Center: Vertices: Foci: Asymptotes2, 0) −4 2 6 8 −4 −6 x 1 16. π 2 17. π 2 333200_10b_AN.qxd 12/9/05 2:44 PM Page A202 A202 Answers to Odd-Numbered Exercises and Tests 6. Ellipse: y 12 9 x 32 16 3, 1 1, 1, 7, 1 3 ± 7, 1 Center: Vertices: Foci: y 6 4 2 −8 −4 −2 2 x −2 −4 7. Circle: Center: y x 22 y 12 1 2 2. 5 y 22 4 5x2 16 1 8. x 32 3 2 45 10. (a) (b) y′ y 6 4 x 12. x 6 4t y 4 7t 11 3π 2 Parabola 18. π π 2 3π 2 3π 2 π 2 Ellipse 19. π 0 3 4 0 2 4 3π 2 Limaçon with inner loop Rose curve 20. Answers will vary. For example: r 1 1 0.25 sin 21. Slope: 0.1511; Change in elevation: 789 feet 22. No; Yes Problem Solving (page 809) 1. (a) 1.2016 radians (b) 2420 feet, 5971 feet 3. y2 4px p 5. (a) Since d1 by definition, the outer bound that the boat can travel is an ellipse. The islands are the foci. ≤ 20, dz (b) Island 1: Island 2: 6, 0; 6, 0 (c) 20 miles; Vertex: y2 64 x2 100 1 (d) 10, 0 7. Answers will vary. 9. Answers will vary. For example: x cost y 2 sint y2 x21 x 1 x (b) r cos 2 sec 11. (a) (c) x 22 9 3, 1 22, 7 4 r 4 sin 13. 14. 15. y 2 4 1 , 22, , 22, 3 4 4 −3 13. Circle 2 −2 3 333200_App_AN.qxd 12/9/05 2:45 PM Page A203 Answers to Odd-Numbered Exercises and Tests A203 15. 4 −6 6 −6 −4 n ≥ 1, n ≤ 1, n 0, For For For a bell is produced. a heart is produced. a rose curve is produced. 4 −4 6 Appendix A Appendix A.1 (page A8) Vocabulary Check (page A8) 1. rational 4. composite 7. terms 2. irrational 5. prime 8. coefficient 3. absolute value 6. variables; constants 9. Zero-Factor Property 29. (a) denotes the set of all real numbers greater than 2, 5 or equal to (b) 2 and less than 5. x (c) Bounded 31. 37. 47. 53. 61. 63 35. 43. 128 75 1 51. 57. 6 y ≥ 0 10 ≤ t ≤ 22 1 45. 5 5 59. 33. 39. 10 41. 5 3 > 3 55. 51 − 1 2 < x ≤ 4 W > 65 1 49. 2 2 $113,356 $112,700 $656 > $500 0.05$112,700 $5635 Because the actual expenses differ from the budget by more than $500, there is failure to meet the “budget variance test.” $37,335 $37,640 $305 < $500 0.05$37,640 $1882 Because the difference between the actual expenses and the budget is less than of the budgeted amount, there is compliance with the “budget variance test.” and less than $500 5% 65. (a) (e) 2 1. (a) 5, 1, 2 2, 2 (b) 0, 5, 1, 2 9, 5, 0, 1, 4, 2, 1
1 7 3, 9, 5, 0, 1, 4, 2, 11 13, 1, 6 (b) 1 (c) 2.01, 13, 1, 6, 0.666 . . . 0.010110111 . . . 6 (b) 3, 8 1 6 3, 8 3, 7.5, 1, 8, 22 11. 0.123 3, 6 (c) 9. 6 (c) (d) 3. (a) 1 (d) (e) 5. (a) (d) 7. 0.625 13. −8 − 7 − 6 −5 −4 15. 3 2 3, 1, 8, 22 (e) 1 < 2.b 17. 19. (a) (b) 21. (a) (b) 23. (a) (b) 25. (a) (b) 27. (a) (b) 1 0 x ≤ 5 equal to 5. denotes the set of all real numbers less than or 0 1 2 3 4 5 6 x (c) Unbounded x < 0 denotes the set of all real numbers less than 0. x (c) Unbounded 1 2 denotes the set of all real numbers greater than 0 − 1 − 2 4, or equal to 4c) Unbounded 2 < x < 2 2 than and less than 2. denotes the set of all real numbers greater x 2 (c) Bounded 1 0 − 1 −2 1 ≤ x < 0 than or equal to denotes the set of all real numbers greater 1 and less than 0. − 1 0 x (c) Bounded Year 1960 1970 1980 1990 2000 240 192 144 96 48 ( Expenditures (in billions) Surplus or deficit (in billions) $0.3 (s) $2.8 (d) $73.8 (d) $221.2 (d) $236.4 (s92.2 $195.6 $590.9 $1253.2 $1788. Year and 4 are the terms; 7 is the coefficient. 3 are the terms; and 8 are the 1 are the terms; 4 and are the coefficients. 2 6 0 (b) 0 (b) 14 2 y ≥ 6 67. 71. 73. 75. 69. x 5 ≤ 3 326 351 25 miles 7x 3x2, 8x, and 11 coefficients. 4x3, x2, and 5 81. (a) is undefined. 77. 10 79. (a) (b) 83. (a) Division by 85. Commutative Property of Addition 87. Multiplicative Inverse Property 89. Distributive Property 91. Multiplicative Identity Property 333200_App_AN.qxd 12/12/05 11:41 AM Page A204 A204 Answers to Odd-Numbered Exercises and Tests 93. Associative Property of Addition 95. Distributive Property 97. 1 2 105. (a) 99. 3 8 n 5n 1 5 101. 48 103. 5x 12 0.5 0.01 0.0001 0.000001 10 500 50,000 5,000,000 (b) The value of 5n 1 b 109. (a) No. If one variable is negative and the other is approaches infinity as n approaches 0. 1 a 107. False. If a b 0. a < b, where then > , (b) positive, the expressions are unequal. u v ≤ u v The expressions are equal when u and v have the same sign. If u and v differ in sign, is less than u v. u v 111. The only even prime number is 2, because its only factors are itself and 1. 113. (a) Negative 115. Yes. a a if a < 0. (b) Negative Appendix A.2 (page A20) Vocabulary Check (page A20) 1. exponent; base 3. square root 5. index; radicand 7. conjugates 9. power; index 2. scientific notation 4. principle nth root 6. simplest form 8. rationalizing 5x3 y 2 222 (b) 59. (a) 0.011 (b) 0.005 61. (a) 4 (b) 2 53 x 63. (a) 22 (b) 3 32 65. (a) 6x2x (b) 18z z2 67. (a) 2x 32x 2 (b) 71. (a) 2x (b) 4y 342 13x 1 (b) 69. (a) 73. (a) 75 11 81. 185x 77. 83. 5 > 32 22 2 2 87. 912 89. 532 79. 85. 91. 2 35 3 21613 93. 8134 99. (a) 3 (b) 3x 12 101. (a) 2 42 (b) 82x 103. 95. 2 x 97.57 seconds 105. (a.93 5.48 7.67 9.53 11.08 12.32 8 9 10 11 12 13.29 14.00 14.50 14.80 14.93 14.96 (b) t → 8.643 14.96 107. True. When dividing variables, you subtract exponents. 109. using the property am an amn: a0 1, a 0, am am amm a0 1. (b) 81 9. (a) 243 64 1600 54 27. 13. 21. 5x6 1 (b) 15. 2.125 23. 1 24y2 (b) 3x2 111. When any positive integer is squared, the units digit is 0, 1, 4, 5, 6, or 9. Therefore, 5233 is not an integer. Appendix A.3 (page A31) Vocabulary Check (page A31) 8 8 8 8 8 1. 4.96 3. 5. (a) 27 9 7. (a) 1 (b) 5 11. (a) (b) 4 6 24 25. (a) 19. 6 (b) 17. 125z3 7 x (b) 29. (a) 33. (a) 2x3 x y2 4 3 31. (a) 1 (b) 1 4x 4 (b) 10 x square miles 35. (a) 33n (b) b5 a 5 gram per cubic centimeter 5.73 107 8.99 105 37. 39. 41. 4,568,000,000 ounces 43. 0.00000000000000000016022 coulomb 45. (a) 50,000 47. (a) 954.448 49. (a) 67,082.039 3 51. (a) 3 2 55. (a) (b) 2 3.077 1010 (b) 39.791 53. (a) (b) 57. (a) 7.550 (b) 200,000 4 (b) (b) 27 8 1 8 n; an; a0 2. descending 1. 3. monomial; binomial; trinomial 4. like terms 5. First terms; Outer terms; Inner terms; Last terms 6. factoring 7. completely factored 2. e 3. b 4. a 1. d 7. 11. (a) 2x3 4x2 3x 20 2x5 14x 1 5. f 15x 4 1 6. c 9. (b) Degree: 5; Leading coefficient: (c) Binomial 7.225 (b) 1 2 333200_App_AN.qxd 12/9/05 2:45 PM Page A205 13. (a) 3x4 2x2 5 (b) Degree: 4; Leading coefficient: (c) Trinomial x5 1 15. (a) 3 (b) Degree: 5; Leading coefficient: 1 (c) Binomial 17. (a) 3 (b) Degree: 0; Leading coefficient: 3 (c) Monomial 19. (a) 4x5 6x4 1 (b) Degree: 5; Leading coefficient: (c) Trinomial 4x3y 21. (a) 4 (b) Degree: 3; Leading coefficient: 4 (c) Monomial 23. Polynomial: 25. Not a polynomial because it includes a term with a nega- 3x3 2x 8 29. 2x 10 8.3x3 29.7x2 11 39. 1 45. 15z 2 5z 2x2 12x 6x 2 7x 5 x2 100 55. x2 4y 2 4x 2 20xy 25y 2 63. 8x3 12x 2y 6xy 2 y 3 m 2 n 2 6m 9 71. 4r 4 25 2.25x2 16 x y 1 53. 43. 49. 59. 67. 75. 37. y 4 y 3 y 2 33. 3x3 6x 2 3x 7.5x3 9x tive exponent 27. Polynomial: 31. 35. 41. 47. 51. 57. 61. 65. 69. 73. 77. 81. 87. 91. 95. 99. 103. 107. 111. 115. 121. 125. 127. 129. 133. 137. 141. 145. 149. 153. 157. 3x3 2x 2 12z 8 4x 4 4x x 2 7x 12 x 4 x 2 1 4x 2 12x 9 x3 3x 2 3x 1 16x6 24x3 9 x2 2xy y2 6x 6y 9 1 4x2 3x 9 9x2 4 1.44x2 7.2x 9 79. 2x2 2x u4 16 83. 85. 3x 2 x2 25 x 5 89. 2xx 2 3 x 1x 6 x 3x 1 x 8 97. 2xx2 4x 10 1 2 3 x 9x 9 4x 1 4x 1 3u 2v3u 2v 2t 12 117. x 2 2 123. y 4 y 2 4y 16 2t 14t 2 2t 1 u 3vu2 3uv 9v2 s 3s 2 x 20x 10 5x 1x 5 x 1x2 2 3 x2 x3 x 23x 4 3x 15x 2 139. 143. 147. 151. 155. 159. 1 2 101. 109. 135. 105. 93. 3 3 3 x 6x 3 24y 34y 3 x 1x 3 x 22 119. 113. 5y 12 x 2x2 2x 4 3u 4v2 x 2x 1 131. y 5 y 4 3x 2x 1 3z 23z 1 2x 1x2 3 3x2 12x 1 2x 13x 2 6x 3x 3 Answers to Odd-Numbered Exercises and Tests A205 x 2x 4x 2x 4 3 4x23 60x 1 2x2 9x 1x 1 3x 1x2 5 x2 3x 12 185. 165. 177. x 12 169. 173. 1 4 x2x 4 163. 2xx 1x 2 x 36x 18 1 81 xx 4x2 1 t 6t 8 181. 5x 2x2 2x 4 51 x23x 24x 3 x 22x 137x 5 3x6 143x 2233x6 20x5 3 14, 14, 2, 2 161. 167. 171. 175. 179. 183. 187. 189. 191. 193. 195. 197. Two possible answers: 2, 199. Two possible answers: P 22x 25,000 201. (a) 500r 2 1000r 500 203. (a) (b) r % 21 2 11, 11, 4, 4, 1, 1 12 2, 4 (b) $85,000 3% 4% 5001 r 2 $525.31 $530.45 $540.80 r 41 2 % 5% 5001 r 2 $546.01 $551.25 (c) The amount increases with increasing r. 205. (a) (b) V 4x3 88x2 468x x (cm) 1 2 3 cm3 V 384 616 720 207. 211. 44x 308 x 209. (a) 3x 2 8x (b) 30x 213 215. 4r 1 217. 46 x6 x 219. (a) hR rR r (b) V 2R r 2 R rh 221. False. 4x2 13x 1 12x3 4x2 3x 1 333200_App_AN.qxd 12/12/05 11:42 AM Page A206 A206 Answers to Odd-Numbered Exercises and Tests a2 b2 a ba b 229. 223. True. 227. 231. 233. Answers will vary. Sample answer: x 3 8x 2 2x 7 x3n y2n is completely factored. x2 3 225. m n xn ynxn yn Appendix A.4 (page A42) Vocabulary Check (page A42) 1. domain 4. smaller 2. rational expression 5. equivalent 6. difference quotient 3. complex 81. (a) 9.09% (b) 1 x 4x h 4 73. 1 x 2 x 71. 75. 77. x 0 x 22x 1, 1 16 79. (a) minute (b) x 16 minute(s) (c) 60 16 15 4 minutes 288MN P NMN 12P; 9.09% 83. (a 10 75 55.9 48.3 45 43.3 42.3 12 14 16 18 20 22 41.7 41.3 41.1 40.9 40.7 40.6 (b) The model is approaching a T-value of 40. 85. False. In order for the simplified expression to be equivalent to the original expression, the domain of the simplified expression needs to be restricted. If n is even, x 1, 1. x 1. 87. Completely factor each polynomial in the numerator and in the denominator. Then conclude that there are no common factors. If n is odd, Appendix A.5 (page A56) Vocabulary Check (page A56) 2. solve ax b 0 1. equation 4. 6. quadratic equation 7. factoring; extracting square roots; completing the 3. identities; conditional 5. extraneous square; Quadratic Formula 3. Conditional equation 9. Conditional equation 1. Identity 7. Identity 9 13. 6 4 21. 5 27. No solution. The x-terms sum to zero. 31. 4 15. 5 23. 25. 9 17. 9 33. 3 35. 0 19. No solution 5. Identity 11. 4 29. 10 37. No solution. The variable is divided out. 39. No solution. The solution is extraneous. 41. 2 45. 0 49. 53. 59. 67. 43. No solution. The solution is extraneous. 47. All real numbers x x2 6x 6 0 51. 55. 57. 20 2, 6 ± 11 2x2 8x 3 0 3x2 90x 10 0 5 61. a 69. 65. 73. ±33 3, 1 2 ±7 0, 1 2 71. 63. 4, 2 3 , 4 1. All real numbers 5. All real numbers x such that 7. All real numbers x such that x 2 x ≥ 1 9. 3. All nonnegative real numbers 3x, x 0 4y , y 1 2 5 19. 13. , x 0 , x 5 3x 2 1 2 xx 3 x 2 x2 1 x 2 , x 2 , x 2 3y y 1 , x 0 15. y 4, y 4 , y 3 23. y 4 y 6 27 2x 3 x 3 x 1 1 2 3 Undef The expressions are equivalent except at x 3. 31. The expression cannot be simplified. 5x 2, x 1 , r 0 35. 33. 1 4 r 1 r , r 1 39. x 6x 1 x2 , x 6 t 3 t 3t 2 , t 2 x 5 x 1 43. 45. 6x 13 x 3 2 x 2 49. x2 3 x 1x 2x 3 2 x x2 1 , x 0 53. The error was incorrect subtraction in the numerator. 57. xx 1, x 1, 0 , x 2 1 2 2x 1 2x , x > 0 61. 2x3 2x2 5 x 112 1 xx h, h 0 67. 63. 1 x2 15 x7 2 x2 3x 1 3 , x 0 11. 17. 21. 25. 29. 37. 41. 47. 51. 55. 59. 65. 69. 333200_App_AN.qxd 12/9/05 2:45 PM Page A207 75. 8, 16 77. 2 ± 14 79. 1 ± 32 2 81. 2 87. 1 ± 83. 6 3 4, 8 85. 11 6, 11 6 89. 2 ± 23 91. 5 ± 89 4 93. 1 2, 1 95. 1 4, 3 4 97. 1 ± 3 99. 7 ± 5 101. 4 ± 25 103. ± 2 3 7 3 105. 4 3 107. 1 2 113. 6 ± 11 115. 119. 123. 1.355, 14.071 0.290, 2.200 ± 2 109. 265 8 ± 3 8 121. 125. 2 7 111. 2 ± 6 2 117. 0.976, 0.643 1.687, 0.488 1 ± 2 127. 6, 12 129. 1 2 ± 3 131. 1 2 133. ± 3 4 97 4 Answers to Odd-Numbered Exercises and Tests A207 195. 500 units 197. False. x3 x 10 3x x2 10 The equation cannot be written in the form ax b 0. 199. False. See Example 14 on page A55. 201. Equivalent equations have the same solution set, and one is derived from the other by steps for generating equivalent equations. 2x 5, 2x 3 8 203. Yes. The student should have subtracted 15x from both sides to make the right side of the equation equal to zero. Factoring out an x shows that there are two solutions, x 0 x 6. and x2 3x 18 0 x2 2x 1 0 x 0, b a x2 22x 112 0 a 9, b 9 207. 211. x 0, 1 205. 209. 213. (a) (b) 137. ±3 139. 6 141. 3, 0 Appendix A.6 (page A66) 135. 143. 149. 157. 165. 173. 0, ± 32 2 3, 1, 1 ± 1 2, ±4 16 159. 3 ± 162 3 ± 21 6 145. 151. ±1 1, 2 147. ± 3, ±1 153. 50 155. 26 2, 5 167. 161. 0 163. 9 ± 14 169. 1 171. 2, 3 2 175. 4, 5 177. 1 ± 31 3 1 17 2 179. 3, 2 181. 3, 3 183. 3, 185. (a) 61.2 inches (b) Yes. The
estimated height of a male with a 19-inch femur is 69.4 inches. (c) Height, x Female femur length Male femur length 60 70 80 90 100 110 15.48 19.80 24.12 28.44 32.76 37.08 14.79 19.28 23.77 28.26 32.75 37.24 (d) 100 inches x 100.59; There would not be a problem because it is not likely for either a male or a female to be 100 inches tall (which is 8 feet 4 inches tall). 187. 189. 191. after about 28 hours y 0.25t 8; 6 inches 6 inches 2 inches 203 3 11.55 inches 193. (a) 1998 (b) During 2007 A P P E N D I X A Vocabulary Check (page A66) 1. solution set 4. solution set 2. graph 5. double 3. negative 6. union Bounded 3. x > 11. Unbounded Unbounded 1 ≤ x ≤ 5. x < 2. 8. f 1. 5. 7. b 13. (a) Yes 15. (a) Yes 17. (a) Yes 19. x < 3 9. d (b) No (b) No (b) Yes 1 2 3 4 5 12. a 10. c (c) Yes (c) No (c) Yes 21. x 11. e (d) No (d) Yes (d) No x < 3 2 3 2 −2 −1 0 1 2 3 x 23. x ≥ 12 25. x > 2 10 11 12 13 14 27 31. x ≥ 4 2 3 4 5 6 35. x ≥ 4 − 6 39 < 15 9 2 − 9 2 − 2 15 29. x < 5 3 4 5 6 7 33. x ≥ 2 0 1 2 3 4 37. 1 < x < 3 −1 416 −4 − 333200_App_AN.qxd 12/9/05 2:45 PM Page A208 A208 Answers to Odd-Numbered Exercises and Tests 45. 6 < x < 6 x −6 −4 −2 0 2 4 6 x 49. No solution 53. x ≤ 3 2, x ≥ 3 − 3 2 −2 −1 1 0 4 < x < 5 57 43. 10.5 ≤ x ≤ 13.5 10.5 13.5 10 11 12 13 14 47. x < 2, x > 2 2 3 −3 51. 0 −1 −2 1 14 ≤ x ≤ 26 26 14 10 15 20 25 30 55. x ≤ 5, x ≥ 11 11 5 10 15 −15 59. 0 −10 −5 x ≤ 29 2 , x ≥ 11 2 − 29 2 − 11 2 −16 −12 −8 −4 x x x x 61. 10 63. −10 10 −10 −10 x > 2 65. 10 x ≤ 2 67. −10 24 −15 10 − 10 −10 6 ≤ x ≤ 22 x ≤ 27 2 , x ≥ 1 2 −a) (b) (a) (b) −5 x ≥ 2 x ≤ 3 2 8 73. −5 10 −2 1 ≤ x ≤ 5 x ≤ 1, x ≥ 7 (a) (b) 5, 77. 3, , 7 75. 2 81. All real numbers within eight units of 10 83. 85. x ≤ 3 x 7 ≥ 3 x 12 < 10 79. 87. x 3 > 4 r > 3.125% 134 ≤ x ≤ 234 89. 93. 97. 99. (a) 91. x > 6 x ≥ 36 95. 5 75 0 150 x ≥ 129 1 ≤ t ≤ 10 t > 16 106.864 square inches ≤ area ≤ 109.464 square inches (b) 101. (a) 103. 105. Might be undercharged or overcharged by $0.19. 107. 13.7 < t < 17.5 (b) 13.7 17.5 t 12 13 14 15 16 17 18 19 20 ≤ h ≤ 80 109. 111. False. c has to be greater than zero. 113. b 10 Appendix A.7 (page A75) 10 1 −10 Vocabulary Check (page A75) 1. numerator 2. reciprocal 1. Change all signs when distributing the minus sign. 2x 3y 4 2x 3y 4 3. Change all signs when distributing the minus sign. 4 14x 1 4 16x 2x 1 numerator and denominator separately. ax y x 9 ax y cannot be simplified. 9. 11. Divide out common factors, not common terms. 2x2 1 5x cannot be simplified. 13. To get rid of negative exponents: 1 a1 b1 1 a1 b1 ab ab ab b a . 15. Factor within grouping symbols before applying exponent to each factor. x2 5x12 xx 512 x12x 512 69. 3 71. 6 −4 8 5. z occurs twice as a factor. 5z6z 30z2 7. The fraction as a whole is multiplied by a, not the 333200_App_AN.qxd 12/9/05 2:45 PM Page A209 17. To add fractions, first find a common denominator. Answers to Odd-Numbered Exercises and Tests A209 , 29. x1 4x4 7x2x13 2x2 x 15 49 16 3x 1 3y 4x xy 21. 25 9 37. 4 3 y x 3x 2 1 2x 2 1 7x 4 3 4x83 7x53 1 x13 7x2 4x 9 x2 33x 14 1 x 323x 274 3x 21215x2 4x 45 2x2 512 55. 47. 51. 19. 27. 35. 41. 45. 49. 53. 59. 23. 1 3 25. 2 33. 1 5x 31. 1, 2 39. 3x22x 13 43. 16 x 5 x 3 x12 5x32 x72 27x2 24x 2 6x 14 4x 3 3x 143 61. (a) (b) (c) 63. 65. x t x 0.5 1.0 1.5 2.0 1.70 1.72 1.78 1.89 2.5 3.0 3.5 4.0 t 2.18 2.57 2.36 2.02 x 0.5 mile 3xx2 8x 20 x 4x2 4 6x2 4x2 8x 20 y2 x 1 xy2 y2 True. x1 y2 1 x True. x 4 x 16 57. x x2 67. Add exponents when multiplying powers with like bases. xn x3n x4n 69. When a binomial is squared, there is also a middle term. xn yn2 x2n 2xnyn y2n x2n y2n 71. The two answers are equivalent and can be obtained by factoring. 1 10 2x 132 2x 13262x 1 10 2x 13212x 4 2x 1323x 1 2x 1323x 1 2x 152 1 6 1 60 1 60 4 60 1 15 2x 332x 1 2 5 (a) (b) 8 15 4 x32x 1 333200_Index_SE.qxd 12/8/05 11:32 AM Page A211 Index A Absolute value of a complex number, 470 inequality, solution of, A63 properties of, A4 of a real number, A4 Acute angle, 283 Addition of a complex number, 163 of fractions with like denominators, A7 with unlike denominators, A7 of matrices, 588 vector, 449 properties of, 451 resultant of, 449 Additive identity for a complex number, 163 for a matrix, 591 for a real number, A6 Additive inverse, A5 for a complex number, 163 for a real number, A6 Adjacent side of a right triangle, 301 Adjoining matrices, 604 Algebraic expression, A5 domain of, A36 equivalent, A36 evaluate, A5 term of, A5 Algebraic function, 218 Algebraic tests for symmetry, 19 Alternative definition of conic, 793 Alternative form of Law of Cosines, 439, 490 Amplitude of sine and cosine curves, 323 Angle(s), 282 acute, 283 between two lines, 729 between two vectors, 461, 492 central, 283 complementary, 285 conversions between radians and degrees, 286 coterminal, 282 degree, 285 of depression, 306 of elevation, 306 initial side, 282 measure of, 283 negative, 282 obtuse, 283 positive, 282 radian, 283 reference, 314 of repose, 351 standard position, 282 supplementary, 285 terminal side, 282 vertex, 282 Angular speed, 287 Aphelion distance, 798 Arc length, 287 Arccosine function, 345 Arcsine function, 343, 345 Arctangent function, 345 Area common formulas for, 7 of an oblique triangle, 434 of a sector of a circle, 289 of a triangle, 622 Heron’s Area Formula, 442, 491 Argument of a complex number, 471 Arithmetic combination, 84 Arithmetic sequence, 653 common difference of, 653 nth partial sum, 657 nth term of, 654 recursion form, 654 sum of a finite, 656, 723 Associative Property of Addition for complex numbers, 164 for matrices, 590 for real numbers, A6 Associative Property of Multiplication for complex numbers, 164 for matrices, 590, 594 for real numbers, A6 Associative Property of scalar multiplication for matrices, 594 Astronomical unit, 796 Asymptote(s) horizontal, 185 of a hyperbola, 755 oblique, 190 of a rational function, 186 slant, 190 vertical, 185 Augmented matrix, 573 Average rate of change, 59 Average value of a population, 261 Index A211 Axis (axes) imaginary, 470 of a parabola, 129, 736 polar, 779 real, 470 rotation of, 763 of symmetry, 129 B Back-substitution, 497 Base, A11 natural, 222 Basic equation, 534 guidelines for solving, 538 Basic Rules of Algebra, A6 Bearings, 355 Bell-shaped curve, 261 Binomial, 683, A23 coefficient, 683 cube of, A25 expanding, 686 square of, A25 Binomial Theorem, 683, 724 Book value, 32 Bounded, A60 Bounded intervals, A2 Branches of a hyperbola, 753 Break-even point, 501 Butterfly curve, 810 C Cardioid, 789 Cartesian plane, 2 Center of a circle, 20 of an ellipse, 744 of a hyperbola, 753 Central angle of a circle, 283 Change-of-base formula, 239 Characteristics of a function from set A to set B, 40 Circle, 20, 789 arc length of, 287 center of, 20 central angle, 283 classifying by discriminant, 767 by general equation, 759 radius of, 20 sector of, 289 area of, 289 333200_Index_SE.qxd 12/8/05 11:32 AM Page A212 A212 Index standard form of the equation of, 20 unit, 294 Circumference, common formulas for, 7 Classification of conics by the discriminant, 767 by general equation, 759 Coded row matrices, 625 Coefficient binomial, 683 correlation, 104 equating, 536 leading, A23 of a polynomial, A23 of a variable term, A5 Coefficient matrix, 573 Cofactor(s) expanding by, 614 of a matrix, 613 Cofunction identities, 374 Collinear points, 13, 623 test for, 623 Column matrix, 572 Combination of n elements taken r at a time, 696 Combined variation, 107 Common difference, 653 Common formulas area, 7 circumference, 7 perimeter, 7 volume, 7 Common logarithmic function, 230 Common ratio, 663 Commutative Property of Addition for complex numbers, 164 for matrices, 590 for real numbers, A6 Commutative Property of Multiplication for complex numbers, 164 for real numbers, A6 Complement of an event, 708 probability of, 708 Complementary angles, 285 Completely factored, A26 Completing the square, A49 Complex conjugates, 165 Complex fraction, A40 Complex number(s), 162 absolute value of, 470 addition of, 163 additive identity, 163 additive inverse, 163 argument of, 471 Associative Property of Addition, 164 Associative Property of Multiplication, 164 Commutative Property of Addition, 164 Commutative Property of Multiplication, 164 Distributive Property, 164 equality of, 162 imaginary part of, 162 modulus of, 471 nth root of, 475, 476 nth roots of unity, 477 polar form, 471 product of two, 472 quotient of two, 472 real part of, 162 standard form of, 162 subtraction of, 163 trigonometric form of, 471 Complex plane, 470 imaginary axis, 470 real axis, 470 Complex zeros occur in conjugate pairs, 173 Component form of a vector v, 448 Components, vector, 463, 464 Composite number, A7 Composition, 86 Compound interest continuous compounding, 223 formulas for, 224 Conditional equation, A46 Conic(s) or conic section(s), 735 alternative definition, 793 classifying by the discriminant, 767 by general equation, 759 degenerate, 735 eccentricity of, 793 locus of, 735 polar equations of, 793, 808 rotation of axes, 763 Conjugate, 173, A17 of a complex number, 165 Conjugate axis of a hyperbola, 755 Consistent system of linear equations, 510 Constant, A5 function, 57, 67 of proportionality, 105 term, A5, A23 of variation, 105 Constraints, 552 Consumer surplus, 546 Continuous compounding, 223 Continuous function, 139, 771 Conversions between degrees and radians, 286 Convex limaçon, 789 Coordinate(s), 2 polar, 779 Coordinate axes, reflection in, 76 Coordinate conversion, 780 Coordinate system, polar, 779 Correlation coefficient, 104 Correspondence, one-to-one, A1 Cosecant function, 295, 301 of any angle, 312 graph of, 335, 338 Cosine curve, amplitude of, 323 Cosine function, 295, 301 of any angle, 312 common angles, 315 domain of, 297 graph of, 325, 338 inverse, 345 period of, 324 range of, 297 special angles, 303 Cotangent function, 295, 301 of any angle, 312 graph of, 334, 338 Coterminal angles, 282 Cramer’s Rule, 619, 620 Critical numbers, 197, 201 Cross multiplying, A48 Cryptogram, 625 Cube of
a binomial, A25 Cube root, A14 Cubic function, 68 Curtate cycloid, 778 Curve butterfly, 810 plane, 771 rose, 788, 789 sine, 321 Cycloid, 775 curate, 778 D Damping factor, 337 Decreasing function, 57 Defined, 47 Definitions of trigonometric functions of any angle, 312 Degenerate conic, 735 Degree, 285 conversion to radians, 286 of a polynomial, A23 DeMoivre’s Theorem, 474 333200_Index_SE.qxd 12/8/05 11:32 AM Page A213 Denominator, A5 Domain rationalizing, 384, A16, A17 Dependent system of linear equations, 510 Dependent variable, 42, 47 Depreciated costs, 32 Descartes’s Rule of Signs, 176 Determinant of a matrix, 606, 611, 614 of a 2 2 matrix, 611 Diagonal matrix, 601, 618 Diagonal of a polygon, 700 Difference common, 653 of functions, 84 quotient, 46, A42 of two squares, A27 of vectors, 449 Differences first, 680 second, 680 Dimpled limaçon, 789 Direct variation, 105 as an nth power, 106 Directed line segment, 447 initial point, 447 length of, 447 magnitude, 447 terminal point, 447 Direction angle of a vector, 453 Directly proportional, 105 to the nth power, 106 Directrix of a parabola, 736 Discrete mathematics, 41 Discriminant, 767 classification of conics by, 767 Distance between a point and a line, 730, 806 between two points in the plane, 4 on the real number line, A4 Distance Formula, 4 Distinguishable permutations, 695 Distributive Property for complex numbers, 164 for matrices, 590, 594 for real numbers, A6 Division of fractions, A7 long, 153 of real numbers, A5 synthetic, 156 Division Algorithm, 154 Divisors, A7 of an algebraic expression, A36 of cosine function, 297 of a function, 40, 47 implied, 44, 47 of a rational function, 184 of sine function, 297 Dot product, 460 properties of, 460, 492 Double-angle formulas, 407, 425 Double inequality, A63 Doyle Log Rule, 505 E Eccentricity of a conic, 793 of an ellipse, 748, 793 of a hyperbola, 793 of a parabola, 793 Effective yield, 251 Elementary row operations, 574 Eliminating the parameter, 773 Elimination Gaussian, 520 with back-substitution, 578 Gauss-Jordan, 579 method of, 507, 508 Ellipse, 744, 793 center of, 744 classifying by discriminant, 767 by general equation, 759 eccentricity of, 748, 793 foci of, 744 latus rectum of, 752 major axis of, 744 minor axis of, 744 standard form of the equation of, 745 vertices of, 744 Endpoints of an interval, A2 Entry of a matrix, 572 main diagonal, 572 Epicycloid, 778 Equal matrices, 587 Equality of complex numbers, 162 properties of, A6 of vectors, 448 Equating the coefficients, 536 Equation(s), 14, A46 basic, 534 conditional, A46 equivalent, A47 generating, A47 Index A213 graph of, 14 identity, A46 of a line, 25 general form, 33 intercept form, 36 point-slope form, 29, 33 slope-intercept form, 25, 33 summary of, 33 two-point form, 29, 33, 624 linear, 16 in one variable, A46 in two variables, 25 parametric, 771 position, 525 quadratic, 16, A49 second-degree polynomial, A49 solution of, 14, A46 solution point, 14 system of, 496 in two variables, 14 Equilibrium point, 514, 546 Equivalent equations, A47 generating, A47 expressions, A36 fractions, A7 generate, A7 inequalities, A61 systems, 509 operations that produce, 520 Evaluate an algebraic expression, A5 Evaluating trigonometric functions of any angle, 315 Even function, 60 trigonometric functions, 298 Even/odd identities, 374 Event(s), 701 complement of, 708 probability of, 708 independent, 707 probability of, 707 mutually exclusive, 705 probability of, 702 the union of two, 705 Existence theorems, 169 Expanding a binomial, 686 by cofactors, 614 Expected value, 726 Experiment, 701 outcome of, 701 sample space of, 701 Exponent(s), A11 properties of, A11 rational, A18 333200_Index_SE.qxd 12/8/05 11:32 AM Page A214 A214 Index Exponential decay model, 257 Exponential equation, solving, 246 Exponential form, A11 Exponential function, 218 f with base a, 218 natural, 222 one-to-one property, 220 Exponential growth model, 257 Exponential notation, A11 Exponentiating, 249 Expression algebraic, A5 fractional, A36 rational, A36 Extended principle of mathematical induction, 675 Extracting square roots, A49 Extraneous solution, A48, A54 F Factor Theorem, 157, 213 Factorial, 644 Factoring, A26 completely, A26 by grouping, A30 polynomials, guidelines for, A30 solving a quadratic equation by, A49 special polynomial forms, A27 Factors of an integer, A7 of a polynomial, 173, 214 Family of functions, 75 Far point, 216 Feasible solutions, 552 Finding a formula for the nth term of a sequence, 678 Finding intercepts of a graph, 17 Finding an inverse function, 97 Finding an inverse matrix, 604 Finding test intervals for a polynomial, 197 Finite sequence, 642 Finite series, 647 First differences, 680 Fixed cost, 31 Fixed point, 397 Focal chord latus rectum, 738 of a parabola, 738 Focus (foci) of an ellipse, 744 of a hyperbola, 753 of a parabola, 736 FOIL Method, A24 Formula(s) change-of-base, 239 for compound interest, 224 double-angle, 407, 425 half-angle, 410 Heron’s Area, 442, 491 for the nth term of a sequence, 678 power-reducing, 409, 425 product-to-sum, 411 Quadratic, A49 reduction, 402 sum and difference, 400, 424 sum-to-product, 412, 426 Four ways to represent a function, 41 Fractal, 726 Fraction(s) addition of with like denominators, A7 with unlike denominators, A7 complex, A40 division of, A7 equivalent, A7 generate, A7 multiplication of, A7 operations of, A7 partial, 533 decomposition, 533 properties of, A7 rules of signs for, A7 subtraction of with like denominators, A7 with unlike denominators, A7 Fractional expression, A36 Frequency, 356 Function(s), 40, 47 algebraic, 218 arithmetic combination of, 84 characteristics of, 40 common logarithmic, 230 composition, 86 constant, 57, 67 continuous, 139, 771 cosecant, 295, 301 cosine, 295, 301 cotangent, 295, 301 cubic, 68 decreasing, 57 defined, 47 difference of, 84 domain of, 40, 47 even, 60 exponential, 218 family of, 75 four ways to represent, 41 graph of, 54 greatest integer, 69 of half-angles, 407 Heaviside, 126 identity, 67 implied domain of, 44, 47 increasing, 57 inverse, 93, 94 cosine, 345 sine, 343, 345 tangent, 345 trigonometric, 345 linear, 66 logarithmic, 229 of multiple angles, 407 name of, 42, 47 natural exponential, 222 natural logarithmic, 233 notation, 42, 47 objective, 552 odd, 60 one-to-one, 96 period of, 297 periodic, 297 piecewise-defined, 43 polynomial, 128 power, 140 product of, 84 quadratic, 128 quotient of, 84 range of, 40, 47 rational, 184 reciprocal, 68 secant, 295, 301 sine, 295, 301 square root, 68 squaring, 67 step, 69 sum of, 84 summary of terminology, 47 tangent, 295, 301 transcendental, 218 trigonometric, 295, 301, 312 undefined, 47 value of, 42, 47 Vertical Line Test, 55 zero of, 56 Fundamental Counting Principle, 692 Fundamental Theorem of Algebra, 169 of Arithmetic, A7 Fundamental trigonometric identities, 304, 374 G Gaussian elimination, 520 with back-substitution, 578 333200_Index_SE.qxd 12/8/05 11:32 AM Page A215 Gaussian model, 257 Gauss-Jordan elimination, 579 General form of the equation of a line, 33 Generalizations about nth roots of real numbers, A15 Generate equivalent fractions, A7 Generating equivalent equations, A47 Geometric sequence, 663 common ratio of, 663 nth term of, 664 sum of a finite, 666, 723 Geometric series, 667 sum of an infinite, 667 Graph, 14 of cosecant function, 335, 338 of cosine function, 325, 338 of cotangent function, 334, 338 of an equation, 14 of a function, 54 of an inequality, 541, A60 in two variables, 541 intercepts of, 17 of inverse cosine function, 345 of an inverse function, 95 of inverse sine function, 345 of inverse tangent function, 345 of a line, 25 point-plotting method, 15 of a rational function, guidelines for analyzing, 187 of secant function, 335, 338 of sine function, 325, 338 special polar, 789 symmetry, 18 of tangent function, 332, 338 Graphical interpretations of solutions, 510 Graphical method, 500 Graphical tests for symmetry, 18 Greatest integer function, 69 Guidelines for analyzing graphs of rational functions, 187 for factoring polynomials, A30 for solving the basic equation, 538 for verifying trigonometric identities, 382 H Half-angle formulas, 410 Half-life, 225 Harmonic motion, simple, 356, 357 Heaviside function, 126 Heron’s Area Formula, 442, 491 Horizontal asymptote, 185 Horizontal components of v, 452 Horizontal line, 33 Horizontal Line Test, 96 Horizontal shift, 74 Horizontal shrink, 78 of a trigonometric function, 324 Horizontal stretch, 78 of a trigonometric function, 324 Horizontal translation of a trigonometric function, 325 Human memory model, 235 Hyperbola, 185, 753, 793 asymptotes of, 755 branches of, 753 center of, 753 classifying by discriminant, 767 by general equation, 759 conjugate axis of, 755 eccentricity of, 793 foci of, 753 standard form of the equation of, 753 transverse axis of, 753 vertices of, 753 Hypocycloid, 810 Hypotenuse of a right triangle, 301 I Idempotent square matrix, 639 Identity, A46 of the complex plane, 470 function, 67 matrix of order n, 594 Imaginary axis of the complex plane, 470 Imaginary number, 162 pure, 162 Imaginary part of a complex number, 162 Imaginary unit i, 162 Implied domain, 44, 47 Improper rational expression, 154 Inclination, 728 and slope, 728, 806 Inclusive or, A7 Inconsistent system of linear equations, 510 Increasing annuity, 668 Increasing function, 57 Independent events, 707 probability of, 707 Independent system of linear equations, 510 Independent variable, 42, 47 Index A215 Index of a radical, A14 of summation, 646 Indirect proof, 568 Inductive, 614 Inequality (inequalities), A2 absolute value, solution of, A63 double, A63 equivalent, A61 graph of, 541, A60 linear, 542, A62 properties of, A61 satisfy, A60 solution of, 541, A60 solution set of, A60 symbol, A2 Infinite geometric series, 667 sum of, 667 Infinite sequence, 642 Infinite series, 647 Infinite wedge, 545 Infinity negative, A3 positive, A3 Initial point, 447 Initial side of an angle, 282 Integer(s) divisors
of, A7 factors of, A7 irreducible over, A26 Intercept form of the equation of a line, 36 Intercepts, 17 finding, 17 Intermediate Value Theorem, 146 Interval bounded, A2 on the real number line, A2 unbounded, A3 Invariant under rotation, 767 Inverse additive, A5 multiplicative, A5 Inverse function, 93 cosine, 345 definition of, 94 finding, 97 graph of, 95 Horizontal Line Test, 96 sine, 343, 345 tangent, 345 Inverse of a matrix, 602 finding an, 604 Inverse properties of logarithms, 230 333200_Index_SE.qxd 12/8/05 11:32 AM Page A216 A216 Index of natural logarithms, 234 of trigonometric functions, 347 Inverse trigonometric functions, 345 Inverse variation, 107 Inversely proportional, 107 Invertible matrix, 603 Irrational number, A1 Irreducible over the integers, A26 over the rationals, 174 over the reals, 174 J Joint variation, 108 Jointly proportional, 108 K Kepler’s Laws, 796 Key points of the graph of a trigonometric function, 322 intercepts, 322 maximum points, 322 minimum points, 322 L Latus rectum of an ellipse, 752 of a parabola, 738 Law of Cosines, 439, 490 alternative form, 439, 490 standard form, 439, 490 Law of Sines, 430, 489 Law of Trichotomy, A3 Leading coefficient of a polynomial, A23 Leading Coefficient Test, 141 Least squares regression line, 104 Lemniscate, 789 Length of a directed line segment, 447 Length of a vector, 448 Like radicals, A17 Like terms of a polynomial, A24 Limaçon, 786, 789 convex, 789 dimpled, 789 with inner loop, 789 Line(s) in the plane graph of, 25 horizontal, 33 inclination of, 728 least squares regression, 104 parallel, 30 perpendicular, 30 slope of, 25, 27 vertical, 33 Linear combination of vectors, 452 Linear depreciation, 32 Linear equation, 16 general form, 33 in one variable, A46 intercept form, 36 point-slope form, 29, 33 slope-intercept form, 25, 33 summary of, 33 two-point form, 29, 182, 624 in two variables, 25 Linear extrapolation, 33 Linear Factorization Theorem, 169, 214 Linear function, 66 Linear inequality, 542, A62 Linear interpolation, 33 Linear programming, 552 problem, solving, 553 Linear speed, 287 Local maximum, 58 Local minimum, 58 Locus, 735 Logarithm(s) change-of-base formula, 239 natural, properties of, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 properties of, 230, 240, 278 inverse, 230 one-to-one, 230 power, 240, 278 product, 240, 278 quotient, 240, 278 Logarithmic equation, solving, 246 Logarithmic function, 229 with base a, 229 common, 230 natural, 233 Logarithmic model, 257 Logistic curve, 262 growth model, 257 Long division, 153 Lower bound, 177 Lower limit of summation, 646 M Magnitude of a directed line segment, 447 of a vector, 448 Main diagonal of a square matrix, 572 Major axis of an ellipse, 744 Marginal cost, 31 Mathematical induction, 673 extended principle of, 675 Principle of, 674 Matrix (matrices), 572 addition, 588 properties of, 590 additive identity, 591 adjoining, 604 augmented, 573 coded row, 625 coefficient, 573 cofactor of, 613 column, 572 determinant of, 606, 611, 614 diagonal, 601, 618 elementary row operations, 574 entry of a, 572 equal, 587 idempotent, 639 identity, 594 inverse of, 602 invertible, 603 minor of, 613 multiplication, 592 properties of, 594 nonsingular, 603 order of a, 572 in reduced row-echelon form, 576 representation of, 587 row, 572 in row-echelon form, 576 row-equivalent, 574 scalar identity, 590 scalar multiplication, 588 singular, 603 square, 572 stochastic, 599 transpose of, 640 uncoded row, 625 zero, 591 Measure of an angle, 283 degree, 285 radian, 283 Method of elimination, 507, 508 of substitution, 496 Midpoint Formula, 5, 124 Midpoint of a line segment, 5 Minor axis of an ellipse, 744 Minor of a matrix, 613 Minors and cofactors of a square matrix, 613 333200_Index_SE.qxd 12/8/05 11:32 AM Page A217 Modulus of a complex number, 471 Monomial, A23 Multiplication of fractions, A7 of matrices, 592 scalar, 588 nth term of an arithmetic sequence, 654 recursion form, 654 of a geometric sequence, 664 of a sequence, finding a formula for, 678 Multiplicative identity of a real number, Number(s) A6 Multiplicative inverse, A5 for a matrix, 602 of a real number, A6 Multiplicity, 143 Multiplier effect, 671 Mutually exclusive events, 705 N n factorial, 644 Name of a function, 42, 47 Natural base, 222 Natural exponential function, 222 Natural logarithm properties of, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 Natural logarithmic function, 233 Near point, 216 Negation, properties of, A6 Negative angle, 282 infinity, A3 of a vector, 449 Newton’s Law of Cooling, 268 Nonnegative number, A1 Nonrigid transformation, 78 Nonsingular matrix, 603 Nonsquare system of linear equations, 524 Normally distributed, 261 Notation exponential, A11 function, 42, 47 scientific, A13 sigma, 646 summation, 646 nth partial sum, 647 of an arithmetic sequence, 657 nth root(s) of a, A14 of a complex number, 475, 476 generalizations about, A15 principal, A14 of unity, 477 complex, 162 composite, A7 critical, 197, 201 imaginary, 162 pure, 162 irrational, A1 nonnegative, A1 prime, A7 rational, A1 real, A1 Number of permutations of n elements, 693 taken r at a time, 694 Number of solutions of a linear system, 522 Numerator, A5 O Objective function, 552 Oblique asymptote, 190 Oblique triangles, 430 area of, 434 Obtuse angle, 283 Odd function, 60 trigonometric functions, 298 One cycle of a sine curve, 321 One-to-one correspondence, A1 One-to-one function, 96 One-to-one property of exponential functions, 220 of logarithms, 230 of natural logarithms, 234 Operations of fractions, A7 Operations that produce equivalent systems, 520 Opposite side of a right triangle, 301 Optimal solution of a linear programming problem, 552 Optimization, 552 Order of a matrix, 572 on the real number line, A2 Ordered pair, 2 Ordered triple, 519 Orientation of a curve, 772 Origin, 2 of polar coordinate system, 779 of the real number line, A1 Index A217 symmetry, 18 Orthogonal vectors, 462 Outcome, 701 P Parabola, 128, 736, 793 axis of, 129, 736 classifying by discriminant, 767 by general equation, 759 directrix of, 736 eccentricity of, 793 focal chord of, 738 focus of, 736 latus rectum of, 738 reflective property, 738 standard form of the equation of, 736, 807 tangent line, 738 vertex of, 129, 133, 736 Parallel lines, 30 Parallelogram law, 449 Parameter, 771 eliminating the, 773 Parametric equation, 771 Partial fraction, 533 decomposition, 533 Pascal’s Triangle, 685 Perfect cube, A15 square, A15 square trinomial, A27, A28 Perihelion distance, 798 Perimeter, common formulas for, 7 Period of a function, 297 of sine and cosine functions, 324 Periodic function, 297 Permutation, 693 distinguishable, 695 of n elements, 693 taken r at a time, 694 Perpendicular lines, 30 Phase shift, 325 Piecewise-defined function, 43 Plane curve, 771 orientation of, 772 Point of diminishing returns, 151 equilibrium, 514, 546 Point-plotting method, 15 Point-slope form, 29, 33 Points of intersection, 500 Polar axis, 779 Polar coordinate system, 779 333200_Index_SE.qxd 12/8/05 11:32 AM Page A218 A218 Index origin of, 779 pole, 779 Polar coordinates, 779 conversion to rectangular, 780 quick tests for symmetry in, 787 test for symmetry in, 786 Polar equations of conics, 793, 808 Polar form of a complex number, 471 Pole, 779 Polynomial(s), A23 coefficient of, A23 completely factored, A26 constant term, A23 degree of, A23 equation, second-degree, A49 factors of, 173, 214 finding test intervals for, 197 guidelines for factoring, A30 irreducible, A26 leading coefficient of, A23 like terms, A24 long division of, 153 prime, A26 prime factor, 174 standard form of, A23 synthetic division, 156 test intervals for, 144 Polynomial function, 128 real zeros of, 143 standard form, 142 test intervals, 197 of x with degree n, 128 Position equation, 525 Positive angle, 282 infinity, A3 Power, A11 Power function, 140 Power property of logarithms, 240, 278 of natural logarithms, 240, 278 Power-reducing formulas, 409, 425 Prime factor of a polynomial, 174 factorization, A7 number, A7 polynomial, A26 Principal nth root of a, A14 of a number, A14 Principal square root of a negative number, 166 of an event, 702 of independent events, 707 of the union of two events, 705 Producer surplus, 546 Product of functions, 84 of trigonometric functions, 407 of two complex numbers, 472 Product property of logarithms, 240, 278 of natural logarithms, 240, 278 Product-to-sum formulas, 411 Projection, of a vector, 464 Proof, 124 by contradiction, 568 indirect, 568 without words, 638 Proper rational expression, 154 Properties of absolute value, A4 of the dot product, 460, 492 of equality, A6 of exponents, A11 of fractions, A7 of inequalities, A61 of inverse trigonometric functions, 347 of logarithms, 230, 240, 278 inverse, 230 one-to-one, 230 power, 240, 278 product, 240, 278 quotient, 240, 278 of matrix addition and scalar multiplication, 590 of matrix multiplication, 594 of natural logarithms, 234, 240, 278 inverse, 234 one-to-one, 234 power, 240, 278 product, 240, 278 quotient, 240, 278 of negation, A6 one-to-one, exponential functions, 220 of radicals, A15 reflective, 738 of sums, 646, 722 of vector addition and scalar multiplication, 451 of zero, A7 Principle of Mathematical Induction, 674 Probability of a complement, 708 Pure imaginary number, 162 Pythagorean identities, 304, 374 Pythagorean Theorem, 4, 370 Q Quadrant, 2 Quadratic equation, 16, A49 solving by completing the square, A49 by extracting square roots, A49 by factoring, A49 using Quadratic Formula, A49 using Square Root Principle, A49 Quadratic Formula, A49 Quadratic function, 128 standard form, 131 Quick tests for symmetry in polar coordinates, 787 Quotient difference, 46 of functions, 84 of two complex numbers, 472 Quotient identities, 304, 374 Quotient property of logarithms, 240, 278 of natural logarithms, 240, 278 R Radian, 283 conversion to degrees, 286 Radical(s)
index of, A14 like, A17 properties of, A15 simplest form, A16 symbol, A14 Radicand, A14 Radius of a circle, 20 Random selection with replacement, 691 without replacement, 691 Range of a function, 40, 47 Rate, 31 Rate of change, 31 average, 59 Ratio, 31 Rational exponent, A18 Rational expression(s), A36 improper, 154 proper, 154 Rational function, 184 asymptotes of, 186 domain of, 184 graph of, guidelines for analyzing, 187 test intervals for, 187 Rational inequality, test intervals, 201 Rational number, A1 333200_Index_SE.qxd 12/8/05 11:32 AM Page A219 Rational Zero Test, 170 Rationalizing a denominator, 384, A16, Rose curve, 788, 789 Rotation A17 Real axis of the complex plane, 470 Real number(s), A1 absolute value of, A4 division of, A5 subset of, A1 subtraction of, A5 Real number line, A1 bounded intervals on, A2 distance between two points, A4 interval on, A2 order on, A2 origin, A1 unbounded intervals on, A3 Real part of a complex number, 162 Real zeros of polynomial functions, 143 Reciprocal function, 68 Reciprocal identities, 304, 374 Rectangular coordinate system, 2 Rectangular coordinates, conversion to polar, 780 Recursion form of the nth term of an arithmetic sequence, 654 Recursion formula, 655 Recursive sequence, 644 Reduced row-echelon form of a matrix, 576 Reducible over the reals, 174 Reduction formulas, 402 Reference angle, 314 Reflection, 76 of a trigonometric function, 324 Reflective property of a parabola, 738 Relation, 40 Relative maximum, 58 Relative minimum, 58 Remainder Theorem, 157, 213 Repeated zero, 143 Representation of matrices, 587 Resultant of vector addition, 449 Right triangle definitions of trigonometric functions, 301 hypotenuse, 301 opposite side, 301 right side of, 301 solving, 306 Rigid transformation, 78 Root(s) of a complex number, 475, 476 cube, A14 principal nth, A14 square, A14 of axes, 763 to eliminate an xy-term, 763 invariants, 767 Row-echelon form, 519 of a matrix, 576 reduced, 576 Row-equivalent, 574 Row matrix, 572 Row operations, 520 Rules of signs for fractions, A7 S Sample space, 701 Satisfy the inequality, A60 Scalar, 588 identity, 590 multiple, 588 Scalar multiplication, 588 properties of, 590 of a vector, 449 properties of, 451 Scatter plot, 3 Scientific notation, A13 Scribner Log Rule, 505 Secant function, 295, 301 of any angle, 312 graph of, 335, 338 Secant line, 59 Second-degree polynomial equation, A49 Second differences, 680 Sector of a circle, 289 area of, 289 Sequence, 642 arithmetic, 653 finite, 642 first differences of, 680 geometric, 663 infinite, 642 nth partial sum, 647 recursive, 644 second differences of, 680 terms of, 642 Series, 647 finite, 647 geometric, 667 infinite, 647 geometric, 667 Sierpinski Triangle, 726 Sigma notation, 646 Sigmoidal curve, 262 Simple harmonic motion, 356, 357 frequency, 356 Index A219 Simplest form, A16 Sine curve, 321 amplitude of, 323 one cycle of, 321 Sine function, 295, 301 of any angle, 312 common angles, 315 curve, 321 domain of, 297 graph of, 325, 338 inverse, 343, 345 period of, 324 range of, 297 special angles, 303 Sines, cosines, and tangents of special angles, 303 Singular matrix, 603 Sketching the graph of an equation by point plotting, 15 Sketching the graph of an inequality in two variables, 541 Slant asymptote, 190 Slope inclination, 728, 806 of a line, 25, 27 Slope-intercept form, 25, 33 Solution(s) of an absolute value inequality, A63 of an equation, 14, A46 extraneous, A48, A54 of an inequality, 541, A60 of a system of equations, 496 graphical interpretations, 510 of a system of inequalities, 543 Solution point, 14 Solution set, A60 Solving an absolute value inequality, A63 an equation, A46 exponential and logarithmic equations, 246 an inequality, A60 a linear programming problem, 553 right triangles, 306 a system of equations, 496 Cramer’s Rule, 619, 620 Gaussian elimination with back-substitution, 578 Gauss-Jordan elimination, 579 graphical method, 500 method of elimination, 507, 508 method of substitution, 496 a system of linear equations, Gaussian elimination, 520 Special products, A25 333200_Index_SE.qxd 12/8/05 11:32 AM Page A220 A220 Index Square of a binomial, A25 of trigonometric functions, 407 Square matrix, 572 determinant of, 614 idempotent, 639 main diagonal of, 572 minors and cofactors of, 613 Square root(s), A14 extracting, A49 function, 68 of a negative number, 166 Square Root Principle, A49 Square system of linear equations, 524 Squaring function, 67 Standard form of a complex number, 162 of the equation of a circle, 20 of the equation of an ellipse, 745 of the equation of a hyperbola, 753 of the equation of a parabola, 736, 807 of Law of Cosines, 439, 490 of a polynomial, A23 of a polynomial function, 142 of a quadratic function, 131 Standard position of an angle, 282 of a vector, 448 Standard unit vector, 452 Step function, 69 Stochastic matrix, 599 Straight-line depreciation, 32 Strategies for solving exponential and logarithmic equations, 246 Strophoid, 810 Subset, A1 Substitution, method of, 496 Substitution Principle, A5 Subtraction of a complex number, 163 of fractions with like denominators, A7 with unlike denominators, A7 of real numbers, A5 Sum(s) of a finite arithmetic sequence, 656, 723 of a finite geometric sequence, 666, 723 of functions, 84 of an infinite geometric series, 667 nth partial, 647 of powers of integers, 679 properties of, 646, 722 of square differences, 104 Sum and difference formulas, 400, 424 Sum and difference of same terms, A25 Sum or difference of two cubes, A27 Summary of a sequence, 642 variable, A5 Terminal point, 447 Terminal side of an angle, 282 Test of equations of lines, 33 of function terminology, 47 for collinear points, 623 for symmetry in polar coordinates, Summation index of, 646 lower limit of, 646 notation, 646 upper limit of, 646 Sum-to-product formulas, 412, 426 Supplementary angles, 285 Surplus consumer, 546 producer, 546 Symmetry, 18 algebraic tests for, 19 graphical tests for, 18 quick tests for, in polar coordinates, 787 test for, in polar coordinates, 786 with respect to the origin, 18 with respect to the x-axis, 18 with respect to the y-axis, 18 Synthetic division, 156 uses of the remainder, 158 System of equations, 496 equivalent, 509 solution of, 496 with a unique solution, 607 System of inequalities, solution of, 543 System of linear equations consistent, 510 dependent, 510 inconsistent, 510 independent, 510 nonsquare, 524 number of solutions, 522 row-echelon form, 519 row operations, 520 square, 524 T Tangent function, 295, 301 of any angle, 312 common angles, 315 graph of, 332, 338 inverse, 345 special angles, 303 Tangent line to a parabola, 738 Term of an algebraic expression, A5 constant, A5, A23 786 Test intervals for a polynomial, 144 for a polynomial inequality, 197 for a rational function, 187 for a rational inequality, 201 Transcendental function, 218 Transformation nonrigid, 78 rigid, 78 Transpose of a matrix, 640 Transverse axis of a hyperbola, 753 Triangle area of, 622 oblique, 430 area of, 434 Trigonometric form of a complex number, 471 argument of, 471 modulus of, 471 Trigonometric function of any angle, 312 evaluating, 315 cosecant, 295, 301 cosine, 295, 301 cotangent, 295, 301 even and odd, 298 horizontal shrink of, 324 horizontal stretch of, 324 horizontal translation of, 325 inverse properties of, 347 key points, 322 intercepts, 322 maximum points, 322 minimum points, 322 product of, 407 reflection of, 324 right triangle definitions of, 301 secant, 295, 301 sine, 295, 301 square of, 407 tangent, 295, 301 unit circle definitions of, 295 Trigonometric identities cofunction identities, 374 even/odd identities, 374 fundamental identities, 304, 374 guidelines for verifying, 382 333200_Index_SE.qxd 12/8/05 11:32 AM Page A221 Pythagorean identities, 304, 374 quotient identities, 304, 374 reciprocal identities, 304, 374 Trigonometric values of common angles, 315 Trigonometry, 282 Trinomial, A23 perfect square, A27, A28 inverse, 107 joint, 108 in sign, 176 Vary directly, 105 as nth power, 106 Vary inversely, 107 Vary jointly, 108 Vector(s) Two-point form of the equation of a line, addition, 449 29, 33, 624 U Unbounded, A60 Unbounded intervals, A3 Uncoded row matrices, 625 Undefined, 47 Unit circle, 294 definitions of trigonometric functions, 295 Unit vector, 448, 621 in the direction of v, 451 standard, 452 Upper bound, 177 Upper limit of summation, 646 Upper and Lower Bound Rules, 177 Uses of the remainder in synthetic division, 158 V Value of a function, 42, 47 Variable, A5 dependent, 42, 47 independent, 42, 47 Variable term, A5 Variation combined, 107 constant of, 105 direct, 105 as an nth power, 106 properties of, 451 resultant of, 449 angle between two, 461, 492 component form of, 448 components, 463, 464 difference of, 449 directed line segment of, 447 direction angle of, 453 dot product of, 460 properties of, 460, 492 equality of, 448 horizontal component of, 452 length of, 448 linear combination of, 452 magnitude of, 448 negative of, 449 orthogonal, 462 parallelogram law, 449 projection, 464 resultant, 449 scalar multiplication of, 449, properties of, 451 standard position of, 448 unit, 448, 621 in the direction of v, 451 standard, 452 v in the plane, 447 vertical component of, 452 zero, 448 Vertex (vertices) of an angle, 282 of an ellipse, 744 Index A221 of a hyperbola, 753 of a parabola, 129, 133, 736 Vertical asymptote, 185 Vertical components of v, 452 Vertical line, 33 Vertical Line Test, 55 Vertical shift, 74 Vertical shrink, 78 Vertical stretch, 78 Volume, common formulas for, 7 W With replacement, 691 Without replacement, 691 Work, 466 x-axis, 2 symmetry, 18 x-coordinate, 2 y-axis, 2 symmetry, 18 y-coordinate, 2 X Y Z Zero(s) of a function, 56 matrix, 591 multiplicity of, 143 of a polynomial function, 143 bounds for, 177 real, 143 properties of, A7 repeated, 143 vector, 448 Zero-Factor Property, A7 333201_AP_FES.qxd 12/5/05 11:46 AM Page ES1 Definition of the Six Trigonometric Functions Right tria
ngle definitions, where Adjacent e t i s o p p O sin opp. hyp. cos adj. hyp. tan opp. adj. csc hyp. opp. sec hyp. adj. cot adj. opp. Circular function definitions, where is any angle 2 2 y θ y r x x sin y r cos x r tan , csc r y sec r x cot 120° 4 135° 150° − ( 1, 0) π 180° y (0, 1) 90° 601, 0) π 3 45° π 4 30° 0° 360° 330 210° 225° π 5 240° π 4 4 3 270° − ) 3 2 π 11 6 315° π 7 300° π 4 5 π 3 3 2 (0, 1)− , ( Reciprocal Identities sin u 1 csc u csc u 1 sin u cos u 1 sec u sec u 1 cos u Quotient Identities tan u sin u cos u cot u cos u sin u tan u 1 cot u cot u 1 tan u Pythagorean Identities sin2 u cos2 u 1 1 tan2 u sec2 u Cofunction Identities u cos u sin 2 cos u sin u 2 tan u cot u 2 Even/Odd Identities sinu sin u cosu cos u tanu tan u 1 cot2 u csc2 u cot 2 sec 2 csc 2 u tan u u csc u u sec u cotu cot u secu sec u cscu csc u Sum and Difference Formulas sinu ± v sin u cos v ± cos u sin v cosu ± v cos u cos v sin u sin v tanu ± v tan u ± tan v 1 tan u tan v Double-Angle Formulas sin 2u 2 sin u cos u cos 2u cos2 u sin2 u tan 2u 2 tan u 1 tan2 u 2 cos2 u 1 1 2 sin2 u 2 Power-Reducing Formulas sin2 u 1 cos 2u cos2 u 1 cos 2u tan2 u 1 cos 2u 1 cos 2u 2 Sum-to-Product Formulas sin u sin v 2 sinu v 2 sin u sin v 2 cosu v 2 cos u cos v 2 cosu v 2 cos u cos v 2 sinu v 2 cosu v 2 sinu v 2 cosu v 2 sinu v 2 cosu v cosu v Product-to-Sum Formulas sin u sin v 1 2 cos u cos v 1 2 sin u cos v 1 2 cos u sin v 1 2 sinu v sinu v sinu v sinu v cosu v cosu v 333201_AP_FES.qxd 12/5/05 11:46 AM Page ES2 FORMULAS FROM GEOMETRY Triangle: h a sin Area 1 2 bh h a θ c b c 2 a2 b2 2ab cos (Law of Cosines) Sector of Circular Ring: Area pw p average radius, w width of ring, in radians Ellipse: Area ab Circumference 2a2 b2 2 Cone: Volume Ah 3 A area of base Right Circular Cone: Volume r2h 3 Lateral Surface Area rr2 h2 Frustum of Right Circular Cone: Volume r 2 rR R2h 3 Lateral Surface Area sR Right Circular Cylinder: r2h Volume Lateral Surface Area 2rh r h s θ r r R p w Sphere: Volume 4 3 Surface Area 4r 2 r3 Wedge: A B sec A area of upper face, B area of base Right Triangle: Pythagorean Theorem c2 a2 b2 Equilateral Triangle: h 3s 2 Area 3s2 4 Parallelogram: Area bh Trapezoid: Area h 2 a b Circle: Area r 2 Circumference 2r Sector of Circle: r2 2 Area s r in radians Circular Ring: Area R2 r2 2pw p average radius, w width of ring 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES1 Linear Function f x mx b y (0, b) ( m( b − , 0 ( m( b − , 0 x f(x) = mx + b, m > 0 f(x) = mx + b, m < 0 GRAPHS OF PARENT FUNCTIONS Absolute Value Function f x x x, x ≥ 0 x, x < 0 y 2 1 −2 −1 (0, 0) f(x) = x x 2 −1 −2 Square Root Function fx x y 4 3 2 1 f(x) = x −1 (0, 0) 2 3 4 −1 , , Domain: Range: bm, 0 x-intercept: 0, b y-intercept: Increasing when Decreasing when m > 0 m < 0 , Domain: 0, Range: 0, 0 Intercept: Decreasing on Increasing on Even function y-axis symmetry , 0 0, 0, Domain: 0, Range: 0, 0 Intercept: Increasing on 0, Greatest Integer Function fx x Quadratic (Squaring) Function fx ax2 Cubic Function fx x3 y f(x) = x[[ ]] 3 2 1 −3 −2 −1 1 2 3 x −2 −1 −3 , Domain: Range: the set of integers x-intercepts: in the interval 0, 0 y-intercept: Constant between each pair of 0, 1 consecutive integers Jumps vertically one unit at each integer value y 3 2 1 −1 −2 −3 f(x) = ax , (x) = ax , a < 0 2 −3 −2 y 3 2 (0, 0) −1 −2 −3 3 2 1 f(x) = x3 , Domain: , Range: 0, 0 Intercept: Increasing on Odd function Origin symmetry , 0, , 0 , Domain: a > 0 : Range a < 0 Range : 0, 0 Intercept: Decreasing on Increasing on Increasing on Decreasing on Even function y-axis symmetry Relative minimum , 0 0, for , 0 0, a > 0 for a > 0 for a < 0 for a < 0 a > 0, a < 0, relative maximum or vertex: 0, 0 x x 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES2 Rational (Reciprocal) Function fx 1 x Exponential Function Logarithmic Function fx ax, a > 0, a 1 fx loga x, a > 0, a 1 y 3 2 1 f(x) = 1 x −1 1 2 3 x y f(x) = ax f(x) = a−x (0, 1) y 1 f(x) = loga x (1, 0) 1 2 x x −1 , 0 0, ) , 0 0, ) Domain: Range: No intercepts Decreasing on Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis , 0 and 0, , Domain: 0, Range: 0, 1 Intercept: Increasing on for fx ax Decreasing on for fx ax , , 0, Domain: , Range: 1, 0 Intercept: Increasing on Vertical asymptote: y-axis Continuous Reflection of graph of 0, fx ax Horizontal asymptote: x-axis Continuous in the line y x Sine Function fx sin x y 3 2 1 f(x) = sin x Cosine Function fx cos x y 3 2 f(x) = cos x Tangent Function fx tan x y f(x) = tan x 3 2 1 π 3π 2 x −2 −3 −2 −3 , Domain: 1, 1 Range: 2 Period: x-intercepts: y-intercept: Odd function Origin symmetry n, 0 0, 0 , Domain: Range: Period: 1, 1 2 n, 0 2 0, 1 x-intercepts: y-intercept: Even function y-axis symmetry n Domain: all x 2 , Range: Period: x-intercepts: y-intercept: Vertical asymptotes: n, 0 0, 0 x n 2 Odd function Origin symmetry 333201_AP_BES.qxd 12/5/05 11:45 AM Page ES3 Cosecant Function fx csc x Secant Function fx sec x Cotangent Function fx cot x y f(x) = csc x = 1 sin x y f(x) = sec x = 1 cos x 3 2 1 3 2 f(x) = cot x = 1 tan x y 3 2 1 − π x π 2π 3π 2π 2π −2 −3 x n , 1 1, 2 Domain: all Range: Period: No intercepts Vertical asymptotes: Odd function Origin symmetry x n Domain: all x n 2 , 1 1, Range: 2 Period: y-intercept: Vertical asymptotes: 0, 1 x n 2 Even function y-axis symmetry x n Domain: all Range: Period: , x-intercepts: 2 n, 0 Vertical asymptotes: Odd function Origin symmetry x n Inverse Sine Function fx arcsin x Inverse Cosine Function fx arccos x Inverse Tangent Function fx arctan x y π 2 −1 x 1 f(x) = arcsin x π 2 − y π f(x) = arccos x −1 x 1 y π 2 π− 2 −2 −1 x 1 2 f(x) = arctan x Range: Domain: 1, 1 , 2 2 0, 0 Intercept: Odd function Origin symmetry 1, 1 Domain: 0, Range: y-intercept: 0, 2 Range: , Domain: , 2 2 0, 0 Intercept: Horizontal asymptotes: y ± 2 Odd function Origin symmetry
participate. The variable could be the amount of extracurricular activities by one high school student. Let X = the amount of extracurricular activities by one high school student. The data are the number of extracurricular activities in which the high school students participate. Examples of the data are 2, 5, 7. 1.1 Find an article online or in a newspaper or magazine that refers to a statistical study or poll. Identify what each of the key terms—population, sample, parameter, statistic, variable, and data—refers to in the study mentioned in the article. Does the article use the key terms correctly? Example 1.2 Determine what the key terms refer to in the following study. A study was conducted at a local high school to analyze the average cumulative GPAs of students who graduated last year. Fill in the letter of the phrase that best describes each of the items below. 1. Population ____ 2. Statistic ____ 3. Parameter ____ 4. Sample ____ 5. Variable ____ 6. Data ____ a) all students who attended the high school last year b) the cumulative GPA of one student who graduated from the high school last year c) 3.65, 2.80, 1.50, 3.90 d) a group of students who graduated from the high school last year, randomly selected e) the average cumulative GPA of students who graduated from the high school last year f) all students who graduated from the high school last year g) the average cumulative GPA of students in the study who graduated from the high school last year This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 9 Solution 1.2 1. f; 2. g; 3. e; 4. d; 5. b; 6. c Example 1.3 Determine what the population, sample, parameter, statistic, variable, and data referred to in the following study. As part of a study designed to test the safety of automobiles, the National Transportation Safety Board collected and reviewed data about the effects of an automobile crash on test dummies (The Data and Story Library, n.d.). Here is the criterion they used. Speed at which Cars Crashed Location of Driver (i.e., dummies) 35 miles/hour Front seat Table 1.1 Cars with dummies in the front seats were crashed into a wall at a speed of 35 miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of 75 cars. Solution 1.3 The population is all cars containing dummies in the front seat. The sample is the 75 cars, selected by a simple random sample. The parameter is the proportion of driver dummies—if they had been real people—who would have suffered head injuries in the population. The statistic is proportion of driver dummies—if they had been real people—who would have suffered head injuries in the sample. The variable X = the number of driver dummies—if they had been real people—who would have suffered head injuries. The data are either: yes, had head injury, or no, did not. Example 1.4 Determine what the population, sample, parameter, statistic, variable, and data referred to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects 500 doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit. Solution 1.4 The population is all medical doctors listed in the professional directory. The parameter is the proportion of medical doctors who have been involved in one or more malpractice suits in the population. The sample is the 500 doctors selected at random from the professional directory. The statistic is the proportion of medical doctors who have been involved in one or more malpractice suits in the sample. The variable X = the number of medical doctors who have been involved in one or more malpractice suits. 10 Chapter 1 \| Sampling and Data The data are either: yes, was involved in one or more malpractice lawsuits; or no, was not. Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average—mean—number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were 1, 0, 1, 3, and 4 glasses of milk. 1.2 \| Data, Sampling, and Variation in Data and Sampling Data may come from a population or from a sample. Lowercase letters like x or y generally are used to represent data values. Most data can be put into the following categories: • Qualitative • Quantitative Qualitative data are the result of categorizing or describing attributes of a population. Qualitative data are also often called categorical data. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O–, or B+. Researchers often prefer to use quantitative data over qualitative data because it lends itself more easily to mathematical analysis. For example, it does not make sense to find an average hair color or blood type. Quantitative data are always numbers. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous. All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get values such as zero, one, two, or three. Data that are not only made up of counting numbers, but that may include fractions, decimals, or irrational numbers, are called quantitative continuous data. Continuous data are often the results of measurements like lengths, weights, or times. A list of the lengths in minutes for all the phone calls that you make in a week, with numbers like 2.4, 7.5, or 11.0, would be quantitative continuous data. Example 1.5 Data Sample of Quantitative Discrete Data The data are the number of books students carry in their backpacks. You sample five students. Two students carry three books, one student carries four books, one student carries two books, and one student carries one book. The numbers of books, 3, 4, 2, and 1, are the quantitative discrete data. 1.5 The data are the number of machines in a gym. You sample five gyms. One gym has 12 machines, one gym has 15 machines, one gym has 10 machines, one gym has 22 machines, and the other gym has 20 machines. What type of data is this? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 11 Example 1.6 Data Sample of Quantitative Continuous Data The data are the weights of backpacks with books in them. You sample the same five students. The weights, in pounds, of their backpacks are 6.2, 7, 6.8, 9.1, 4.3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data. 1.6 The data are the areas of lawns in square feet. You sample five houses. The areas of the lawns are 144 sq. ft., 160 sq. ft., 190 sq. ft., 180 sq. ft., and 210 sq. ft. What type of data is this? Example 1.7 You go to the supermarket and purchase three cans of soup (19 ounces tomato bisque, 14.1 ounces lentil, and 19 ounces Italian wedding), two packages of nuts (walnuts and peanuts), four different kinds of vegetable (broccoli, cauliflower, spinach, and carrots), and two desserts (16 ounces pistachio ice cream and 32 ounces chocolate chip cookies). Name data sets that are quantitative discrete, quantitative continuous, and qualitative. Solution 1.7 A possible solution • One example of a quantitative discrete data set would be three cans of soup, two packages of nuts, four kinds of vegetables, and two desserts because you count them. • The weights of the soups (19 ounces, 14.1 ounces, 19 ounces) are quantitative continuous data because you measure weights as precisely as possible. • Types of soups, nuts, vegetables, and desserts are qualitative data because they are categorical. Try to identify additional data sets in this example. Example 1.8 The data are the colors of backpacks. Again, you sample the same five students. One student has a red backpack, two students have black backpacks, one student has a green backpack, and one student has a gray backpack. The colors red, black, black, green, and gray are qualitative data. 1.8 The data are the colors of houses. You sample five houses. The colors of the houses are white, yellow, white, red, and white. What type of data is this? NOTE You may collect data as numbers and report it categorically. For example, the quiz scores for each student are recorded throughout the term. At the end of the term, the quiz scores are reported as A, B, C, D, or F. 12 Chapter 1 \| Sampling and Data Example 1.9 Work collaboratively to determine the correct data type: quantitative or qualitative. Indicate whether quantitative data are continuous or discrete. Hint: Data that are discrete often start with the words the number of. • • • • • the number of pairs of shoes you own the type of car you drive the distance from your home to the nearest grocery store the number of classes you take per school year the type of calculator you use • weights of sumo wrestlers • number of correct answers on a quiz • IQ scores (This may cause some discussion.) Solution 1.9 Items a, d, and g are quantitative discrete; items c, f, and h are quantitative continuous; items b a
nd e are qualitative or categorical. 1.9 Determine the correct data type, quantitative or qualitative, for the number of cars in a parking lot. Indicate whether quantitative data are continuous or discrete. Example 1.10 A statistics professor collects information about the classification of her students as freshmen, sophomores, juniors, or seniors. The data she collects are summarized in the pie chart Figure 1.2. What type of data does this graph show? Figure 1.3 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 13 Solution 1.10 This pie chart shows the students in each year, which is qualitative or categorical data. 1.10 A large school district keeps data of the number of students who receive test scores on an end of the year standardized exam. The data he collects are summarized in the histogram. The class boundaries are 50 to less than 60, 60 to less than 70, 70 to less than 80, 80 to less than 90, and 90 to less than 100. Figure 1.4 Qualitative Data Discussion Below are tables comparing the number of part-time and full-time students at De Anza College and Foothill College enrolled for the spring 2010 quarter. The tables display counts, frequencies, and percentages or proportions, relative frequencies. For instance, to calculate the percentage of part time students at De Anza College, divide 9,200/22,496 to get .4089. Round to the nearest thousandth—third decimal place and then multiply by 100 to get the percentage, which is 40.9 percent. So, the percent columns make comparing the same categories in the colleges easier. Displaying percentages along with the numbers is often helpful, but it is particularly important when comparing sets of data that do not have the same totals, such as the total enrollments for both colleges in this example. Notice how much larger the percentage for part-time students at Foothill College is compared to De Anza College. De Anza College Foothill College Number Percent Number Percent Full-time 9,200 40.90% Full-time 4,059 28.60% Part-time 13,296 59.10% Part-time 10,124 71.40% Table 1.2 Fall Term 2007 (Census day) 14 Chapter 1 \| Sampling and Data De Anza College Foothill College Total 22,496 100% Total 14,183 100% Table 1.2 Fall Term 2007 (Census day) Tables are a good way of organizing and displaying data. But graphs can be even more helpful in understanding the data. Two graphs that are used to display qualitative data are pie charts and bar graphs. In a pie chart, categories of data are shown by wedges in a circle that represent the percent of individuals/items in each category. We use pie charts when we want to show parts of a whole. In a bar graph, the length of the bar for each category represents the number or percent of individuals in each category. Bars may be vertical or horizontal. We use bar graphs when we want to compare categories or show changes over tim A Pareto chart consists of bars that are sorted into order by category size (largest to smallest). Look at Figure 1.5 and Figure 1.6 and determine which graph (pie or bar) you think displays the comparisons better. It is a good idea to look at a variety of graphs to see which is the most helpful in displaying the data. We might make different choices of what we think is the best graph depending on the data and the context. Our choice also depends on what we are using the data for. Figure 1.5 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 15 Figure 1.6 Percentages That Add to More (or Less) Than 100 Percent Sometimes percentages add up to be more than 100 percent (or less than 100 percent). In the graph, the percentages add to more than 100 percent because students can be in more than one category. A bar graph is appropriate to compare the relative size of the categories. A pie chart cannot be used. It also could not be used if the percentages added to less than 100 percent. Characteristic/Category Students studying technical subjects Students studying non-technical subjects Percent 40.9% 48.6% Students who intend to transfer to a four-year educational institutional 61.0% TOTAL 150.5% Table 1.3 De Anza College Year 2010 16 Chapter 1 \| Sampling and Data Figure 1.7 Omitting Categories/Missing Data The table displays Ethnicity of Students but is missing the Other/Unknown category. This category contains people who did not feel they fit into any of the ethnicity categories or declined to respond. Notice that the frequencies do not add up to the total number of students. In this situation, create a bar graph and not a pie chart. Frequency Percent Asian Black Filipino Hispanic 8,794 1,412 1,298 4,180 Native American 146 Pacific Islander 236 5,978 White TOTAL 36.1% 5.8% 5.3% 17.1% .6% 1.0% 24.5% 22,044 out of 24,382 90.4% out of 100% Table 1.4 Ethnicity of Students at De Anza College Fall Term 2007 (Census Day) This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 17 Figure 1.8 The following graph is the same as the previous graph but the Other/Unknown percent (9.6 percent) has been included. The Other/Unknown category is large compared to some of the other categories (Native American, .6 percent, Pacific Islander 1.0 percent). This is important to know when we think about what the data are telling us. This particular bar graph in Figure 1.9 can be difficult to understand visually. The graph in Figure 1.10 is a Pareto chart. The Pareto chart has the bars sorted from largest to smallest and is easier to read and interpret. Figure 1.9 Bar Graph with Other/Unknown Category 18 Chapter 1 \| Sampling and Data Figure 1.10 Pareto Chart With Bars Sorted by Size Pie Charts: No Missing Data The following pie charts have the Other/Unknown category included since the percentages must add to 100 percent. The chart in Figure 1.11b is organized by the size of each wedge, which makes it a more visually informative graph than the unsorted, alphabetical graph in Figure 1.11a. Figure 1.11 Marginal Distributions in Two-Way Tables Below is a two-way table, also called a contingency table, showing the favorite sports for 50 adults: 20 women and 30 men. Football Basketball Tennis Total Men Women Total Table 1.5 20 5 25 8 7 15 2 8 10 30 20 50 This is a two-way table because it displays information about two categorical variables, in this case, gender and sports. Data This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 19 of this type (two variable data) are referred to as bivariate data. Because the data represent a count, or tally, of choices, it is a two-way frequency table. The entries in the total row and the total column represent marginal frequencies or marginal distributions. Note—The term marginal distributions gets its name from the fact that the distributions are found in the margins of frequency distribution tables. Marginal distributions may be given as a fraction or decimal: For example, the total for men could be given as .6 or 3/5 since 30 / 50 = .6 = 3 / 5. Marginal distributions require bivariate data and only focus on one of the variables represented in the table. In other words, the reason 20 is a marginal frequency in this two-way table is because it represents the margin or portion of the total population that is women (20/50). The reason 25 is a marginal frequency is because it represents the portion of those sampled who favor football (25/50). Note: The values that make up the body of the table (e.g., 20, 8, 2) are called joint frequencies. Conditional Distributions in Two-Way Tables The distinction between a marginal distribution and a conditional distribution is that the focus is on only a particular subset of the population (not the entire population). For example, in the table, if we focused only on the subpopulation of women who prefer football, then we could calculate the conditional distributions as shown in the two-way table below. Football Basketball Tennis Total Men Women Total Table 1.6 20 5 25 8 7 15 2 8 10 30 20 50 To find the first sub-population of women who prefer football, read the value at the intersection of the Women row and Football column which is 5. Then, divide this by the total population of football players which is 25. So, the subpopulation of football players who are women is 5/25 which is .2. Similarly, to find the subpopulation of women who play football, use the value of 5 which is the number of women who play football. Then, divide this by the total population of women which is 20. So, the subpopulation of women who play football is 5/20 which is .25. Presenting Data After deciding which graph best represents your data, you may need to present your statistical data to a class or other group in an oral report or multimedia presentation. When giving an oral presentation, you must be prepared to explain exactly how you collected or calculated the data, as well as why you chose the categories, scales, and types of graphs that you are showing. Although you may have made numerous graphs of your data, be sure to use only those that actually demonstrate the stated intentions of your statistical study. While preparing your presentation, be sure that all colors, text, and scales are visible to the entire audience. Finally, make sure to allow time for your audience to ask questions and be prepared to answer them. Example 1.11 Suppose the guidance counselors at De Anza and Foothill need to make an oral presentation of the student data presented in Figures 1.5 and 1.6. Under what context should they choose to display the pie graph? When might they choose the bar graph? For each graph, explain which features they should point out and the potential display problems that might exist. Solution 1.11 The guidance counselors should use the pie graph if the desired information is the percentage of each school’s enrollment. They should use the bar graph if knowing the exact numbers
of students and the relative sizes of each category at each school are important points to be made. For the pie graph, they should point out which color represents part-time students and which represents full-time students. They should also be sure that the numbers and colors are visible when displayed. For the bar graph, they should point out the scale and the total numbers for each category, and they should be sure that the numbers, colors, and scale marks are all displayed clearly. 20 Chapter 1 \| Sampling and Data 1.11 Suppose you were asked to give an oral presentation of the data graphed in the pie chart in Figure 1.11(b). What features would you point out on the graph? What potential display problems with the graph should you check before giving your presentation? Sampling Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population. A sample should have the same characteristics as the population it is representing. Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of random sampling. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a simple random sample. Each method has pros and cons. In a simple random sample, each group has the same chance of being selected. In other words, each sample of the same size has an equal chance of being selected. For example, suppose Lisa wants to form a four-person study group (herself and three other people) from her pre-calculus class, which has 31 members not including Lisa. To choose a simple random sample of size three from the other members of her class, Lisa could put all 31 names in a hat, shake the hat, close her eyes, and pick out three names. A more technological way is for Lisa to first list the last names of the members of her class together with a two-digit number, as in Table 1.7. ID Name ID Name ID Name 00 01 02 03 Anselmo Bautista Bayani 11 12 13 King Legeny Lisa 22 23 24 Roquero Roth Rowell Cheng 14 Lundquist 25 Salangsang 04 Cuarismo 15 Macierz 26 Slade 05 Cuningham 16 Motogawa 27 Stratcher 06 07 08 09 10 Fontecha 17 Okimoto Hong Hoobler Jiao Khan 18 19 20 21 Patel Price Quizon Reyes 28 29 30 31 Tallai Tran Wai Wood Table 1.7 Class Roster Lisa can use a table of random numbers (found in many statistics books and mathematical handbooks), a calculator, or a computer to generate random numbers. The most common random number generators are five digit numbers where each digit is a unique number from 0 to 9. For this example, suppose Lisa chooses to generate random numbers from a calculator. The numbers generated are as follows: .94360, .99832, .14669, .51470, .40581, .73381, .04399. Lisa reads two-digit groups until she has chosen three class members (That is, she reads .94360 as the groups 94, 43, 36, 60.) Each random number may only contribute one class member. If she needed to, Lisa could have generated more random numbers. The table below shows how Lisa reads two-digit numbers form each random number. Each two-digit number in the table would represent each student in the roster above in Table 1.7. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 21 Random number Numbers read by Lisa .94360 .99832 .14669 .51470 .40581 .73381 .04399 94 99 14 51 40 73 04 43 98 46 14 05 33 39 36 83 66 47 58 38 39 60 32 69 70 81 81 99 Table 1.8 Lisa randomly generated the decimals in the Random Number column. She then used each consecutive number in each decimal to make the numbers she read. Some of the read numbers correspond with the ID numbers given to the students in her class (e.g., 14 = Lundquist in Table 1.7) The random numbers .94360 and .99832 do not contain appropriate two digit numbers. However the third random number, .14669, contains 14 (the fourth random number also contains 14), the fifth random number contains 05, and the seventh random number contains 04. The two-digit number 14 corresponds to Lundquist, 05 corresponds to Cuningham, and 04 corresponds to Cuarismo. Besides herself, Lisa’s group will consist of Lundquist, Cuningham, and Cuarismo. To generate random numbers perform the following steps: • Press MATH. • Arrow over to PRB. • Press 5:randInt(0, 30). • Press ENTER for the first random number. • Press ENTER two more times for the other two random numbers. If there is a repeat press ENTER again. Note—randInt(0, 30, 3) will generate three random numbers. Figure 1.12 Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample. Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample. To choose a stratified sample, divide the population into groups called strata and then the sample is selected by picking the 22 Chapter 1 \| Sampling and Data same number of values from each strata until the desired sample size is reached. For example, you could stratify (group) your high school student population by year (freshmen, sophomore, juniors, and seniors) and then choose a proportionate simple random sample from each stratum (each year) to get a stratified random sample. To choose a simple random sample from each year, number each student of the first year, number each student of the second year, and do the same for the remaining years. Then use simple random sampling to choose proportionate numbers of students from the first year and do the same for each of the remaining years. Those numbers picked from the first year, picked from the second year, and so on represent the students who make up the stratified sample. To choose a cluster sample, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four homeroom classes from your student population, the four classes make up the cluster sample. Each class is a cluster. Number each cluster, and then choose four different numbers using random sampling. All the students of the four classes with those numbers are the cluster sample. So, unlike a stratified example, a cluster sample may not contain an equal number of randomly chosen students from each class. A type of sampling that is non-random is convenience sampling. Convenience sampling involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others. Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased. They may favor a certain group. It is better for the person conducting the survey to select the sample respondents. When you analyze data, it is important to be aware of sampling errors and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause nonsampling errors. A defective counting device can cause a nonsampling error. In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error. In statistics, a sampling bias is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others. Remember, each member of the population should have an equally likely chance of being chosen. When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied. For instance, if a survey of all students is conducted only during noon lunchtime hours is biased. This is because the students who do not have a noon lunchtime would not be included. Critical Evaluation We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include the following: • Problems with samples: —A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not reliable. Reliability in statistical measures must also be considered when analyzing data. Reliability refers to the consistency of a measure. A measure is reliable when the same results are produced given the same circumstances. • Self-selected samples—Responses only by people who choose to respond, such as internet surveys, are often unreliable. • Sample size issues—: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples include crash testing cars or medical testing for rare conditions. • Undue influence—: collecting data or asking questions in a way that influences the response. • Non-response or refusal of subject to participate: —The collected responses may no longer be representative of the population. Often, people with strong positive or negative opinions may answer surveys, which can affect the results. • Causality: —A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable. • Self-funded or self-interest studies—: A study performed by a person
or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done. • Misleading use of data—: These can be improperly displayed graphs, incomplete data, or lack of context. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 23 As a class, determine whether or not the following samples are representative. If they are not, discuss the reasons. 1. To find the average GPA of all students in a high school, use all honor students at the university as the sample. 2. To find out the most popular cereal among young people under the age of 10, stand outside a large supermarket for three hours and speak to every twentieth child under age 10 who enters the supermarket. 3. To find the average annual income of all adults in the United States, sample U.S. congressmen. Create a cluster sample by considering each state as a stratum (group). By using simple random sampling, select states to be part of the cluster. Then survey every U.S. congressman in the cluster. 4. To determine the proportion of people taking public transportation to work, survey 20 people in New York City. Conduct the survey by sitting in Central Park on a bench and interviewing every person who sits next to you. 5. To determine the average cost of a two-day stay in a hospital in Massachusetts, survey 100 hospitals across the state using simple random sampling. Example 1.12 A study is done to determine the average tuition that private high school students pay per semester. Each student in the following samples is asked how much tuition he or she paid for the fall semester. What is the type of sampling in each case? a. A sample of 100 high school students is taken by organizing the students’ names by classification (freshman, sophomore, junior, or senior) and then selecting 25 students from each. b. A random number generator is used to select a student from the alphabetical listing of all high school students in the fall semester. Starting with that student, every 50th student is chosen until 75 students are included in the sample. c. A completely random method is used to select 75 students. Each high school student in the fall semester has the same probability of being chosen at any stage of the sampling process. d. The freshman, sophomore, junior, and senior years are numbered one, two, three, and four, respectively. A random number generator is used to pick two of those years. All students in those two years are in the sample. e. An administrative assistant is asked to stand in front of the library one Wednesday and to ask the first 100 undergraduate students he encounters what they paid for tuition the fall semester. Those 100 students are the sample. Solution 1.12 a. stratified, b. systematic, c. simple random, d. cluster, e. convenience 1.12 You are going to use the random number generator to generate different types of samples from the data. This table displays six sets of quiz scores (each quiz counts 10 points) for an elementary statistics class. 24 Chapter 1 \| Sampling and Data #1 #2 #3 #4 #5 #6 5 10 10 10 8 9 7 8 7 8 10 9 8 10 9 9 10 9 8 10 9 8 6 9 5 10 9 10 Table 1.9 Scores for quizzes #1-6 for 10 students in a statistics class. Each quiz is out of 10 points. Instructions: Use the Random Number Generator to pick samples. 1. Create a stratified sample by column. Pick three quiz scores randomly from each column. a. Number each row one through 10. b. On your calculator, press Math and arrow over to PRB. c. For column 1, Press 5:randInt( and enter 1,10). Press ENTER. Record the number. Press ENTER 2 more times (even the repeats). Record these numbers. Record the three quiz scores in column one that correspond to these three numbers. d. Repeat for columns two through six. e. These 18 quiz scores are a stratified sample. 2. Create a cluster sample by picking two of the columns. Use the column numbers: one through six. a. Press MATH and arrow over to the PRB function. b. Press 5:randInt (“and then enter “1,6). Press ENTER. c. Record the number the calculator displays into the first column. Then, press ENTER. d. Record the next number the calculator displays into the second column. e. Repeat steps (c) and (d) nine more times until there are a total of 20 quiz scores for the cluster sample. 3. Create a simple random sample of 15 quiz scores. a. Use the numbering one through 60. b. Press MATH. Arrow over to PRB. Press 5:randInt(1, 60). c. Press ENTER 15 times and record the numbers. d. Record the quiz scores that correspond to these numbers. e. These 15 quiz scores are the systematic sample. 4. Create a systematic sample of 12 quiz scores. a. Use the numbering one through 60. b. Press MATH. Arrow over to PRB. Press 5:randInt(1, 60). c. Press ENTER. Record the number and the first quiz score. From that number, count ten quiz scores and This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 25 record that quiz score. Keep counting ten quiz scores and recording the quiz score until you have a sample of 12 quiz scores. You may wrap around (go back to the beginning). Example 1.13 Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). a. A soccer coach selects six players from a group of boys aged eight to ten, seven players from a group of boys aged 11 to 12, and three players from a group of boys aged 13 to 14 to form a recreational soccer team. b. A pollster interviews all human resource personnel in five different high tech companies. c. A high school educational researcher interviews 50 high school female teachers and 50 high school male teachers. d. A medical researcher interviews every third cancer patient from a list of cancer patients at a local hospital. e. A high school counselor uses a computer to generate 50 random numbers and then picks students whose names correspond to the numbers. f. A student interviews classmates in his algebra class to determine how many pairs of jeans a student owns, on average. Solution 1.13 a. stratified b. cluster c. stratified d. systematic e. simple random f. convenience 1.13 Determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). A high school principal polls 50 freshmen, 50 sophomores, 50 juniors, and 50 seniors regarding policy changes for after school activities. If we were to examine two samples representing the same population, even if we used random sampling methods for the samples, they would not be exactly the same. Just as there is variation in data, there is variation in samples. As you become accustomed to sampling, the variability will begin to seem natural. Example 1.14 Suppose ABC high school has 10,000 upperclassman (junior and senior level) students (the population). We are interested in the average amount of money a upperclassmen spends on books in the fall term. Asking all 10,000 upperclassmen is an almost impossible task. Suppose we take two different samples. First, we use convenience sampling and survey ten upperclassman students from a first term organic chemistry class. Many of these students are taking first term calculus in addition to the organic chemistry class. The amount of money they spend on books is as follows: $128, $87, $173, $116, $130, $204, $147, $189, $93, $153. The second sample is taken using a list of seniors who take P.E. classes and taking every fifth seniors on the list, for a total of ten seniors. They spend the following: $50, $40, $36, $15, $50, $100, $40, $53, $22, $22. It is unlikely that any student is in both samples. 26 Chapter 1 \| Sampling and Data a. Do you think that either of these samples is representative of (or is characteristic of) the entire 10,000 part-time student population? Solution 1.14 a. No. The first sample probably consists of science-oriented students. Besides the chemistry course, some of them are also taking first-term calculus. Books for these classes tend to be expensive. Most of these students are, more than likely, paying more than the average part-time student for their books. The second sample is a group of senior citizens who are, more than likely, taking courses for health and interest. The amount of money they spend on books is probably much less than the average parttime student. Both samples are biased. Also, in both cases, not all students have a chance to be in either sample. b. Since these samples are not representative of the entire population, is it wise to use the results to describe the entire population? Solution 1.14 b. No. For these samples, each member of the population did not have an equally likely chance of being chosen. Now, suppose we take a third sample. We choose ten different part-time students from the disciplines of chemistry, math, English, psychology, sociology, history, nursing, physical education, art, and early childhood development. We assume that these are the only disciplines in which part-time students at ABC College are enrolled and that an equal number of part-time students are enrolled in each of the disciplines. Each student is chosen using simple random sampling. Using a calculator, random numbers are generated and a student from a particular discipline is selected if he or she has a corresponding number. The students spend the following amounts: $180, $50, $150, $85, $260, $75, $180, $200, $200, $150. c. Is the sample biased? Solution 1.14 c. The sample is unbiased, but a larger sample would be recommended to increase the likelihood that the sample will be close to representative of the population. However, for a biased sampling technique, even a large sample runs the risk of not being representative of the population. Students often ask if it is good enough to take a sample, instead of surveyi
ng the entire population. If the survey is done well, the answer is yes. 1.14 A local radio station has a fan base of 20,000 listeners. The station wants to know if its audience would prefer more music or more talk shows. Asking all 20,000 listeners is an almost impossible task. The station uses convenience sampling and surveys the first 200 people they meet at one of the station’s music concert events. Twenty-four people said they’d prefer more talk shows, and 176 people said they’d prefer more music. Do you think that this sample is representative of (or is characteristic of) the entire 20,000 listener population? Variation in Data Variation is present in any set of data. For example, 16-ounce cans of beverage may contain more or less than 16 ounces of liquid. In one study, eight 16 ounce cans were measured and produced the following amount (in ounces) of beverage: 15.8, 16.1, 15.2, 14.8, 15.8, 15.9, 16.0, 15.5. Measurements of the amount of beverage in a 16-ounce can may vary because different people make the measurements or because the exact amount, 16 ounces of liquid, was not put into the cans. Manufacturers regularly run tests to determine if the amount of beverage in a 16-ounce can falls within the desired range. Be aware that as you take data, your data may vary somewhat from the data someone else is taking for the same purpose. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 27 This is completely natural. However, if two or more of you are taking the same data and get very different results, it is time for you and the others to reevaluate your data-taking methods and your accuracy. Variation in Samples It was mentioned previously that two or more samples from the same population, taken randomly, and having close to the same characteristics of the population will likely be different from each other. Suppose Doreen and Jung both decide to study the average amount of time students at their high school sleep each night. Doreen and Jung each take samples of 500 students. Doreen uses systematic sampling and Jung uses cluster sampling. Doreen's sample will be different from Jung's sample. Even if Doreen and Jung used the same sampling method, in all likelihood their samples would be different. Neither would be wrong, however. Think about what contributes to making Doreen’s and Jung’s samples different. If Doreen and Jung took larger samples, that is, the number of data values is increased, their sample results (the average amount of time a student sleeps) might be closer to the actual population average. But still, their samples would be, in all likelihood, different from each other. This is called sampling variability. In other words, it refers to how much a statistic varies from sample to sample within a population. The larger the sample size, the smaller the variability between samples will be. So, the large sample size makes for a better, more reliable statistic. Size of a Sample The size of a sample (often called the number of observations) is important. The examples you have seen in this book so far have been small. Samples of only a few hundred observations, or even smaller, are sufficient for many purposes. In polling, samples that are from 1,200–1,500 observations are considered large enough and good enough if the survey is random and is well done. You will learn why when you study confidence intervals. Be aware that many large samples are biased. For example, internet surveys are invariably biased, because people choose to respond or not. 28 Chapter 1 \| Sampling and Data Divide into groups of two, three, or four. Your instructor will give each group one six-sided die. Try this experiment twice. Roll one fair die (six-sided) 20 times. Record the number of ones, twos, threes, fours, fives, and sixes you get in Table 1.10 and Table 1.11 (frequency is the number of times a particular face of the die occurs) Face on Die Frequency 1 2 3 4 5 6 Table 1.10 First Experiment (20 rolls) Face on Die Frequency 1 2 3 4 5 6 Table 1.11 Second Experiment (20 rolls) Did the two experiments have the same results? Probably not. If you did the experiment a third time, do you expect the results to be identical to the first or second experiment? Why or why not? Which experiment had the correct results? They both did. The job of the statistician is to see through the variability and draw appropriate conclusions. 1.3 \| Frequency, Frequency Tables, and Levels of Measurement Once you have a set of data, you will need to organize it so that you can analyze how frequently each datum occurs in the set. However, when calculating the frequency, you may need to round your answers so that they are as precise as possible. Answers and Rounding Off A simple way to round off answers is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results, if possible. If it becomes necessary to round off intermediate results, carry them to at least twice as many decimal places as the final answer. Expect that some of your answers will vary from the text due to rounding errors. It is not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 29 is more helpful to leave an answer as an unreduced fraction. Levels of Measurement The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement. Not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement. They are as follows (from lowest to highest level): • Nominal scale level • Ordinal scale level • Interval scale level • Ratio scale level Data that is measured using a nominal scale is qualitative (categorical). Categories, colors, names, labels, and favorite foods along with yes or no responses are examples of nominal level data. Nominal scale data are not ordered. For example, trying to classify people according to their favorite food does not make any sense. Putting pizza first and sushi second is not meaningful. Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, but there is no agreed upon order of these brands, even though people may have personal preferences. Nominal scale data cannot be used in calculations. Data that is measured using an ordinal scale is similar to nominal scale data but there is a big difference. The ordinal scale data can be ordered. An example of ordinal scale data is a list of the top five national parks in the United States. The top five national parks in the United States can be ranked from one to five but we cannot measure differences between the data. Another example of using the ordinal scale is a cruise survey where the responses to questions about the cruise are excellent, good, satisfactory, and unsatisfactory. These responses are ordered from the most desired response to the least desired. But the differences between two pieces of data cannot be measured. Like the nominal scale data, ordinal scale data cannot be used in calculations. Data that is measured using the interval scale is similar to ordinal level data because it has a definite ordering but there is a difference between data. The differences between interval scale data can be measured though the data does not have a starting point. Temperature scales like Celsius (C) and Fahrenheit (F) are measured by using the interval scale. In both temperature measurements, 40° is equal to 100° minus 60°. Differences make sense. But 0 degrees does not because, in both scales, 0 is not the absolute lowest temperature. Temperatures like –10 °F and –15 °C exist and are colder than 0. Interval level data can be used in calculations, but one type of comparison cannot be done. 80 °C is not four times as hot as 20 °C (nor is 80 °F four times as hot as 20 °F). There is no meaning to the ratio of 80 to 20 (or four to one). Data that is measured using the ratio scale takes care of the ratio problem and gives you the most information. Ratio scale data is like interval scale data, but it has a 0 point and ratios can be calculated. For example, four multiple choice statistics final exam scores are 80, 68, 20 and 92 (out of a possible 100 points). The exams are machine-graded. The data can be put in order from lowest to highest 20, 68, 80, 92. The differences between the data have meaning. The score 92 is more than the score 68 by 24 points. Ratios can be calculated. The smallest score is 0. So 80 is four times 20. The score of 80 is four times better than the score of 20. Frequency Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows: 5, 6, 3, 3, 2, 4, 7, 5, 2, 3, 5, 6, 5, 4, 4, 3, 5, 2, 5, 3. Table 1.12 lists the different data values in ascending order and their frequencies. DATA VALUE FREQUENCY 2 3 3 5 Table 1.12 Frequency Table of Student Work Hours 30 Chapter 1 \| Sampling and Data DATA VALUE FREQUENCY 4 5 6 7 3 6 2 1 Table 1.12 Frequency Table of Student Work Hours A frequency is the number of times a value of the data occurs. According to Table 1.12, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample. A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes. To find the relative frequencies, divide each frequency by the total number of students in the sample, in this case, 20. Relative frequencies can be written as fractions, percents, or decim
als. DATA VALUE FREQUENCY RELATIVE FREQUENCY or .15 or .25 or .15 or .30 or .10 or .05 3 20 5 20 3 20 6 20 2 20 1 20 Table 1.13 Frequency Table of Student Work Hours with Relative Frequencies The sum of the values in the relative frequency column of Table 1.13 is 20 20 , or 1. Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row, as shown in Table 1.14. In the first row, the cumulative frequency is simply .15 because it is the only one. In the second row, the relative frequency was .25, so adding that to .15, we get a relative frequency of .40. Continue adding the relative frequencies in each row to get the rest of the column. DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 2 3 or .15 3 20 .15 Table 1.14 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 31 DATA VALUE FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY or .25 or .15 or .30 or .10 or .05 5 20 3 20 6 20 2 20 1 20 .15 + .25 = .40 .40 + .15 = .55 .55 + .30 = .85 .85 + .10 = .95 .95 + .05 = 1.00 Table 1.14 Frequency Table of Student Work Hours with Relative and Cumulative Relative Frequencies The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated. NOTE Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be one. However, they each should be close to one. Table 1.15 represents the heights, in inches, of a sample of 100 male semiprofessional soccer players. HEIGHTS (INCHES) FREQUENCY RELATIVE FREQUENCY 59.95–61.95 61.95–63.95 63.95–65.95 65.95–67.95 67.95–69.95 69.95–71.95 71.95–73.95 73.95–75.95 5 3 15 40 17 12 7 1 5 100 3 100 15 100 40 100 17 100 12 100 7 100 1 100 = .05 = .03 = .15 = .40 = .17 = .12 = .07 = .01 CUMULATIVE RELATIVE FREQUENCY .05 .05 + .03 = .08 .08 + .15 = .23 .23 + .40 = .63 .63 + .17 = .80 .80 + .12 = .92 .92 + .07 = .99 .99 + .01 = 1.00 Total = 100 Total = 1.00 Table 1.15 Frequency Table of Soccer Player Height 32 Chapter 1 \| Sampling and Data The data in this table have been grouped into the following intervals: • 59.95–61.95 inches • 61.95–63.95 inches • 63.95–65.95 inches • 65.95–67.95 inches • 67.95–69.95 inches • 69.95–71.95 inches • 71.95–73.95 inches • 73.95–75.95 inches NOTE This example is used again in Descriptive Statistics, where the method used to compute the intervals will be explained. In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95, seven players whose heights fall within the interval 71.95–73.95, and one player whose heights fall within the interval 73.95–75.95. All heights fall between the endpoints of an interval and not at the endpoints. Example 1.15 From Table 1.15, find the percentage of heights that are less than 65.95 inches. Solution 1.15 If you look at the first, second, and third rows, the heights are all less than 65.95 inches. There are 5 + 3 + 15 = 23 players whose heights are less than 65.95 inches. The percentage of heights less than 65.95 inches is then 23 100 or 23 percent. This percentage is the cumulative relative frequency entry in the third row. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 33 1.15 Table 1.16 shows the amount, in inches, of annual rainfall in a sample of towns. Rainfall (Inches) Frequency Relative Frequency Cumulative Relative Frequency 2.95–4.97 4.97–6.99 6.99–9.01 9.01–11.03 11.03–13.05 13.05–15.07 6 7 15 8 9 5 = .12 = .14 = .30 = .16 = .18 = .10 6 50 7 50 15 50 8 50 9 50 5 50 .12 .12 + .14 = .26 .26 + .30 = .56 .56 + .16 = .72 .72 + .18 = .90 .90 + .10 = 1.00 Total = 50 Total = 1.00 Table 1.16 From Table 1.16, find the percentage of rainfall that is less than 9.01 inches. Example 1.16 From Table 1.15, find the percentage of heights that fall between 61.95 and 65.95 inches. Solution 1.16 Add the relative frequencies in the second and third rows: .03 + .15 = .18 or 18 percent. 1.16 From Table 1.16, find the percentage of rainfall that is between 6.99 and 13.05 inches. Example 1.17 Use the heights of the 100 male semiprofessional soccer players in Table 1.15. Fill in the blanks and check your answers. a. The percentage of heights that are from 67.95–71.95 inches is ________. b. The percentage of heights that are from 67.95–73.95 inches is ________. c. The percentage of heights that are more than 65.95 inches is ________. d. The number of players in the sample who are between 61.95 and 71.95 inches tall is ________. e. What kind of data are the heights? 34 Chapter 1 \| Sampling and Data f. Describe how you could gather this data (the heights) so that the data are characteristic of all male semiprofessional soccer players. Remember, you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row. Solution 1.17 a. 29 percent b. 36 percent c. 77 percent d. 87 e. quantitative continuous f. get rosters from each team and choose a simple random sample from each 1.17 From Table 1.16, find the number of towns that have rainfall between 2.95 and 9.01 inches. In your class, have someone conduct a survey of the number of siblings (brothers and sisters) each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions: 1. What percentage of the students in your class have no siblings? 2. What percentage of the students have from one to three siblings? 3. What percentage of the students have fewer than three siblings? Example 1.18 Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows: 2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10. Table 1.17 was produced. DATA FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 3 4 3 1 3 19 1 19 .1579 .2105 Table 1.17 Frequency of Commuting Distances This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 35 DATA FREQUENCY RELATIVE FREQUENCY CUMULATIVE RELATIVE FREQUENCY 5 7 10 12 13 15 18 20 3 2 3 2 1 1 1 1 3 19 2 19 4 19 2 19 1 19 1 19 1 19 1 19 .1579 .2632 .4737 .7895 .8421 .8948 .9474 1.0000 Table 1.17 Frequency of Commuting Distances a. Is the table correct? If it is not correct, what is wrong? b. True or False: Three percent of the people surveyed commute three miles. If the statement is not correct, what should it be? If the table is incorrect, make the corrections. c. What fraction of the people surveyed commute five or seven miles? d. What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)? Solution 1.18 a. No. The frequency column sums to 18, not 19. Not all cumulative relative frequencies are correct. b. False. The frequency for three miles should be one; for two miles (left out), two. The cumulative relative frequency column should read 1052, 01579, 02105, 03684, 04737, 06316, 07368, 07895, 08421, 09474, 1.0000. c. d. 5 19 7 19 , 12 19 , 7 19 1.18 Table 1.16 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year? 36 Chapter 1 \| Sampling and Data Example 1.19 Table 1.18 contains the total number of deaths worldwide as a result of earthquakes for the period from 2000 to 2012. Year Total Number of Deaths 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total Table 1.18 231 21,357 11,685 33,819 228,802 88,003 6,605 712 88,011 1,790 320,120 21,953 768 823,856 Answer the following questions: a. What is the frequency of deaths measured from 2006 through 2009? b. What percentage of deaths occurred after 2009? c. What is the relative frequency of deaths that occurred in 2003 or earlier? d. What is the percentage of deaths that occurred in 2004? e. What kind of data are the numbers of deaths? f. The Richter scale is used to quantify the energy produced by an earthquake. Examples of Richter scale numbers are 2.3, 4.0, 6.1, and 7.0. What kind of data are these numbers? Solution 1.19 a. 97,118 (11.8 percent) b. 41.6 percent c. 67,092/823,356 or 0.081 or 8.1 percent d. 27.8 percent e. quantitative discrete f. quantitative continuous This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 37 1.19 Table 1.19 contains the total number of fatal motor vehicle traffic crashes in the United States for the period from 1994–2011. Year Total Number of Crashes Year Total Number of Crashes 1994 36,254 1995 37,241 1996 37,494 1997 37,324 1998 37,107 1999 37,140 2000 37,526 2001 37,862 2002 38,491 2003 38,477 Table 1.19 2004 38,444 2005 39,252 2006 38,648 2007 37,435 2008 34,172 2009 30,862 2010 30,296 2011 29,757 Total 653,782 Answer the following questions: a. What is the frequency of deaths measured from 2000 through 2004? b. What percentage of deaths occurred after 2006? c. What is the relative frequency of deaths that occurred in 2000 or before? d. What is the percentage of deaths that occurred in 2011? e. What is the cumulative rela
tive frequency for 2006? Explain what this number tells you about the data. 1.4 \| Experimental Design and Ethics Does aspirin reduce the risk of heart attacks? Is one brand of fertilizer more effective at growing roses than another? Is fatigue as dangerous to a driver as speeding? Questions like these are answered using randomized experiments. In this module, you will learn important aspects of experimental design. Proper study design ensures the production of reliable, accurate data. The purpose of an experiment is to investigate the relationship between two variables. In an experiment, there is the explanatory variable which affects the response variable. In a randomized experiment, the researcher manipulates the explanatory variable and then observes the response variable. Each value of the explanatory variable used in an experiment is called a treatment. You want to investigate the effectiveness of vitamin E in preventing disease. You recruit a group of subjects and ask them if they regularly take vitamin E. You notice that the subjects who take vitamin E exhibit better health on average than those who do not. Does this prove that vitamin E is effective in disease prevention? It does not. There are many differences between the two groups compared in addition to vitamin E consumption. People who take vitamin E regularly often take other steps to improve their health: exercise, diet, other vitamin supplements. Any one of these factors could be influencing health. As described, this study does not prove that vitamin E is the key to disease prevention. Additional variables that can cloud a study are called lurking variables. In order to prove that the explanatory variable is causing a change in the response variable, it is necessary to isolate the explanatory variable. The researcher must design her experiment in such a way that there is only one difference between groups being compared: the planned treatments. This is accomplished by the random assignment of experimental units to treatment groups. When subjects are assigned treatments 38 Chapter 1 \| Sampling and Data randomly, all of the potential lurking variables are spread equally among the groups. At this point the only difference between groups is the one imposed by the researcher. Different outcomes measured in the response variable, therefore, must be a direct result of the different treatments. In this way, an experiment can prove a cause-and-effect connection between the explanatory and response variables. Confounding occurs when the effects of multiple factors on a response cannot be separated, for instance, if a student guesses on the even-numbered questions on an exam and sits in a favorite spot on exam day. Why does the student get a high test scores on the exam? It could be the increased study time or sitting in the favorite spot or both. Confounding makes it difficult to draw valid conclusions about the effect of each factor on the outcome. The way around this is to test several outcomes with one method (treatment). This way, we know which treatment really works. The power of suggestion can have an important influence on the outcome of an experiment. Studies have shown that the expectation of the study participant can be as important as the actual medication. In one study of performance-enhancing substances, researchers noted the following: Results showed that believing one had taken the substance resulted in [performance] times almost as fast as those associated with consuming the substance itself. In contrast, taking the substance without knowledge yielded no significant performance increment.[1] When participation in a study prompts a physical response from a participant, it is difficult to isolate the effects of the explanatory variable. To counter the power of suggestion, researchers set aside one treatment group as a control group. This group is given a placebo treatment, a treatment that cannot influence the response variable. The control group helps researchers balance the effects of being in an experiment with the effects of the active treatments. Of course, if you are participating in a study and you know that you are receiving a pill that contains no actual medication, then the power of suggestion is no longer a factor. Blinding in a randomized experiment designed to reduce bias by hiding information. When a person involved in a research study is blinded, he does not know who is receiving the active treatment(s) and who is receiving the placebo treatment. A double-blind experiment is one in which both the subjects and the researchers involved with the subjects are blinded. Sometimes, it is neither possible nor ethical for researchers to conduct experimental studies. For example, if you want to investigate whether malnutrition affects elementary school performance in children, it would not be appropriate to assign an experimental group to be malnourished. In these cases, observational studies or surveys may be used. In an observational study, the researcher does not directly manipulate the independent variable. Instead, he or she takes recordings and measurements of naturally occurring phenomena. By sorting these data into control and experimental conditions, the relationship between the dependent and independent variables can be drawn. In a survey, a researcher’s measurements consist of questionnaires that are answered by the research participants. Example 1.20 Researchers want to investigate whether taking aspirin regularly reduces the risk of a heart attack. 400 men between the ages of 50 and 84 are recruited as participants. The men are divided randomly into two groups: one group will take aspirin, and the other group will take a placebo. Each man takes one pill each day for three years, but he does not know whether he is taking aspirin or the placebo. At the end of the study, researchers count the number of men in each group who have had heart attacks. Identify the following values for this study: population, sample, experimental units, explanatory variable, response variable, treatments. Solution 1.20 The population is men aged 50 to 84. The sample is the 400 men who participated. The experimental units are the individual men in the study. The explanatory variable is oral medication. The treatments are aspirin and a placebo. The response variable is whether a subject had a heart attack. 1. McClung, M. and Collins, D. (2007 June). "Because I know it will!" Placebo effects of an ergogenic aid on athletic performance. Journal of Sport & Exercise Psychology, 29(3), 382-94. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 39 Example 1.21 The Smell & Taste Treatment and Research Foundation conducted a study to investigate whether smell can affect learning. Subjects completed mazes multiple times while wearing masks. They completed the pencil and paper mazes three times wearing floral-scented masks, and three times with unscented masks. Participants were assigned at random to wear the floral mask during the first three trials or during the last three trials. For each trial, researchers recorded the time it took to complete the maze and the subject’s impression of the mask’s scent: positive, negative, or neutral. a. Describe the explanatory and response variables in this study. b. What are the treatments? c. d. Identify any lurking variables that could interfere with this study. Is it possible to use blinding in this study? Solution 1.21 a. The explanatory variable is scent, and the response variable is the time it takes to complete the maze. b. There are two treatments: a floral-scented mask and an unscented mask. c. All subjects experienced both treatments. The order of treatments was randomly assigned so there were no differences between the treatment groups. Random assignment eliminates the problem of lurking variables. d. Subjects will clearly know whether they can smell flowers or not, so subjects cannot be blinded in this study. Researchers timing the mazes can be blinded, though. The researcher who is observing a subject will not know which mask is being worn. Example 1.22 A researcher wants to study the effects of birth order on personality. Explain why this study could not be conducted as a randomized experiment. What is the main problem in a study that cannot be designed as a randomized experiment? Solution 1.22 The explanatory variable is birth order. You cannot randomly assign a person’s birth order. Random assignment eliminates the impact of lurking variables. When you cannot assign subjects to treatment groups at random, there will be differences between the groups other than the explanatory variable. 1.22 You are concerned about the effects of texting on driving performance. Design a study to test the response time of drivers while texting and while driving only. How many seconds does it take for a driver to respond when a leading car hits the brakes? a. Describe the explanatory and response variables in the study. b. What are the treatments? c. What should you consider when selecting participants? d. Your research partner wants to divide participants randomly into two groups: one to drive without distraction and one to text and drive simultaneously. Is this a good idea? Why or why not? e. Identify any lurking variables that could interfere with this study. f. How can blinding be used in this study? 40 Ethics Chapter 1 \| Sampling and Data The widespread misuse and misrepresentation of statistical information often gives the field a bad name. Some say that “numbers don’t lie,” but the people who use numbers to support their claims often do. A recent investigation of famous social psychologist, Diederik Stapel, has led to the retraction of his articles from some of the world’s top journals including, Journal of Experimental Social Psychology, Social Psychology, Basic and Applied Social Psychology, British Journal of Social Psychology, and the magazine Science. D
iederik Stapel is a former professor at Tilburg University in the Netherlands. Over the past two years, an extensive investigation involving three universities where Stapel has worked concluded that the psychologist is guilty of fraud on a colossal scale. Falsified data taints over 55 papers he authored and 10 Ph.D. dissertations that he supervised. Stapel did not deny that his deceit was driven by ambition. But it was more complicated than that, he told me. He insisted that he loved social psychology but had been frustrated by the messiness of experimental data, which rarely led to clear conclusions. His lifelong obsession with elegance and order, he said, led him to concoct results that journals found attractive. “It was a quest for aesthetics, for beauty—instead of the truth,” he said. He described his behavior as an addiction that drove him to carry out acts of increasingly daring fraud.[2] The committee investigating Stapel concluded that he is guilty of several practices including • creating datasets, which largely confirmed the prior expectations, • altering data in existing datasets, • changing measuring instruments without reporting the change, and • misrepresenting the number of experimental subjects. Clearly, it is never acceptable to falsify data the way this researcher did. Sometimes, however, violations of ethics are not as easy to spot. Researchers have a responsibility to verify that proper methods are being followed. The report describing the investigation of Stapel’s fraud states that, “statistical flaws frequently revealed a lack of familiarity with elementary statistics.”[3] Many of Stapel’s co-authors should have spotted irregularities in his data. Unfortunately, they did not know very much about statistical analysis, and they simply trusted that he was collecting and reporting data properly. Many types of statistical fraud are difficult to spot. Some researchers simply stop collecting data once they have just enough to prove what they had hoped to prove. They don’t want to take the chance that a more extensive study would complicate their lives by producing data contradicting their hypothesis. Professional organizations, like the American Statistical Association, clearly define expectations for researchers. There are even laws in the federal code about the use of research data. When a statistical study uses human participants, as in medical studies, both ethics and the law dictate that researchers should be mindful of the safety of their research subjects. The U.S. Department of Health and Human Services oversees federal regulations of research studies with the aim of protecting participants. When a university or other research institution engages in research, it must ensure the safety of all human subjects. For this reason, research institutions establish oversight committees known as Institutional Review Boards (IRB). All planned studies must be approved in advance by the IRB. Key protections that are mandated by law include the following: • Risks to participants must be minimized and reasonable with respect to projected benefits. • Participants must give informed consent. This means that the risks of participation must be clearly explained to the subjects of the study. Subjects must consent in writing, and researchers are required to keep documentation of their consent. • Data collected from individuals must be guarded carefully to protect their privacy. These ideas may seem fundamental, but they can be very difficult to verify in practice. Is removing a participant’s name from the data record sufficient to protect privacy? Perhaps the person’s identity could be discovered from the data that remains. What happens if the study does not proceed as planned and risks arise that were not anticipated? When is informed consent really necessary? Suppose your doctor wants a blood sample to check your cholesterol level. Once the sample has been tested, you expect the lab to dispose of the remaining blood. At that point the blood becomes biological waste. Does a 2. Bhattacharjee, Y. (2013, April 26). The mind of a con man. The New York Times. Retrieved from http://www.nytimes.com/2013/04/28/magazine/diederik-stapels-audacious-academic-fraud.html?_r=3&src=dayp&. 3. Tillburg University. (2012, Nov. 28). Flawed science: the fraudulent research practices of social psychologist Diederik Stapel. Retrieved from https://www.tilburguniversity.edu/upload/3ff904d7-547b-40ae-85febea38e05a34a_Final%20report%20Flawed%20Science.pdf. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 41 researcher have the right to take it for use in a study? It is important that students of statistics take time to consider the ethical questions that arise in statistical studies. is fraud in statistical studies? You might be surprised—and disappointed. There is a website How prevalent (http://openstaxcollege.org/l/40introone) dedicated to cataloging retractions of study articles that have been proven fraudulent. A quick glance will show that the misuse of statistics is a bigger problem than most people realize. Vigilance against fraud requires knowledge. Learning the basic theory of statistics will empower you to analyze statistical studies critically. Example 1.23 Describe the unethical behavior in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A researcher is collecting data in a community. a. She selects a block where she is comfortable walking because she knows many of the people living on the street. b. No one seems to be home at four houses on her route. She does not record the addresses and does not return at a later time to try to find residents at home. c. She skips four houses on her route because she is running late for an appointment. When she gets home, she fills in the forms by selecting random answers from other residents in the neighborhood. Solution 1.23 a. By selecting a convenient sample, the researcher is intentionally selecting a sample that could be biased. Claiming that this sample represents the community is misleading. The researcher needs to select areas in the community at random. b. c. Intentionally omitting relevant data will create bias in the sample. Suppose the researcher is gathering information about jobs and child care. By ignoring people who are not home, she may be missing data from working families that are relevant to her study. She needs to make every effort to interview all members of the target sample. It is never acceptable to fake data. Even though the responses she uses are real responses provided by other participants, the duplication is fraudulent and can create bias in the data. She needs to work diligently to interview everyone on her route. 1.23 Describe the unethical behavior, if any, in each example and describe how it could impact the reliability of the resulting data. Explain how the problem should be corrected. A study is commissioned to determine the favorite brand of fruit juice among teens in California. a. The survey is commissioned by the seller of a popular brand of apple juice. b. There are only two types of juice included in the study: apple juice and cranberry juice. c. Researchers allow participants to see the brand of juice as samples are poured for a taste test. d. Twenty-five percent of participants prefer Brand X, 33 percent prefer Brand Y and 42 percent have no preference between the two brands. Brand X references the study in a commercial saying “Most teens like Brand X as much as or more than Brand Y.” 1.5 \| Data Collection Experiment 42 Chapter 1 \| Sampling and Data 1.1 Data Collection Experiment Student Learning Outcomes • The student will demonstrate the systematic sampling technique. • The student will construct relative frequency tables. • The student will interpret results and their differences from different data groupings. Movie Survey Get a class roster/list. Randomly mark a person’s name, and then mark every fourth name on the list until you get 12 names. You may have to go back to the start of the list. For each name marked, record the number of movies they saw at the theater last month. Order the Data Complete the two relative frequency tables below using your class data. Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0 1 2 3 4 5 6 7+ Table 1.20 Frequency of Number of Movies Viewed Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0–1 2–3 4–5 6–7+ Table 1.21 Frequency of Number of Movies Viewed 1. Using the tables, find the percent of data that is at most two. Which table did you use and why? 2. Using the tables, find the percent of data that is at most three. Which table did you use and why? 3. Using the tables, find the percent of data that is more than two. Which table did you use and why? 4. Using the tables, find the percent of data that is more than three. Which table did you use and why? Discussion Questions This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 43 1. 2. Is one of the tables more correct than the other? Why or why not? In general, how could you group the data differently? Are there any advantages to either way of grouping the data? 3. Why did you switch between tables, if you did, when answering the question above? 1.6 \| Sampling Experiment 44 Chapter 1 \| Sampling and Data 1.2 Sampling Experiment Student Learning Outcomes • The student will demonstrate the simple random, systematic, stratified, and cluster sampling techniques. • The student will explain the details of each procedure used. In this lab, you will be asked to pick several random samples of restaurants. In each case, describe your procedure briefly, including how you might have used the random number generator, and then list the restaurants in the sample you obtained. NOTE The following section contains restauran
ts stratified by city into columns and grouped horizontally by entree cost (clusters). Restaurants Stratified by City and Entree Cost Entree Cost San Jose Under $10 $10 to under $15 $15 to under $20 Over $20 El Abuelo Taq, Pasta Mia, Emma’s Express, Bamboo Hut Emperor’s Guard, Creekside Inn Agenda, Gervais, Miro’s Blake’s, Eulipia, Hayes Mansion, Germania Palo Alto Senor Taco, Tuscan Garden, Taxi’s Ming’s, P.A. Joe’s, Stickney’s Scott’s Seafood, Poolside Grill, Fish Market Sundance Mine, Maddalena’s, Sally's Los Gatos Mary’s Patio, Mount Everest, Sweet Pea’s, Andele Taqueria Lindsey’s, Willow Street Toll House Charter House, La Maison Du Cafe Mountain View Maharaja, New Ma’s, Thai-Rific, Garden Fresh Amber Indian, La Fiesta, Fiesta del Mar, Dawit Austin’s, Shiva’s, Mazeh Le Petit Bistro Cupertino Hobees, Hung Fu, Samrat, China Express Santa Barb. Grill, Mand. Gourmet, Bombay Oven, Kathmandu West Fontana’s, Blue Pheasant Hamasushi, Helios Sunnyvale Chekijababi, Taj India, Full Throttle, Tia Juana, Lemon Grass Pacific Fresh, Charley Brown’s, Cafe Cameroon, Faz, Aruba’s Lion & Compass, The Palace, Beau Sejour Santa Clara Rangoli, Armadillo Willy’s, Thai Pepper, Pasand Arthur’s, Katie’s Cafe, Pedro’s, La Galleria Birk’s, Truya Sushi, Valley Plaza Lakeside, Mariani’s Table 1.22 Restaurants Used in Sample A Simple Random Sample Pick a simple random sample of 15 restaurants. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 45 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ Table 1.23 A Systematic Sample Pick a systematic sample of 15 restaurants. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 2. __________ 7. __________ 12. __________ 3. __________ 8. __________ 13. __________ 4. __________ 9. __________ 14. __________ 5. __________ 10. __________ 15. __________ Table 1.24 A Stratified Sample Pick a stratified sample, by city, of 20 restaurants. Use 25 percent of the restaurants from each stratum. Round to the nearest whole number. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ Table 1.25 A Stratified Sample Pick a stratified sample, by entree cost, of 21 restaurants. Use 25 percent of the restaurants from each stratum. Round 46 Chapter 1 \| Sampling and Data to the nearest whole number. 1. Describe your procedure. 2. Complete the table with your sample. 1. __________ 6. __________ 11. __________ 16. __________ 2. __________ 7. __________ 12. __________ 17. __________ 3. __________ 8. __________ 13. __________ 18. __________ 4. __________ 9. __________ 14. __________ 19. __________ 5. __________ 10. __________ 15. __________ 20. __________ 21. __________ Table 1.26 A Cluster Sample Pick a cluster sample of restaurants from two cities. The number of restaurants will vary. 1. Describe your procedure. 2. Complete the table with your sample. 1. ________ 6. ________ 11. ________ 16. ________ 21. ________ 2. ________ 7. ________ 12. ________ 17. ________ 22. ________ 3. ________ 8. ________ 13. ________ 18. ________ 23. ________ 4. ________ 9. ________ 14. ________ 19. ________ 24. ________ 5. ________ 10. ________ 15. ________ 20. ________ 25. ________ Table 1.27 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 47 KEY TERMS average also called mean; a number that describes the central tendency of the data blinding not telling participants which treatment a subject is receiving categorical variable variables that take on values that are names or labels cluster sampling a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters; every individual in the chosen clusters is included in the sample continuous random variable a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV control group a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups convenience sampling a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data cumulative relative frequency the term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number) Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage) discrete random variable a random variable (RV) whose outcomes are counted double-blinding the act of blinding both the subjects of an experiment and the researchers who work with the subjects experimental unit any individual or object to be measured explanatory variable the independent variable in an experiment; the value controlled by researchers frequency the number of times a value of the data occurs informed consent any human subject in a research study must be cognizant of any risks or costs associated with the study; the subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits; consent must be given freely by an informed, fit participant institutional review board a committee tasked with oversight of research programs that involve human subjects lurking variable variable a variable that has an effect on a study even though it is neither an explanatory variable nor a response mathematical models a description of a phenomenon using mathematical concepts, such as equations, inequalities, distributions, etc. nonsampling error an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis numerical Variable variables that take on values that are indicated by numbers observational study a study in which the independent variable is not manipulated by the researcher parameter a number that is used to represent a population characteristic and that generally cannot be determined easily placebo an inactive treatment that has no real effect on the explanatory variable 48 Chapter 1 \| Sampling and Data population all individuals, objects, or measurements whose properties are being studied probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur proportion the number of successes divided by the total number in the sample qualitative data see data quantitative data see data random assignment the act of organizing experimental units into treatment groups using random methods random sampling selected a method of selecting a sample that gives every member of the population an equal chance of being relative frequency the ratio of the number of times a value of the data occurs in the set of all outcomes to the number of all outcomes to the total number of outcomes reliability the consistency of a measure; a measure is reliable when the same results are produced given the same circumstances representative sample a subset of the population that has the same characteristics as the population response variable experiment the dependent variable in an experiment; the value that is measured for change at the end of an sample a subset of the population studied sampling bias not all members of the population are equally likely to be selected sampling error the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error sampling with replacement once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual sampling without replacement a member of the population may be chosen for inclusion in a sample only once; if chosen, the member is not returned to the population before the next selection simple random sampling a straightforward method for selecting a random sample; give each member of the population a number Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter statistical models a description of a phenomenon using probability distributions that describe the expected behavior of the phenomenon and the variability in the expected observations stratified sampling a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum surve
y a study in which data is collected as reported by individuals. systematic sampling a method for selecting a random sample; list the members of the population Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with the one that was randomly selected. If necessary, return to the beginning of the population list to complete your sample treatments different values or components of the explanatory variable applied in an experiment validity refers to how much a measure or conclusion accurately reflects real world This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 49 variable a characteristic of interest for each person or object in a population CHAPTER REVIEW 1.1 Definitions of Statistics, Probability, and Key Terms The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. 1.2 Data, Sampling, and Variation in Data and Sampling Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete. Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampling methods include simple random sampling, stratified sampling, cluster sampling, and systematic sampling. Convenience sampling is a nonrandom method of choosing a sample that often produces biased data. Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the reliability of a sample. Statistical data needs to be critically analyzed, not simply accepted. 1.3 Frequency, Frequency Tables, and Levels of Measurement Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round your final answer to one more decimal place than was present in the original data. This means that if you have data measured to the nearest tenth of a unit, report the final statistic to the nearest hundredth. Expect that some of your answers will vary from the text due to rounding errors. In addition to rounding your answers, you can measure your data using the following four levels of measurement: • Nominal scale level data that cannot be ordered nor can it be used in calculations • Ordinal scale level data that can be ordered; the differences cannot be measured • Interval scale level data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio • Ratio scale level data with a starting point that can be ordered; the differences have meaning and ratios can be calculated When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these. 1.4 Experimental Design and Ethics A poorly designed study will not produce reliable data. There are certain key components that must be included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active treatment is not applied. Participants in the control group receive a placebo treatment that looks exactly like the active treatments but cannot influence the response variable. To preserve the integrity of the placebo, both researchers and subjects may be blinded. When a study is designed properly, the only difference between treatment groups is the one imposed by the researcher. Therefore, when groups respond differently to different treatments, the difference must be due to the influence of the explanatory variable. “An ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.”[4] Ethical violations in statistics are not always easy to spot. Professional 4. Gelman, A. (2013, May 1). Open data and open methods. Ethics and Statistics. Retrieved from http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf. 50 Chapter 1 \| Sampling and Data associations and federal agencies post guidelines for proper conduct. It is important that you learn basic statistical procedures so that you can recognize proper data analysis. PRACTICE 1.1 Definitions of Statistics, Probability, and Key Terms 1. Below is a two-way table showing the types of college sports played by men and women. Soccer Basketball Lacrosse Total Women Men Total 8 4 12 Table 1.28 8 12 20 4 4 8 20 20 40 Given these data, calculate the marginal distributions of college sports for the people surveyed. 2. Below is a two-way table showing the types of college sports played by men and women. Soccer Basketball Lacrosse Total Women Men Total 8 4 12 Table 1.29 8 12 20 4 4 8 20 20 40 Given these data, calculate the conditional distributions for the subpopulation of women who play college sports. Use the following information to answer the next five exercises. Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time in months patients live once they start the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected. Researcher A 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 Researcher B 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 Determine what the key terms refer to in the example for Researcher A. 3. population 4. sample 5. parameter 6. statistic 7. variable This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 51 1.2 Data, Sampling, and Variation in Data and Sampling 8. Number of times per week is what type of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous Use the following information to answer the next four exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Antonio, Texas. The first house in the neighborhood around the park was selected randomly, and then the resident of every eighth house in the neighborhood around the park was interviewed. 9. The sampling method was a. simple random; b. systematic; c. stratified; d. cluster 10. Duration (amount of time) is what type of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous 11. The colors of the houses around the park are what kind of data? a. qualitative (categorical); b. quantitative discrete; c. quantitative continuous 12. The population is ________. 52 Chapter 1 \| Sampling and Data 13. Table 1.30 contains the total number of deaths worldwide as a result of earthquakes from 2000–2012. Year Total Number of Deaths 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 Total 231 21,357 11,685 33,819 228,802 88,003 6,605 712 88,011 1,790 320,120 21,953 768 823,856 Table 1.30 Use Table 1.30 to answer the following questions. a. What is the proportion of deaths between 2007–2012? b. What percent of deaths occurred before 2001? c. What is the percent of deaths that occurred in 2003 or after 2010? d. What is the fraction of deaths that happened before 2012? e. What kind of data is the number of deaths? f. Earthquakes are quantified according to the amount of energy they produce (examples are 2.1, 5.0, 6.7). What type of data is that? g. What contributed to the large number of deaths in 2010? In 2004? Explain. h. If you were asked to present these data in an oral presentation, what type of graph would you choose to present and why? Explain what features you would point out on the graph during your presentation. For the following four exercises, determine the type of sampling used (simple random, stratified, systematic, cluster, or convenience). 14. A group of test subjects is divided into twelve groups; then four of the groups are chosen at random. 15. A market researcher polls every tenth person who walks into a store. 16. The first 50 people who walk into a sporting event are polled on their television preferences. 17. A computer generates 100 random numbers, and 100 people whose names correspond with the numbers on the list are chosen. Use the following information to answer the next seven exercises: Studies are often done by pharmaceutical companies to determine the effectiveness of a treatment program. Suppose that a new viral antibody drug is currently under study. It is given to patients once the virus's symptoms have revealed themselves. Of interest is the average (mean) length of time
in months patients live once starting the treatment. Two researchers each follow a different set of 40 patients with the viral disease from the start of treatment until their deaths. The following data (in months) are collected: Researcher A: 3; 4; 11; 15; 16; 17; 22; 44; 37; 16; 14; 24; 25; 15; 26; 27; 33; 29; 35; 44; 13; 21; 22; 10; 12; 8; 40; 32; 26; 27; 31; 34; 29; 17; 8; 24; 18; 47; 33; 34 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 53 Researcher B: 3; 14; 11; 5; 16; 17; 28; 41; 31; 18; 14; 14; 26; 25; 21; 22; 31; 2; 35; 44; 23; 21; 21; 16; 12; 18; 41; 22; 16; 25; 33; 34; 29; 13; 18; 24; 23; 42; 33; 29 18. Complete the tables using the data provided. Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency .5–6.5 6.5–12.5 12.5–18.5 18.5–24.5 24.5–30.5 30.5–36.5 36.5–42.5 42.5–48.5 Table 1.31 Researcher A Survival Length (in months) Frequency Relative Frequency Cumulative Relative Frequency .5–6.5 6.5–12.5 12.5–18.5 18.5–24.5 24.5–30.5 30.5–36.5 36.5-45.5 Table 1.32 Researcher B 19. Determine what the key term data refers to in the above example for Researcher A. 20. List two reasons why the data may differ. 21. Can you tell if one researcher is correct and the other one is incorrect? Why? 22. Would you expect the data to be identical? Why or why not? 23. Suggest at least two methods the researchers might use to gather random data. 24. Suppose that the first researcher conducted his survey by randomly choosing one state in the nation and then randomly picking 40 patients from that state. What sampling method would that researcher have used? 25. Suppose that the second researcher conducted his survey by choosing 40 patients he knew. What sampling method would that researcher have used? What concerns would you have about this data set, based upon the data collection method? Use the following data to answer the next five exercises: Two researchers are gathering data on hours of video games played by school-aged children and young adults. They each randomly sample different groups of 150 students from the same school. They collect the following data: 54 Chapter 1 \| Sampling and Data Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0–2 2–4 4–6 6–8 8–10 10–12 26 30 49 25 12 8 .17 .20 .33 .17 .08 .05 Table 1.33 Researcher A .17 .37 .70 .87 .95 1 Hours Played per Week Frequency Relative Frequency Cumulative Relative Frequency 0–2 2–4 4–6 6–8 8–10 10–12 48 51 24 12 11 4 .32 .34 .16 .08 .07 .03 Table 1.34 Researcher B 26. Give a reason why the data may differ. .32 .66 .82 .90 .97 1 27. Would the sample size be large enough if the population is the students in the school? 28. Would the sample size be large enough if the population is school-aged children and young adults in the United States? 29. Researcher A concludes that most students play video games between four and six hours each week. Researcher B concludes that most students play video games between two and four hours each week. Who is correct? 30. Suppose you were asked to present the data from researchers A and B in an oral presentation. When would a pie graph be appropriate? When would a bar graph more desirable? Explain which features you would point out on each type of graph and what potential display problems you would try to avoid. 31. As part of a way to reward students for participating in the survey, the researchers gave each student a gift card to a video game store. Would this affect the data if students knew about the award before the study? Use the following data to answer the next five exercises: A pair of studies was performed to measure the effectiveness of a new software program designed to help stroke patients regain their problem-solving skills. Patients were asked to use the software program twice a day, once in the morning, and once in the evening. The studies observed 200 stroke patients recovering over a period of several weeks. The first study collected the data in Table 1.35. The second study collected the data in Table 1.36. Group Showed Improvement No Improvement Deterioration Used program Did not use program Table 1.35 142 72 43 110 15 18 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 55 Group Showed Improvement No Improvement Deterioration Used program Did not use program Table 1.36 105 89 74 99 19 12 32. Given what you know, which study is correct? 33. The first study was performed by the company that designed the software program. The second study was performed by the American Medical Association. Which study is more reliable? 34. Both groups that performed the study concluded that the software works. Is this accurate? 35. The company takes the two studies as proof that their software causes mental improvement in stroke patients. Is this a fair statement? 36. Patients who used the software were also a part of an exercise program whereas patients who did not use the software were not. Does this change the validity of the conclusions from Exercise 1.34? 37. Is a sample size of 1,000 a reliable measure for a population of 5,000? 38. Is a sample of 500 volunteers a reliable measure for a population of 2,500? 39. A question on a survey reads: "Do you prefer the delicious taste of Brand X or the taste of Brand Y?" Is this a fair question? 40. Is a sample size of two representative of a population of five? 41. Is it possible for two experiments to be well run with similar sample sizes to get different data? 1.3 Frequency, Frequency Tables, and Levels of Measurement 42. What type of measure scale is being used? Nominal, ordinal, interval or ratio. Incomes measured in dollars a. High school soccer players classified by their athletic ability: superior, average, above average b. Baking temperatures for various main dishes: 350, 400, 325, 250, 300 c. The colors of crayons in a 24-crayon box d. Social security numbers e. f. A satisfaction survey of a social website by number: 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied g. Preferred TV shows: comedy, drama, science fiction, sports, news h. Time of day on an analog watch i. The distance in miles to the closest grocery store j. The dates 1066, 1492, 1644, 1947, and 1944 k. The heights of 21–65-year-old women l. Common letter grades: A, B, C, D, and F 1.4 Experimental Design and Ethics 43. Design an experiment. Identify the explanatory and response variables. Describe the population being studied and the experimental units. Explain the treatments that will be used and how they will be assigned to the experimental units. Describe how blinding and placebos may be used to counter the power of suggestion. 44. Discuss potential violations of the rule requiring informed consent. Inmates in a correctional facility are offered good behavior credit in return for participation in a study. a. b. A research study is designed to investigate a new children’s allergy medication. c. Participants in a study are told that the new medication being tested is highly promising, but they are not told that only a small portion of participants will receive the new medication. Others will receive placebo treatments and traditional treatments. 56 Chapter 1 \| Sampling and Data HOMEWORK 1.1 Definitions of Statistics, Probability, and Key Terms 45. For each of the following situations, indicate whether it would be best modeled with a mathematical model or a statistical model. Explain your answers. a. driving time from New York to Florida b. departure time of a commuter train at rush hour c. distance from your house to school d. e. weight of a bag of rice at the store temperature of a refrigerator at any given time For each of the following eight exercises, identify: a. the population, b. the sample, c. the parameter, d. the statistic, e. the variable, and f. the data. Give examples where appropriate. 46. A fitness center is interested in the mean amount of time a client exercises in the center each week. 47. Ski resorts are interested in the mean age that children take their first ski and snowboard lessons. They need this information to plan their ski classes optimally. 48. A cardiologist is interested in the mean recovery period of her patients who have had heart attacks. 49. Insurance companies are interested in the mean health costs each year of their clients, so that they can determine the costs of health insurance. 50. A politician is interested in the proportion of voters in his district who think he is doing a new good job. 51. A marriage counselor is interested in the proportion of clients she counsels who stay married. 52. Political pollsters may be interested in the proportion of people who will vote for a particular cause. 53. A marketing company is interested in the proportion of people who will buy a particular product. Use the following information to answer the next three exercises: A Lake Tahoe Community College instructor is interested in the mean number of days Lake Tahoe Community College math students are absent from class during a quarter. 54. What is the population she is interested in? a. all Lake Tahoe Community College students b. all Lake Tahoe Community College English students c. all Lake Tahoe Community College students in her classes d. all Lake Tahoe Community College math students 55. Consider the following X = number of days a Lake Tahoe Community College math student is absent. In this case, X is an example of which of the following? a. variable b. population c. statistic d. data 56. The instructor’s sample produces a mean number of days absent of 3.5 days. This value is an example of which of the following? a. parameter b. data statistic c. d. variable 1.2 Data, Sampling, and Variation in Data and Sampling For the following exercises, identify the type of data that would be used to describe a response (quantitative discrete, quantitative continuous, or qualitative), and give an exa
mple of the data. 57. number of tickets sold to a concert This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 57 58. percent of body fat 59. favorite baseball team 60. time in line to buy groceries 61. number of students enrolled at Evergreen Valley College 62. most-watched television show 63. brand of toothpaste 64. distance to the closest movie theatre 65. age of executives in Fortune 500 companies 66. number of competing computer spreadsheet software packages Use the following information to answer the next two exercises: A study was done to determine the age, number of times per week, and the duration (amount of time) of resident use of a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every 8th house in the neighborhood around the park was interviewed. 67. Number of times per week is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 68. Duration (amount of time) is what type of data? a. qualitative b. quantitative discrete c. quantitative continuous 69. Airline companies are interested in the consistency of the number of babies on each flight, so that they have adequate safety equipment. Suppose an airline conducts a survey. Over Thanksgiving weekend, it surveys six flights from Boston to Salt Lake City to determine the number of babies on the flights. It determines the amount of safety equipment needed by the result of that study. a. Using complete sentences, list three things wrong with the way the survey was conducted. b. Using complete sentences, list three ways that you would improve the survey if it were to be repeated. 70. Suppose you want to determine the mean number of students per statistics class in your state. Describe a possible sampling method in three to five complete sentences. Make the description detailed. 71. Suppose you want to determine the mean number of cans of soda drunk each month by students in their twenties at your school. Describe a possible sampling method in three to five complete sentences. Make the description detailed. 72. List some practical difficulties involved in getting accurate results from a telephone survey. 73. List some practical difficulties involved in getting accurate results from a mailed survey. 74. With your classmates, brainstorm some ways you could overcome these problems if you needed to conduct a phone or mail survey. 75. The instructor takes her sample by gathering data on five randomly selected students from each Lake Tahoe Community College math class. The type of sampling she used is which of the following? a. cluster sampling b. c. d. convenience sampling stratified sampling simple random sampling 76. A study was done to determine the age, number of times per week, and the duration (amount of time) of residents using a local park in San Jose. The first house in the neighborhood around the park was selected randomly and then every eighth house in the neighborhood around the park was interviewed. The sampling method was which of the following? simple random systematic stratified a. b. c. d. cluster 58 Chapter 1 \| Sampling and Data 77. Name the sampling method used in each of the following situations: a. A woman in the airport is handing out questionnaires to travelers asking them to evaluate the airport’s service. She does not ask travelers who are hurrying through the airport with their hands full of luggage, but instead asks all travelers who are sitting near gates and not taking naps while they wait. b. A teacher wants to know if her students are doing homework, so she randomly selects rows two and five and then calls on all students in row two and all students in row five to present the solutions to homework problems to the class. c. The marketing manager for an electronics chain store wants information about the ages of its customers. Over the next two weeks, at each store location, 100 randomly selected customers are given questionnaires to fill out asking for information about age, as well as about other variables of interest. d. The librarian at a public library wants to determine what proportion of the library users are children. The librarian has a tally sheet on which she marks whether books are checked out by an adult or a child. She records this data for every fourth patron who checks out books. e. A political party wants to know the reaction of voters to a debate between the candidates. The day after the debate, the party’s polling staff calls 1,200 randomly selected phone numbers. If a registered voter answers the phone or is available to come to the phone, that registered voter is asked whom he or she intends to vote for and whether the debate changed his or her opinion of the candidates. 78. A random survey was conducted of 3,274 people of the microprocessor generation—people born since 1971, the year the microprocessor was invented. It was reported that 48 percent of those individuals surveyed stated that if they had $2,000 to spend, they would use it for computer equipment. Also, 66 percent of those surveyed considered themselves relatively savvy computer users. a. Do you consider the sample size large enough for a study of this type? Why or why not? b. Based on your gut feeling, do you believe the percents accurately reflect the U.S. population for those individuals born since 1971? If not, do you think the percents of the population are actually higher or lower than the sample statistics? Why? Additional information: The survey, reported by Intel Corporation, was filled out by individuals who visited the Los Angeles Convention Center to see the Smithsonian Institute's road show called “America’s Smithsonian.” c. With this additional information, do you feel that all demographic and ethnic groups were equally represented at the event? Why or why not? d. With the additional information, comment on how accurately you think the sample statistics reflect the population parameters. 79. The Well-Being Index is a survey that follows trends of U.S. residents on a regular basis. There are six areas of health and wellness covered in the survey: Life Evaluation, Emotional Health, Physical Health, Healthy Behavior, Work Environment, and Basic Access. Some of the questions used to measure the Index are listed below. Identify the type of data obtained from each question used in this survey: qualitative, quantitative discrete, or quantitative continuous. a. Do you have any health problems that prevent you from doing any of the things people your age can normally do? b. During the past 30 days, for about how many days did poor health keep you from doing your usual activities? c. d. Do you have health insurance coverage? In the last seven days, on how many days did you exercise for 30 minutes or more? 80. In advance of the 1936 presidential election, a magazine released the results of an opinion poll predicting that the republican candidate Alf Landon would win by a large margin. The magazine sent post cards to approximately 10,000,000 prospective voters. These prospective voters were selected from the subscription list of the magazine, from automobile registration lists, from phone lists, and from club membership lists. Approximately 2,300,000 people returned the postcards. a. Think about the state of the United States in 1936. Explain why a sample chosen from magazine subscription lists, automobile registration lists, phone books, and club membership lists was not representative of the population of the United States at that time. b. What effect does the low response rate have on the reliability of the sample? c. Are these problems examples of sampling error or nonsampling error? d. During the same year, another pollster conducted a poll of 30,000 prospective voters. These researchers used a method they called quota sampling to obtain survey answers from specific subsets of the population. Quota sampling is an example of which sampling method described in this module? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 59 81. Crime-related and demographic statistics for 47 US states in 1960 were collected from government agencies, including the FBI's Uniform Crime Report. One analysis of this data found a strong connection between education and crime indicating that higher levels of education in a community correspond to higher crime rates. Which of the potential problems with samples discussed in Data, Sampling, and Variation in Data and Sampling could explain this connection? 82. A website that allows anyone to create and respond to polls had a question posted on April 15 which asked: “Do you feel happy paying your taxes when members of the Obama administration are allowed to ignore their tax liabilities?”[5] As of April 25, 11 people responded to this question. Each participant answered “NO!” Which of the potential problems with samples discussed in this module could explain this connection? 83. A scholarly article about response rates begins with the following quote: “Declining contact and cooperation rates in random digit dial (RDD) national telephone surveys raise serious concerns about the validity of estimates drawn from such research.”[6] The Pew Research Center for People and the Press admits “The percentage of people we interview—out of all we try to interview—has been declining over the past decade or more.”[7] a. What are some reasons for the decline in response rate over the past decade? b. Explain why researchers are concerned with the impact of the declining response rate on public opinion polls. 1.3 Frequency, Frequency Tables, and Levels of Measurement 84. Fifty part-time students were asked how many courses they were taking this term. The (incomplete) results are shown below. # of Courses Frequency Relative Frequency Cumulative Relative Frequency 1 2 3 30 15 .6 Table 1.37 Part-time Student Course Loads a. Fill in the
blanks in Table 1.37. b. What percent of students take exactly two courses? c. What percent of students take one or two courses? lastbaldeagle. Retrieved from http://www.youpolls.com/details.aspx?id=12328. 5. 6. Keeter, S., et al. (2006). Gauging the impact of growing nonresponse on estimates from a national RDD telephone survey. Public Opinion Quarterly, 70(5). Retrieved from http://hbanaszak.mjr.uw.edu.pl/TempTxt/Links/ GAUGING%20THE%20IMPACT%20OF%20GROWING.pdf. 7. Pew Research Center. (n.d.). Frequently asked questions. Retrieved from http://www.pewresearch.org/methodology/us-survey-research/frequently-asked-questions/#dont-you-have-trouble-getting-people-to-answer-your-polls. 60 Chapter 1 \| Sampling and Data 85. Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table 1.38. # Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency 0 1 3 6 7 27 18 3 1 .4500 .0500 .0167 Table 1.38 Flossing Frequency for Adults with Gum Disease a. Fill in the blanks in Table 1.38. b. What percent of adults flossed six times per week? c. What percent flossed at most three times per week? .9333 86. Nineteen immigrants to the United States were asked how many years, to the nearest year, they have lived in the United States The data are as follows: 2, 5, 7, 2, 2, 10, 20, 15, 0, 7, 0, 20, 5, 12, 15, 12, 4, 5, 10. Table 1.39 was produced. Data Frequency Relative Frequency Cumulative Relative Frequency 0 2 4 5 7 10 12 15 20 19 3 19 1 19 3 19 2 19 2 19 2 19 1 19 1 19 .1053 .2632 .3158 .4737 .5789 .6842 .7895 .8421 1.0000 Table 1.39 Frequency of Immigrant Survey Responses a. Fix the errors in Table 1.39. Also, explain how someone might have arrived at the incorrect number(s). b. Explain what is wrong with this statement: “47 percent of the people surveyed have lived in the United States for 5 years.” c. Fix the statement in b to make it correct. d. What fraction of the people surveyed have lived in the United States five or seven years? e. What fraction of the people surveyed have lived in the United States at most 12 years? f. What fraction of the people surveyed have lived in the United States fewer than 12 years? g. What fraction of the people surveyed have lived in the United States from five to 20 years, inclusive? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 61 87. How much time does it take to travel to work? Table 1.40 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly. 24.0 24.3 25.9 18.9 27.5 17.9 21.8 20.9 16.7 27.3 18.2 24.7 20.0 22.6 23.9 18.0 31.4 22.3 24.0 25.5 24.7 24.6 28.1 24.9 22.6 23.6 23.4 25.7 24.8 25.5 21.2 25.7 23.1 23.0 23.9 26.0 16.3 23.1 21.4 21.5 27.0 27.0 18.6 31.7 23.3 30.1 22.9 23.3 21.7 18.6 Table 1.40 88. A business magazine published data on the best small firms in 2012. These were firms which had been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion. Table 1.41 shows the ages of the chief executive officers for the first 60 ranked firms. Age Frequency Relative Frequency Cumulative Relative Frequency 40–44 45–49 50–54 55–59 60–64 65–69 70–74 Table 1.41 3 11 13 16 10 6 1 a. What is the frequency for CEO ages between 54 and 65? b. What percentage of CEOs are 65 years or older? c. What is the relative frequency of ages under 50? d. What is the cumulative relative frequency for CEOs younger than 55? e. Which graph shows the relative frequency and which shows the cumulative relative frequency? (a) Figure 1.13 (b) 62 Chapter 1 \| Sampling and Data Use the following information to answer the next two exercises: Table 1.42 contains data on hurricanes that have made direct hits on the United States. Between 1851-2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm. Category Number of Direct Hits Relative Frequency Cumulative Frequency 1 2 3 4 5 109 72 71 18 3 .3993 .2637 .2601 .0110 .3993 .6630 .9890 1.0000 Total = 273 Table 1.42 Frequency of Hurricane Direct Hits 89. What is the relative frequency of direct hits that were category 4 hurricanes? .0768 .0659 .2601 a. b. c. d. not enough information to calculate 90. What is the relative frequency of direct hits that were AT MOST a category 3 storm? a. b. c. d. .3480 .9231 .2601 .3370 1.4 Experimental Design and Ethics 91. How does sleep deprivation affect your ability to drive? A recent study measured the effects on 19 professional drivers. Each driver participated in two experimental sessions: one after normal sleep and one after 27 hours of total sleep deprivation. The treatments were assigned in random order. In each session, performance was measured on a variety of tasks including a driving simulation. Use key terms from this module to describe the design of this experiment. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 63 92. An advertisement for Acme Investments displays the two graphs in Figure 1.14 to show the value of Acme’s product in comparison with the Other Guy’s product. Describe the potentially misleading visual effect of these comparison graphs. How can this be corrected? (a) (b) Figure 1.14 As the graphs show, Acme consistently outperforms the Other Guys! 93. The graph in Figure 1.15 shows the number of complaints for six different airlines as reported to the U.S. Department of Transportation in February 2013. Alaska, Pinnacle, and Airtran Airlines have far fewer complaints reported than American, Delta, and United. Can we conclude that American, Delta, and United are the worst airline carriers since they have the most complaints? Figure 1.15 94. An epidemiologist is studying the spread of the common cold among college students. He is interested in how the temperature of the dorm room correlates with the incidence of new infections. How can he design an observational study to answer this question? If he chooses to use surveys in his measurements, what type of questions should he include in the survey? BRINGING IT TOGETHER: HOMEWORK 64 Chapter 1 \| Sampling and Data 95. Seven hundred and seventy-one distance learning students at Long Beach City College responded to surveys in the 2010–11 academic year. Highlights of the summary report are listed in Table 1.43. Have computer at home Unable to come to campus for classes Age 41 or over 96% 65% 24% Would like LBCC to offer more DL courses 95% Took DL classes due to a disability Live at least 16 miles from campus 17% 13% Took DL courses to fulfill transfer requirements 71% Table 1.43 LBCC Distance Learning Survey Results a. What percent of the students surveyed do not have a computer at home? b. About how many students in the survey live at least 16 miles from campus? c. If the same survey were done at Great Basin College in Elko, Nevada, do you think the percentages would be the same? Why? 96. Several online textbook retailers advertise that they have lower prices than on-campus bookstores. However, an important factor is whether the Internet retailers actually have the textbooks that students need in stock. Students need to be able to get textbooks promptly at the beginning of the college term. If the book is not available, then a student would not be able to get the textbook at all, or might get a delayed delivery if the book is back ordered. A college newspaper reporter is investigating textbook availability at online retailers. He decides to investigate one textbook for each of the following seven subjects: calculus, biology, chemistry, physics, statistics, geology, and general engineering. He consults textbook industry sales data and selects the most popular nationally used textbook in each of these subjects. He visits websites for a random sample of major online textbook sellers and looks up each of these seven textbooks to see if they are available in stock for quick delivery through these retailers. Based on his investigation, he writes an article in which he draws conclusions about the overall availability of all college textbooks through online textbook retailers. Write an analysis of his study that addresses the following issues: Is his sample representative of the population of all college textbooks? Explain why or why not. Describe some possible sources of bias in this study, and how it might affect the results of the study. Give some suggestions about what could be done to improve the study. REFERENCES 1.1 Definitions of Statistics, Probability, and Key Terms The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html. 1.2 Data, Sampling, and Variation in Data and Sampling Gallup. Retrieved from http://www.well-beingindex.com/. Gallup. Retrieved from http://www.gallup.com/poll/110548/gallup-presidential-election-trialheat-trends-19362004.aspx#4. Gallup. Retrieved from http://www.gallup.com/175196/gallup-healthways-index-methodology.aspx. Data from http://www.bookofodds.com/Relationships-Society/Articles/A0374-How-George-Gallup-Picked-the-President. LBCC Distance Learning (DL) Program. Retrieved from http://de.lbcc.edu/reports/2010-11/future/highlights.html#focus. Lusinchi, D. (2012). “President” Landon and the 1936 Literary Digest poll: Were automobile and telephone owners to blame? Social Science History 36(1), 23-54. Retrieved from https://muse.jhu.edu/article/471582/pdf. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Datafiles/USCrime.html. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 65 The Mercury News. Retrieved from http://www.mercurynews.com/. Virtual Laboratories in Probability and Statistics. Retrieved from http://www.math.uah.edu/stat/data/L
iteraryDigest.html. The Mercury News. Retrieved from http://www.mercurynews.com/. 1.3 Frequency, Frequency Tables, and Levels of Measurement Levels of Measurement. Retrieved from http://cnx.org/content/m10809/latest/. National Hurricane Center. Retrieved from http://www.nhc.noaa.gov/gifs/table5.gif. ThoughtCo. Retrieved from https://www.thoughtco.com/levels-of-measurement-in-statistics-3126349. U.S. Census Bureau. Retrieved from https://www.census.gov/quickfacts/table/PST045216/00. Levels of measurement. Retrieved from https://www.cos.edu/Faculty/georgew/Tutorial/Data_Levels.htm. 1.4 Experimental Design and Ethics Econoclass.com. Retrieved from http://www.econoclass.com/misleadingstats.html. Bloomberg Businessweek. Retrieved from www.businessweek.com. Ethics in statistics. Retrieved from http://cnx.org/content/m15555/latest/. Forbes. Retrieved from www.forbes.com. Forbes. http://www.forbes.com/best-small-companies/list/. Harvard School of Public Health. Retrieved from https://www.hsph.harvard.edu/nutritionsource/vitamin-e/. Jacskon, M.L., et al. (2013). Cognitive components of simulated driving performance: Sleep loss effect and predictors. Accident Analysis and Prevention Journal 50, 438-44. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/22721550. International Business Times. Retrieved from http://www.ibtimes.com/daily-dose-aspirin-helps-reduce-heart-attacksstudy-300443. National Highway Traffic Safety Administration. Retrieved from http://www-fars.nhtsa.dot.gov/Main/index.aspx. Athleteinme.com. Retrieved from http://www.athleteinme.com/ArticleView.aspx?id=1053. The Data and Story Library. Retrieved from http://lib.stat.cmu.edu/DASL/Stories/ScentsandLearning.html. U.S. Department of Health and Human Services. Retrieved from https://www.hhs.gov/ohrp/regulations-and-policy/ regulations/45-cfr-46/index.html. U.S. Department of Transportation. Retrieved from http://www.dot.gov/airconsumer/april-2013-air-travel-consumer-report. U.S. Geological Survey. Retrieved from http://earthquake.usgs.gov/earthquakes/eqarchives/year/. SOLUTIONS 1 soccer = 12/40 = ; basketball = 20/40 = ; lacrosse = 8/40 = 0.2 2 women who play soccer = 8/20 = ; women who play basketball = 8/20 = ; women who play lacrosse = 4/20 = ; 3 patients with the virus 5 The average length of time (in months) patients live after treatment. 7 X = the length of time (in months) patients live after treatment 9 b 11 a 13 a. .5242 b. .03 percent 66 c. 6.86 percent d. 823,088 823,856 e. quantitative discrete f. quantitative continuous Chapter 1 \| Sampling and Data g. In both years, underwater earthquakes produced massive tsunamis. h. Answers may vary. Sample answer: A bar graph with one bar for each year, in order, would be best since it would show the change in the number of deaths from year to year. In my presentation, I would point out that the scale of the graph is in thousands, and I would discuss which specific earthquakes were responsible for the greatest numbers of deaths in those years. 15 systematic 17 simple random 19 values for X, such as 3, 4, 11, and so on 21 No, we do not have enough information to make such a claim. 23 Take a simple random sample from each group. One way is by assigning a number to each patient and using a random number generator to randomly select patients. 25 This would be convenience sampling and is not random. 27 Yes, the sample size of 150 would be large enough to reflect a population of one school. 29 Even though the specific data support each researcher’s conclusions, the different results suggest that more data need to be collected before the researchers can reach a conclusion. 30 Answers may vary. Sample answer: A pie graph would be best for showing the percentage of students that fall into each Hours Played category. A bar graph would be more desirable if knowing the total numbers of students in each category is important. I would be sure that the colors used on the two pie graphs are the same for each category and are clearly distinguishable when displayed. The percentages should be legible, and the pie graph should be large enough to show the smaller sections clearly. For the bar graph, I would display the bars in chronological order and make sure that the colors used for each researcher’s data are clearly distinguishable. The numbers and the scale should be legible and clear when the bar graph is displayed. 32 There is not enough information given to judge if either one is correct or incorrect. 34 The software program seems to work because the second study shows that more patients improve while using the software than not. Even though the difference is not as large as that in the first study, the results from the second study are likely more reliable and still show improvement. 36 Yes, because we cannot tell if the improvement was due to the software or the exercise; the data is confounded, and a reliable conclusion cannot be drawn. New studies should be performed. 38 No, even though the sample is large enough, the fact that the sample consists of volunteers makes it a self-selected sample, which is not reliable. 40 No, even though the sample is a large portion of the population, two responses are not enough to justify any conclusions. Because the population is so small, it would be better to include everyone in the population to get the most accurate data. 42 a. ordinal b. interval c. nominal d. nominal e. ratio f. ordinal g. nominal This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 67 h. interval i. j. ratio interval k. ratio l. ordinal 44 a. Inmates may not feel comfortable refusing participation, or may feel obligated to take advantage of the promised benefits. They may not feel truly free to refuse participation. b. Parents can provide consent on behalf of their children, but children are not competent to provide consent for themselves. c. All risks and benefits must be clearly outlined. Study participants must be informed of relevant aspects of the study in order to give appropriate consent. 45 a. statistical model: The time any journey takes from New York to Florida is variable and depends on traffic and other driving conditions. b. statistical model: Although trains try to leave on time, the exact time of departure differs slightly from day to day. c. mathematical model: The distance from your house to school is the same every day and can be precisely determined. statistical model: The temperature of a refrigerator fluctuates as the compressor turns on and off. statistical model: The fill weight of a bag of rice is different for each bag. Manufacturers spend considerable effort to minimize the variance from bag to bag. d. e. 47 a. all children who take ski or snowboard lessons b. a group of these children c. d. the population mean age of children who take their first snowboard lesson the sample mean age of children who take their first snowboard lesson e. X = the age of one child who takes his or her first ski or snowboard lesson f. values for X, such as 3, 7, and so on 49 a. the clients of the insurance companies b. a group of the clients c. d. the mean health costs of the clients the mean health costs of the sample e. X = the health costs of one client f. values for X, such as 34, 9, 82, and so on 51 a. all the clients of this counselor b. a group of clients of this marriage counselor c. d. the proportion of all her clients who stay married the proportion of the sample of the counselor’s clients who stay married e. X = the number of couples who stay married f. yes, no 68 53 a. all people (maybe in a certain geographic area, such as the United States) b. a group of the people c. d. the proportion of all people who will buy the product the proportion of the sample who will buy the product e. X = the number of people who will buy it Chapter 1 \| Sampling and Data f. buy, not buy 55 a 57 quantitative discrete, 150 59 qualitative, Oakland A’s 61 quantitative discrete, 11,234 students 63 qualitative, Crest 65 quantitative continuous, 47.3 years 67 b 69 a. The survey was conducted using six similar flights. The survey would not be a true representation of the entire population of air travelers. Conducting the survey on a holiday weekend will not produce representative results. b. Conduct the survey during different times of the year. Conduct the survey using flights to and from various locations. Conduct the survey on different days of the week. 71 Answers will vary. Sample Answer: You could use a systematic sampling method. Stop the tenth person as they leave one of the buildings on campus at 9:50 in the morning. Then stop the tenth person as they leave a different building on campus at 1:50 in the afternoon. 73 Answers will vary. Sample Answer: Many people will not respond to mail surveys. If they do respond to the surveys, you can’t be sure who is responding. In addition, mailing lists can be incomplete. 75 b 77 convenience; cluster; stratified ; systematic; simple random 79 a. qualitative b. quantitative discrete c. quantitative discrete d. qualitative 81 Causality: The fact that two variables are related does not guarantee that one variable is influencing the other. We cannot assume that crime rate impacts education level or that education level impacts crime rate. Confounding: There are many factors that define a community other than education level and crime rate. Communities with high crime rates and high education levels may have other lurking variables that distinguish them from communities with lower crime rates and lower education levels. Because we cannot isolate these variables of interest, we cannot draw valid conclusions about the connection between education and crime. Possible lurking variables include police expenditures, unemployment levels, region, average age, and size. 83 a. Possible reasons: increased use of caller id, decreased use of landlines, increased use of private numbers, voice mail, privacy managers, hectic nature of p
ersonal schedules, decreased willingness to be interviewed b. When a large number of people refuse to participate, then the sample may not have the same characteristics of the population. Perhaps the majority of people willing to participate are doing so because they feel strongly about the This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 1 \| Sampling and Data 69 subject of the survey. 85 a. # Flossing per Week Frequency Relative Frequency Cumulative Relative Frequency 0 1 3 6 7 27 18 11 3 1 .4500 .3000 .1833 .0500 .0167 .4500 .7500 .9333 .9833 1 Table 1.44 b. 5.00 percent c. 93.33 percent 87 The sum of the travel times is 1,173.1. Divide the sum by 50 to calculate the mean value: 23.462. Because each state’s travel time was measured to the nearest tenth, round this calculation to the nearest hundredth: 23.46. 89 b 91 Explanatory variable: amount of sleep Response variable: performance measured in assigned tasks Treatments: normal sleep and 27 hours of total sleep deprivation Experimental Units: 19 professional drivers Lurking variables: none – all drivers participated in both treatments Random assignment: treatments were assigned in random order; this eliminated the effect of any learning that may take place during the first experimental session Control/Placebo: completing the experimental session under normal sleep conditions Blinding: researchers evaluating subjects’ performance must not know which treatment is being applied at the time 93 You cannot assume that the numbers of complaints reflect the quality of the airlines. The airlines shown with the greatest number of complaints are the ones with the most passengers. You must consider the appropriateness of methods for presenting data; in this case displaying totals is misleading. 94 He can observe a population of 100 college students on campus. He can collect data about the temperature of their dorm rooms and track how many of them catch a cold. If he uses a survey, the temperature of the dorm rooms can be determined from the survey. He can also ask them to self-report when they catch a cold. 96 Answers will vary. Sample answer: The sample is not representative of the population of all college textbooks. Two reasons why it is not representative are that he only sampled seven subjects and he only investigated one textbook in each subject. There are several possible sources of bias in the study. The seven subjects that he investigated are all in mathematics and the sciences; there are many subjects in the humanities, social sciences, and other subject areas, for example: literature, art, history, psychology, sociology, business, that he did not investigate at all. It may be that different subject areas exhibit different patterns of textbook availability, but his sample would not detect such results. He also looked only at the most popular textbook in each of the subjects he investigated. The availability of the most popular textbooks may differ from the availability of other textbooks in one of two ways: • The most popular textbooks may be more readily available online, because more new copies are printed, and more students nationwide are selling back their used copies • The most popular textbooks may be harder to find available online, because more student demand exhausts the supply more quickly. In reality, many college students do not use the most popular textbook in their subject, and this study gives no useful information about the situation for those less popular textbooks. He could improve this study by • expanding the selection of subjects he investigates so that it is more representative of all subjects studied by college 70 students, and Chapter 1 \| Sampling and Data • expanding the selection of textbooks he investigates within each subject to include a mixed representation of both the most popular and less popular textbooks. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 71 2 \| DESCRIPTIVE STATISTICS Figure 2.1 When you have a large amount of data, you will need to organize it in a way that makes sense. These ballots from an election are rolled together with similar ballots to keep them organized. (credit: William Greeson) Introduction By the end of this chapter, the student should be able to do the following: Chapter Objectives • Display data graphically and interpret the following graphs: stem-and-leaf plots, line graphs, bar graphs, frequency polygons, time series graphs, histograms, box plots, and dot plots • Recognize, describe, and calculate the measures of location of data with quartiles and percentiles • Recognize, describe, and calculate the measures of the center of data with mean, median, and mode • Recognize, describe, and calculate the measures of the spread of data with variance, standard deviation, and range Once you have a data collection, what will you do with it? Data can be described and presented in many different formats. For example, suppose you are interested in buying a house in a particular area. You may have no clue about the house prices, so you might ask your real estate agent to give you a sample data set of prices. Looking at all the prices in the sample often is overwhelming. A better way might be to look at the median price and the variation of prices. The median and variation 72 Chapter 2 \| Descriptive Statistics are just two ways that you will learn to describe data. Your agent might also provide you with a graph of the data. In this chapter, you will study numerical and graphical ways to describe and display your data. This area of statistics is called descriptive statistics. You will learn how to calculate and, even more important, how to interpret these measurements and graphs. A statistical graph is a tool that helps you learn about the shape or distribution of a sample or a population. A graph can be a more effective way of presenting data than a mass of numbers because we can see where data values cluster and where there are only a few data values. Newspapers and the internet use graphs to show trends and to enable readers to compare facts and figures quickly. Statisticians often graph data first to get a picture of the data. Then more formal tools may be applied. Some of the types of graphs that are used to summarize and organize data are the dot plot, the bar graph, the histogram, the stem-and-leaf plot, the frequency polygon—a type of broken line graph—the pie chart, and the box plot. In this chapter, we will briefly look at stem-and-leaf plots, line graphs, and bar graphs as well as frequency polygons, time series graphs, and dot plots. Our emphasis will be on histograms and box plots. NOTE This book contains instructions for constructing a histogram and a box plot for the TI-83+ and TI-84 calculators. The Texas Instruments (TI) website (http://education.ti.com/educationportal/sites/US/sectionHome/ support.html) provides additional instructions for using these calculators. 2.1 \| Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The stem consists of the leading digit(s), while the leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem. Make sure the leaves show a space between values, so that the exact data values may be easily determined. The frequency of data values for each stem provides information about the shape of the distribution. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 73 Example 2.1 For Susan Dean's spring precalculus class, scores for the first exam were as follows (smallest to largest): 33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100 Stem Leaf 10 0 Table 2.1 Stem-andLeaf Graph The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26 percent ⎛ ⎝ were in the 90s or 100, a fairly high number of As. ⎞ ⎠ 8 31 2.1 For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest): 32, 32, 33, 34, 38, 40, 42, 42, 43, 44, 46, 47, 47, 48, 48, 48, 49, 50, 50, 51, 52, 52, 52, 53, 54, 56, 57, 57, 60, 61 Construct a stemplot for the data. The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes, for example, writing 50 instead of 500, while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later. Example 2.2 The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data. 1.1, 1.5, 2.3, 2.5, 2.7, 3.2, 3.3, 3.3, 3.5, 3.8, 4.0, 4.2, 4.5, 4.5, 4.7, 4.8, 5.5, 5.6, 6.5, 6.7, 12.3 Do the data seem to have any concentration of values? The leaves are to the right of the decimal. 74 Chapter 2 \| Descriptive Statistics Solution 2.2 The value 12.3 may be an outlier. Values appear to concentrate at 3 and 4 kilometers. Stem Leaf 1 2 3 4 5 6 7 8 9 10 11 12 Table 2.2 2.2 The data below show the distances (in miles) from the h
omes of high school students to the school. Create a stemplot using the following data and identify any outliers. 0.5, 0.7, 1.1, 1.2, 1.2, 1.3, 1.3, 1.5, 1.5, 1.7, 1.7, 1.8, 1.9, 2.0, 2.2, 2.5, 2.6, 2.8, 2.8, 2.8, 3.5, 3.8, 4.4, 4.8, 4.9, 5.2, 5.5, 5.7, 5.8, 8.0 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 75 Example 2.3 A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Table 2.3 and Table 2.4 show the ages of presidents at their inauguration and at their death. Construct a sideby-side stem-and-leaf plot using these data. President Age President Age President Age Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren 57 61 57 57 58 57 61 54 Tyler Polk Taylor Fillmore Pierce Buchanan 51 49 64 50 48 65 Lincoln A. Johnson Grant Hayes Garfield Arthur Cleveland 52 56 46 54 49 51 47 Hoover 54 F. Roosevelt 51 Truman Eisenhower Kennedy L. Johnson Nixon 60 62 43 55 56 61 52 69 B. Harrison 55 Ford McKinley 55 54 Carter Reagan T. Roosevelt 42 G.H.W. Bush 64 Taft Wilson Harding Coolidge 51 56 55 51 Clinton G. W. Bush Obama 47 54 47 W. H. Harrison 68 Cleveland Table 2.3 Presidential Ages at Inauguration President Age President Age President Age Hoover 90 F. Roosevelt 63 Truman 88 Eisenhower 78 Kennedy L. Johnson Nixon 46 64 81 93 93 Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren 67 90 83 85 73 80 78 79 Lincoln A. Johnson Grant Hayes Garfield Arthur Cleveland 56 66 63 70 49 56 71 B. Harrison 67 Ford W. H. Harrison 68 Cleveland Tyler Polk Taylor 71 53 65 McKinley T. Roosevelt 60 Taft 72 Reagan 71 58 Table 2.4 Presidential Age at Death 76 Chapter 2 \| Descriptive Statistics President Age President Age President Age Fillmore Pierce Buchanan 74 64 77 Wilson Harding Coolidge 67 57 60 Table 2.4 Presidential Age at Death Solution 2.3 Ages at Inauguration Ages at Death Table 2.5 Notice that the leaf values increase in order, from right to left, for leaves shown to the left of the stem, while the leaf values increase in order from left to right, for leaves shown to the right of the stem. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 77 2.3 The table shows the number of wins and losses a sports team has had in 42 seasons. Create a side-by-side stemand-leaf plot of these wins and losses. Losses Wins Year Losses Wins Year 48 48 36 36 46 35 31 29 31 41 46 50 31 42 43 40 34 50 57 50 52 1968–1969 41 1969–1970 39 1970–1971 44 1971–1972 39 1972–1973 25 1973–1974 40 1974–1975 36 1975–1976 26 1976–1977 32 1977–1978 19 1978–1979 54 1979–1980 57 1980–1981 49 1981–1982 47 1982–1983 54 1983–1984 69 1984–1985 56 1985–1986 52 1986–1987 45 1987–1988 35 1988–1989 29 34 34 46 46 36 47 51 53 51 41 36 32 51 40 39 42 48 32 25 32 30 Table 2.6 41 43 38 43 57 42 46 56 50 31 28 25 33 35 28 13 26 30 37 47 53 1989–1990 1990–1991 1991–1992 1992–1993 1993–1994 1994–1995 1995–1996 1996–1997 1997–1998 1998–1999 1999–2000 2000–2001 2001–2002 2002–2003 2003–2004 2004–2005 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in Example 2.4, the x-axis (horizontal axis) consists of data values and the y-axis (vertical axis) consists of frequency points. The frequency points are connected using line segments. Example 2.4 In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table 2.7 and in Figure 2.2. 78 Chapter 2 \| Descriptive Statistics Number of Times Teenager Is Reminded Frequency 0 1 2 3 4 5 Table 2.7 2 5 8 14 7 4 Figure 2.2 2.4 In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table 2.8. Construct a line graph. Number of Times in Shop Frequency 0 1 2 3 Table 2.8 7 10 14 9 Bar graphs consist of bars that are separated from each other. The bars can be rectangles, or they can be rectangular boxes, used in three-dimensional plots, and they can be vertical or horizontal. The bar graph shown in Example 2.5 has age- This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 79 groups represented on the x-axis and proportions on the y-axis. Example 2.5 By the end of 2011, a social media site had more than 146 million users in the United States. Table 2.9 shows three age-groups, the number of users in each age-group, and the proportion (percentage) of users in each agegroup. Construct a bar graph using this data. Age-Groups Number of Site Users Proportion (%) of Site Users 13–25 26–44 45–64 Table 2.9 65,082,280 53,300,200 27,885,100 45% 36% 19% Solution 2.5 Figure 2.3 80 Chapter 2 \| Descriptive Statistics 2.5 The population in Park City is made up of children, working-age adults, and retirees. Table 2.10 shows the three age-groups, the number of people in the town from each age-group, and the proportion (%) of people in each agegroup. Construct a bar graph showing the proportions. Age-Groups Number of People Proportion of Population Children 67,059 Working-age adults 152,198 Retirees 131,662 Table 2.10 19% 43% 38% Example 2.6 The columns in Table 2.11 contain the race or ethnicity of students in U.S. public schools for the class of 2011, percentages for the Advanced Placement (AP) examinee population for that class, and percentages for the overall student population. Create a bar graph with the student race or ethnicity (qualitative data) on the x-axis and the AP examinee population percentages on the y-axis. Race/Ethnicity 1 = Asian, Asian American, or Pacific Islander 2 = Black or African American 3 = Hispanic or Latino 4 = American Indian or Alaska Native 5 = White 6 = Not reported/other Table 2.11 AP Examinee Population Overall Student Population 10.3% 9.0% 17.0% 0.6% 57.1% 6.0% 5.7% 14.7% 17.6% 1.1% 59.2% 1.7% This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 81 Solution 2.6 Figure 2.4 2.6 Park City is broken down into six voting districts. The table shows the percentage of the total registered voter population that lives in each district as well as the percentage of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district. District Registered Voter Population Overall City Population 1 2 3 4 5 6 15.5% 12.2% 9.8% 17.4% 22.8% 22.3% Table 2.12 19.4% 15.6% 9.0% 18.5% 20.7% 16.8% Example 2.7 Table 2.13 is a two-way table showing the types of pets owned by men and women. 82 Chapter 2 \| Descriptive Statistics Dogs Cats Fish Total Men 4 Women 4 Total 8 Table 2.13 2 6 8 2 2 4 8 12 20 Given these data, calculate the marginal distributions of pets for the people surveyed. Solution 2.7 Dogs = 8/20 = 0.4 Cats = 8/20 = 0.4 Fish = 4/20 = 0.2 Note—The sum of all the marginal distributions must equal one. In this case, 0.4 + 0.4 + 0.2 = 1; therefore, the solution checks. Example 2.8 Table 2.14 is a two-way table showing the types of pets owned by men and women. Dogs Cats Fish Total Men 4 Women 4 Total 8 Table 2.14 2 6 8 2 2 4 8 12 20 Given these data, calculate the conditional distributions for the subpopulation of men who own each pet type. Solution 2.8 Men who own dogs = 4/8 = 0.5 Men who own cats = 2/8 = 0.25 Men who own fish = 2/8 = 0.25 Note—The sum of all the conditional distributions must equal one. In this case, 0.5 + 0.25 + 0.25 = 1; therefore, the solution checks. 2.2 \| Histograms, Frequency Polygons, and Time Series Graphs For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 83 A histogram consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is more or less a number line, labeled with what the data represents, for example, distance from your home to school. The vertical axis is labeled either frequency or relative frequency (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data. The shape of the data refers to the shape of the distribution, whether normal, approximately normal, or skewed in some direction, whereas the center is thought of as the middle of a data set, and the spread indicates how far the values are dispersed about the center. In a skewed distribution, the mean is pulled toward the tail of the distribution. The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. Remember, frequency is defined as the number of times an answer occurs. If • f = frequency, • n = total number of data values (or the sum of the individual frequencies), and • RF = relative frequency, then RF = f n . For example, if three students in Mr. Ahab's English class of 40 students received from ninety to 100 percent, then f = 3, n = 40, and RF = f n = 0.075. Thus, 7.5 percent of the students received 90 to 100 percent. Ninety to 100 percent is a = 3 40 quantitative measures. To construct a histogram, first decide how many bars or intervals, also called classes, represent the data. Many histograms consist of five to 15 bars or classes for clarity. The width of each bar is also referred to as the bin size, which may be calculated by dividing the range of the data values by the desired number of bins (or bars). There is not a set procedure for determining the number of bars or
bar width/bin size; however, consistency is key when determining which data values to place inside each interval. Example 2.9 The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are continuous data since height is measured. 60, 60.5, 61, 61, 61.5, 63.5, 63.5, 63.5, 64, 64, 64, 64, 64, 64, 64, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 64.5, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 66.5, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 67.5, 68, 68, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69, 69.5, 69.5, 69.5, 69.5, 69.5, 70, 70, 70, 70, 70, 70, 70.5, 70.5, 70.5, 71, 71, 71, 72, 72, 72, 72.5, 72.5, 73, 73.5, 74 The smallest data value is 60, and the largest data value is 74. To make sure each is included in an interval, we can use 59.95 as the smallest value and 74.05 as the largest value, subtracting and adding .05 to these values, respectively. We have a small range here of 14.1 (74.05 – 59.95), so we will want a fewer number of bins; let’'s say eight. So, 14.1 divided by eight bins gives a bin size (or interval size) of approximately 1.76. NOTE We will round up to two and make each bar or class interval two units wide. Rounding up to two is a way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals. 84 Chapter 2 \| Descriptive Statistics The boundaries are as follows: • 59.95 • 59.95 + 2 = 61.95 • 61.95 + 2 = 63.95 • 63.95 + 2 = 65.95 • 65.95 + 2 = 67.95 • 67.95 + 2 = 69.95 • 69.95 + 2 = 71.95 • 71.95 + 2 = 73.95 • 73.95 + 2 = 75.95 The heights 60 through 61.5 inches are in the interval 59.95–61.95. The heights that are 63.5 are in the interval 61.95–63.95. The heights that are 64 through 64.5 are in the interval 63.95–65.95. The heights 66 through 67.5 are in the interval 65.95–67.95. The heights 68 through 69.5 are in the interval 67.95–69.95. The heights 70 through 71 are in the interval 69.95–71.95. The heights 72 through 73.5 are in the interval 71.95–73.95. The height 74 is in the interval 73.95–75.95. The following histogram displays the heights on the x-axis and relative frequency on the y-axis. Figure 2.5 Interval Frequency Relative Frequency 59.95–61.95 61.95–63.95 63.95–65.95 65.95–67.95 67.95–69.95 69.95–71.95 71.95–73.95 Table 2.15 5 3 15 40 17 12 7 5/100 = 0.05 3/100 = 0.03 15/100 = 0.15 40/100 = 0.40 17/100 = 0.17 12/100 = 0.12 7/100 = 0.07 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 85 Interval Frequency Relative Frequency 73.95–75.95 1 1/100 = 0.01 Table 2.15 2.9 The following data are the shoe sizes of 50 male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Use six bars on the histogram. 9, 9, 9.5, 9.5, 10, 10, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 10.5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 11.5, 12, 12, 12, 12, 12, 12, 12, 12.5, 12.5, 12.5, 12.5, 14 Example 2.10 The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is discrete data since books are counted. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6 Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books. Calculate the width of each bar/bin size/interval size. Solution 2.10 The smallest data value is 1, and the largest data value is 6. To make sure each is included in an interval, we can use 0.5 as the smallest value and 6.5 as the largest value by subtracting and adding 0.5 to these values. We have a small range here of 6 (6.5 –– 0.5), so we will want a fewer number of bins; let’'s say six this time. So, six divided by six bins gives a bin size (or interval size) of one. Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. In the previous example, we added and subtracted .05, while this time, we added and subtracted .5. Given a data set, you will be able to determine what is appropriate and reasonable. The following histogram displays the number of books on the x-axis and the frequency on the y-axis. 86 Chapter 2 \| Descriptive Statistics Figure 2.6 Go to Appendix G. There are calculator instructions for entering data and for creating a customized histogram. Create the histogram for Example 2.10. • Press Y=. Press CLEAR to delete any equations. • Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2. • • Into L1, enter 1, 2, 3, 4, 5, 6. Note that these values represent the numbers of books. Into L2, enter 11, 10, 16, 6, 5, 2. Note that these numbers represent the frequencies for the numbers of books. • Press WINDOW. Set Xmin = .5, Xscl = (6.5 – .5)/6, Ymin = –1, Ymax = 20, Yscl = 1, Xres = 1. The window settings are chosen to accurately and completely show the data value range and the frequency range. • Press second Y=. Start by pressing 4:Plotsoff ENTER. • Press second Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the third picture (histogram). Press ENTER. • Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (second 2). • Press GRAPH. • Use the TRACE key and the arrow keys to examine the histogram. 2.10 The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 Twenty student athletes play one sport. Twenty-two student athletes play two sports. Eight student athletes play three sports. Calculate a desired bin size for the data. Create a histogram and clearly label the endpoints of the intervals. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 87 Example 2.11 Using this data set, construct a histogram. Number of Hours My Classmates Spent Playing Video Games on Weekends 9.95 19.5 5.5 23 20 Table 2.16 10 22.5 11 21.9 15 2.25 7.5 10 24 22.9 16.75 15 20.75 23.75 18.8 0 12.75 17.5 18 20.5 Solution 2.11 Figure 2.7 Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram. 2.11 The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram. 22, 35, 15, 26, 40, 28, 18, 20, 25, 34, 39, 42, 24, 22, 19, 27, 22, 34, 40, 20, 38, 28 88 Chapter 2 \| Descriptive Statistics Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think would be appropriate. You may want to experiment with the number of intervals. Frequency Polygons Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons. To construct a frequency polygon, first examine the data and decide on the number of intervals and resulting interval size, for both the x-axis and y-axis. The x-axis will show the lower and upper bound for each interval, containing the data values, whereas the y-axis will represent the frequencies of the values. Each data point represents the frequency for each interval. For example, if an interval has three data values in it, the frequency polygon will show a 3 at the upper endpoint of that interval. After choosing the appropriate intervals, begin plotting the data points. After all the points are plotted, draw line segments to connect them. Example 2.12 A frequency polygon was constructed from the frequency table below. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.17 59.5 69.5 79.5 89.5 99.5 5 10 30 40 15 5 15 45 85 100 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 89 Figure 2.8 Notice that each point represents frequency for a particular interval. These points are located halfway between the lower bound and upper bound. In fact, the horizontal axis, or x-axis, shows only these midpoint values. For the interval 49.5–59.5 the value 54.5 is represented by a point, showing the correct frequency of 5. For the interval occurring before 49.5–59.5, (as well as 39.5–49.5), the value of the midpoint, or 44.5, is represented by a point, showing a frequency of 0, since we do not have any values in that range. The same idea applies to the last interval of 99.5–109.5, which has a midpoint of 104.5 and correctly shows a point representing a frequency of 0. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side. 2.12 Construct a frequency polygon of U.S. presidents’ ages at inauguration s
hown in Table 2.18. Age at Inauguration Frequency 41.5–46.5 46.5–51.5 51.5–56.5 56.5–61.5 61.5–66.5 66.5–71.5 Table 2.18 4 11 14 9 4 2 Frequency polygons are useful for comparing distributions. This comparison is achieved by overlaying the frequency polygons drawn for different data sets. 90 Chapter 2 \| Descriptive Statistics Example 2.13 We will construct an overlay frequency polygon comparing the scores from Example 2.12 with the students’ final numeric grades. Frequency Distribution for Calculus Final Test Scores Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.19 59.5 69.5 79.5 89.5 99.5 5 10 30 40 15 5 15 45 85 100 Frequency Distribution for Calculus Final Grades Lower Bound Upper Bound Frequency Cumulative Frequency 49.5 59.5 69.5 79.5 89.5 Table 2.20 59.5 69.5 79.5 89.5 99.5 10 10 30 45 5 10 20 50 95 100 Figure 2.9 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 91 Suppose that we want to study the temperature range of a region for an entire month. Every day at noon, we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected. One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph. Constructing a Time Series Graph To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By using the axes in that way, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur. 92 Chapter 2 \| Descriptive Statistics Example 2.14 The following data show the Annual Consumer Price Index each month for 10 years. Construct a time series graph for the Annual Consumer Price Index data only. Year Jan Feb Mar Apr May Jun Jul 2003 181.7 183.1 184.2 183.8 183.5 183.7 183.9 2004 185.2 186.2 187.4 188.0 189.1 189.7 189.4 2005 190.7 191.8 193.3 194.6 194.4 194.5 195.4 2006 198.3 198.7 199.8 201.5 202.5 202.9 203.5 2007 202.416 203.499 205.352 206.686 207.949 208.352 208.299 2008 211.080 211.693 213.528 214.823 216.632 218.815 219.964 2009 211.143 212.193 212.709 213.240 213.856 215.693 215.351 2010 216.687 216.741 217.631 218.009 218.178 217.965 218.011 2011 220.223 221.309 223.467 224.906 225.964 225.722 225.922 2012 226.665 227.663 229.392 230.085 229.815 229.478 229.104 Table 2.21 Year Aug Sep Oct Nov Dec Annual 2003 184.6 185.2 185.0 184.5 184.3 184.0 2004 189.5 189.9 190.9 191.0 190.3 188.9 2005 196.4 198.8 199.2 197.6 196.8 195.3 2006 203.9 202.9 201.8 201.5 201.8 201.6 2007 207.917 208.490 208.936 210.177 210.036 207.342 2008 219.086 218.783 216.573 212.425 210.228 215.303 2009 215.834 215.969 216.177 216.330 215.949 214.537 2010 218.312 218.439 218.711 218.803 219.179 218.056 2011 226.545 226.889 226.421 226.230 225.672 224.939 2012 230.379 231.407 231.317 230.221 229.601 229.594 Table 2.22 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 93 Solution 2.14 Figure 2.10 The annual amounts are plotted for each year. Then, consecutive points are connected with a line. 2.14 The following table is a portion of a data set from a banking website. Use the table to construct a time series graph for CO2 emissions for the United States. CO2 Emissions Ukraine United Kingdom United States 2003 352,259 2004 343,121 2005 339,029 2006 327,797 2007 328,357 2008 323,657 2009 272,176 Table 2.23 540,640 540,409 541,990 542,045 528,631 522,247 474,579 5,681,664 5,790,761 5,826,394 5,737,615 5,828,697 5,656,839 5,299,563 Uses of a Time Series Graph Time series graphs are important tools in various applications of statistics. When a researcher records values of the same variable over an extended period of time, it is sometimes difficult for him or her to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot. 2.3 \| Measures of the Location of the Data The common measures of location are quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1, is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile. To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the 94 Chapter 2 \| Descriptive Statistics test scores are the same as or greater than your test score. Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75th percentile. That translates into a score of at least 1220. Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant. The median is a number that measures the center of the data. You can think of the median as the middle value, but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data: 1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1 Ordered from smallest to largest: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order. Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two. 6.8 + 7.2 2 = 7 The median is seven. Half of the values are smaller than seven and half of the values are larger than seven. Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as 6.8 + 7.2 and equals 7. 2 So, the median, or second quartile ( Q2 ), is 7. The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( Q1 ) will be the middle value, or 2. Using the same procedure, we can see that the median of the upper half, or the third quartile ( Q3 ) will be the middle value of the upper half, or 9. The quartiles are illustrated below: Figure 2.11 The interquartile range is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (Q3) and the first quartile (Q1) IQR = Q3 – Q1. The IQR for this data set is calculated as 9 minus 2, or 7. The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than 1.5 × IQR below the first quartile or more than 1.5 × IQR above the third quartile. Potential outliers always require further This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 95 investigation. NOTE A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality, or they may be a key to understanding the data. Example 2.15 For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars. 389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000 Solution 2.15 Order the following data from
smallest to largest: 114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000 M = 488,800 Q1 = 230,500 + 387,000 2 = 308,750 Q3 = 639,000 + 659,000 2 = 649,000 IQR = 649,000 – 308,750 = 340,250 (1.5)(IQR) = (1.5)(340,250) = 510,375 Q1 – (1.5)(IQR) = 308,750 – 510,375 = –201,625 Q3 + (1.5)(IQR) = 649,000 + 510,375 = 1,159,375 No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier. 2.15 For the 11 salaries, calculate the IQR and determine if any salaries are outliers. The following salaries are in dollars. $33,000; $64,500; $28,000; $54,000; $72,000; $68,500; $69,000; $42,000; $54,000; $120,000; $40,500 In the example above, you just saw the calculation of the median, first quartile, and third quartile. These three values are part of the five number summary. The other two values are the minimum value (or min) and the maximum value (or max). The five number summary is used to create a box plot. 2.15 Find the interquartile range for the following two data sets and compare them. Test Scores for Class A: 69, 96, 81, 79, 65, 76, 83, 99, 89, 67, 90, 77, 85, 98, 66, 91, 77, 69, 80, 94 Test Scores for Class B: 90, 72, 80, 92, 90, 97, 92, 75, 79, 68, 70, 80, 99, 95, 78, 73, 71, 68, 95, 100 96 Chapter 2 \| Descriptive Statistics Example 2.16 Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were as follows: Amount of Sleep per School Night (Hours) Frequency Relative Frequency Cumulative Relative Frequency 4 5 6 7 8 9 10 Table 2.24 2 5 7 12 14 7 3 .04 .10 .14 .24 .28 .14 .06 .04 .14 .28 .52 .80 .94 1.00 Find the 28th percentile. Notice the .28 in the Cumulative Relative Frequency column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28th percentile is between the last six and the first seven. The 28th percentile is 6.5. Find the median. Look again at the Cumulative Relative Frequency column and find .52. The median is the 50th percentile or the second quartile. Fifty percent of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and 11 of the 7s. The median or 50th percentile is between the 25th, or seven, and 26th, or seven, values. The median is seven. Find the third quartile. The third quartile is the same as the 75th percentile. You can eyeball this answer. If you look at the Cumulative Relative Frequency column, you find .52 and .80. When you have all the fours, fives, sixes, and sevens, you have 52 percent of the data. When you include all the 8s, you have 80 percent of the data. The 75th percentile, then, must be an eight. Another way to look at the problem is to find 75 percent of 50, which is 37.5, and round up to 38. The third quartile, Q3, is the 38th value, which is an eight. You can check this answer by counting the values. There are 37 values below the third quartile and 12 values above. 2.16 Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65th percentile. Amount of Time Spent on Route (Hours) Frequency Relative Frequency Cumulative Relative Frequency 2 3 4 5 Table 2.25 12 14 10 4 .30 .35 .25 .10 .30 .65 .90 1.00 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 97 Example 2.17 Using Table 2.24: a. Find the 80th percentile. b. Find the 90th percentile. c. Find the first quartile. What is another name for the first quartile? Solution 2.17 Using the data from the frequency table, we have the following: a. The 80th percentile is between the last eight and the first nine in the table (between the 40th and 41st values). Therefore, we need to take the mean of the 40th an 41st values. The 80th percentile = 8 + 9 2 = 8.5. b. The 90th percentile will be the 45th data value (location is 0.90(50) = 45), and the 45th data value is nine. c. Q1 is also the 25th percentile. The 25th percentile location calculation: P25 = .25(50) = 12.5 ≈ 13, the 13th data value. Thus, the 25th percentile is six. 2.17 Refer to Table 2.25. Find the third quartile. What is another name for the third quartile? Your instructor or a member of the class will ask everyone in class how many sweaters he or she owns. Answer the following questions: 1. How many students were surveyed? 2. What kind of sampling did you do? 3. Construct two different histograms. For each, starting value = ________ and ending value = ________. 4. Find the median, first quartile, and third quartile. 5. Construct a table of the data to find the following: a. The 10th percentile b. The 70th percentile c. The percentage of students who own fewer than four sweaters A Formula for Finding the kth Percentile If you were to do a little research, you would find several formulas for calculating the kth percentile. Here is one of them. k = the kth percentile. It may or may not be part of the data. i = the index (ranking or position of a data value) n = the total number of data • Order the data from smallest to largest. 98 Chapter 2 \| Descriptive Statistics • Calculate i = k 100 (n + 1). • • If i is an integer, then the kth percentile is the data value in the ith position in the ordered set of data. If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example. Example 2.18 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the 70th percentile. b. Find the 83rd percentile. Solution 2.18 a. b. k = 70 i = the index n = 29 i = k 100 (n + 1) = ( 70 100 k = 83rd percentile i = the index n = 29 i = k 100 (n + 1) = ( 83 100 )(29 + 1) = 21. This equation tells us that i, or the position of the data value in the data set, is 21. So, we will count over to the 21st position, which shows a data value of 64. )(29 + 1) = 24.9, which is not an integer. Round it down to 24 and up to 25. The age in the 24th position is 71, and the age in the 25th position is 72. Average 71 and 72. The 83rd percentile is 71.5 years. 2.18 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 Calculate the 20th percentile and the 55th percentile. NOTE You can calculate percentiles using calculators and computers. There are a variety of online calculators. A Formula for Finding the Percentile of a Value in a Data Set • Order the data from smallest to largest. • x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile. • y = the number of data values equal to the data value for which you want to find the percentile. • n = the total number of data. • Calculate x + .5y n (100). Then round to the nearest integer. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 99 Example 2.19 Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the percentile for 58. b. Find the percentile for 25. Solution 2.19 a. Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58. x = 18 and y = 1. x + .5y n (100) = 18 + .5(1) 29 (100) = 63.80. Fifty-eight is the 64th percentile. b. Counting from the bottom of the list, there are three data values less than 25. There is one value of 25. x = 3 and y = 1. x + .5y n (100) = 3 + .5(1) 29 (100) = 12.07. Twenty-five is the 12th percentile. 2.19 Listed are 30 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 Find the percentiles for 47 and 31. Interpreting Percentiles, Quartiles, and Median A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15 percent of data values are less than or equal to the 15th percentile. • Low percentiles always correspond to lower data values. • High percentiles always correspond to higher data values. A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person's height doesn't carry any value judgment. Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text. GUIDELINE When writing the interpretation of a percentile in the context of the given data, make sure the sentence contains the following information: • Information about the context of the situation being considered • The data value (value of the variable) that represents the percentile • The percentage of individuals or items with data values below the percentile •
The percentage of individuals or items with data values above the percentile 100 Chapter 2 \| Descriptive Statistics Example 2.20 On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation. Solution 2.20 • Twenty-five percent of students finished the exam in 35 minutes or less. • Seventy-five percent of students finished the exam in 35 minutes or more. • A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. If you take too long, you might not be able to finish. 2.20 For the 100-meter dash, the third quartile for times for finishing the race was 11.5 seconds. Interpret the third quartile in the context of the situation. Example 2.21 On a 20-question math test, the 70th percentile for number of correct answers was 16. Interpret the 70th percentile in the context of this situation. Solution 2.21 • Seventy percent of students answered 16 or fewer questions correctly. • Thirty percent of students answered 16 or more questions correctly. • A higher percentile could be considered good, as answering more questions correctly is desirable. 2.21 On a 60-point written assignment, the 80th percentile for the number of points earned was 49. Interpret the 80th percentile in the context of this situation. Example 2.22 At a high school, it was found that the 30th percentile of number of hours that students spend studying per week is seven hours. Interpret the 30th percentile in the context of this situation. Solution 2.22 • Thirty percent of students study seven or fewer hours per week. • Seventy percent of students study seven or more hours per week. • In this example, there is not necessarily a good or bad value judgment associated with a higher or lower percentile, since the time a student studies per week is dependent on his/her needs. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 101 2.22 During a season, the 40th percentile for points scored per player in a game is eight. Interpret the 40th percentile in the context of this situation. Example 2.23 A middle school is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown: 0 minutes, 40 minutes, 60 minutes, 30 minutes, 60 minutes, 10 minutes, 45 minutes, 30 minutes, 300 minutes, 90 minutes, 30 minutes, 120 minutes, 60 minutes, 0 minutes, 20 minutes Find the five values that make up the five number summary. Min = 0 Q1 = 20 Med = 40 Q3 = 60 Max = 300 Listing the data in ascending order gives the following: Figure 2.12 The minimum value is 0. The maximum value is 300. Since there are an odd number of data values, the median is the middle value of this data set as it is arranged in ascending order, or 40. The first quartile is the median of the lower half of the scores and does not include the median. The lower half has seven data values; the median of the lower half will equal the middle value of the lower half, or 20. The third quartile is the median of the upper half of the scores and does not include the median. The upper half also has seven data values; so the median of the upper half will equal the middle value of the upper half, or 60. If you were the principal, would you be justified in purchasing new fitness equipment? Since 75 percent of the students exercise for 60 minutes or less daily, and since the IQR is 40 minutes (60 – 20 = 40), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment. However, the principal needs to be careful. The value 300 appears to be a potential outlier. Q3 + 1.5(IQR) = 60 + (1.5)(40) = 120. The value 300 is greater than 120, so it is a potential outlier. If we delete it and calculate the five values, we get the following values: Min = 0 Q1 = 20 102 Chapter 2 \| Descriptive Statistics Q3 = 60 Max = 120 We still have 75 percent of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample, and the principal should survey more students to be sure of his survey results. 2.4 \| Box Plots Box plots, also called box-and-whisker plots or box-whisker plots, give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. As mentioned previously, a box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value. We use these values to compare how close other data values are to them. To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box, and the third quartile marks the other end of the box. Approximately the middle 50 percent of the data fall inside the box. The whiskers extend from the ends of the box to the smallest and largest data values. A box plot easily shows the range of a data set, which is the difference between the largest and smallest data values (or the difference between the maximum and minimum). Unless the median, first quartile, and third quartile are the same value, the median will lie inside the box or between the first and third quartiles. The box plot gives a good, quick picture of the data. NOTE You may encounter box-and-whisker plots that have dots marking outlier values. In those cases, the whiskers are not extending to the minimum and maximum values. Consider, again, this data set: 1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5 The first quartile is two, the median is seven, and the third quartile is nine. The smallest value is one, and the largest value is 11.5. The following image shows the constructed box plot. NOTE See the calculator instructions on the TI website (https://education.ti.com/en/professional-development/ webinars-and-tutorials/technology-tutorials) or in the appendix. Figure 2.13 The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line. NOTE It is important to start a box plot with a scaled number line. Otherwise, the box plot may not be useful. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 103 Example 2.24 The following data are the heights of 40 students in a statistics class: 59, 60, 61, 62, 62, 63, 63, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 65, 65, 66, 66, 67, 67, 68, 68, 69, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 77. Construct a box plot with the following properties. Calculator instructions for finding the five number summary follow this example: • Minimum value = 59 • Maximum value = 77 • Q1: First quartile = 64.5 • Q2: Second quartile or median = 66 • Q3: Third quartile = 70 Figure 2.14 a. Each quarter has approximately 25 percent of the data. b. The spreads of the four quarters are 64.5 – 59 = 5.5 (first quarter), 66 – 64.5 = 1.5 (second quarter), 70 – 66 = 4 (third quarter), and 77 – 70 = 7 (fourth quarter). So, the second quarter has the smallest spread, and the fourth quarter has the largest spread. c. Range = maximum value – minimum value = 77 – 59 = 18. d. Interquartile Range: IQR = Q3 – Q1 = 70 – 64.5 = 5.5. e. The interval 59–65 has more than 25 percent of the data, so it has more data in it than the interval 66–70, which has 25 percent of the data. f. The middle 50 percent (middle half) of the data has a range of 5.5 inches. To find the minimum, maximum, and quartiles: Enter data into the list editor (Pres STAT 1:EDIT). If you need to clear the list, arrow up to the name L1, press CLEAR, and then arrow down. Put the data values into the list L1. Press STAT and arrow to CALC. Press 1:1-VarStats. Enter L1. Press ENTER. Use the down and up arrow keys to scroll. Smallest value = 59. Largest value = 77. Q1: First quartile = 64.5. Q2: Second quartile or median = 66. Q3: Third quartile = 70. 104 Chapter 2 \| Descriptive Statistics To construct the box plot: Press 4:Plotsoff. Press ENTER. Arrow down and then use the right arrow key to go to the fifth picture, which is the box plot. Press ENTER. Arrow down to Xlist: Press 2nd 1 for L1. Arrow down to Freq: Press ALPHA. Press 1. Press Zoom. Press 9: ZoomStat. Press TRACE and use the arrow keys to examine the box plot. 2.24 The following data are the number of pages in 40 books on a shelf. Construct a box plot using a graphing calculator and state the interquartile range. 136, 140, 178, 190, 205, 215, 217, 218, 232, 234, 240, 255, 270, 275, 290, 301, 303, 315, 317, 318, 326, 333, 343, 349, 360, 369, 377, 388, 391, 392, 398, 400, 402, 405, 408, 422, 429, 450, 475, 512 For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same. For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like the following: Figure 2.15 In this case, at least 25 percent of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25 percent of the values are equal to five. The top 25 percent of the values fall between five and seven, inclusive. Example 2.25 Test s
cores for Mr. Ramirez's class held during the day are as follows: 99, 56, 78, 55.5, 32, 90, 80, 81, 56, 59, 45, 77, 84.5, 84, 70, 72, 68, 32, 79, 90. Test scores for Ms. Park's class held during the evening are as follows: 98, 78, 68, 83, 81, 89, 88, 76, 65, 45, 98, 90, 80, 84.5, 85, 79, 78, 98, 90, 79, 81, 25.5. a. Find the smallest and largest values, the median, and the first and third quartile for Mr. Ramirez's class. b. Find the smallest and largest values, the median, and the first and third quartile for Ms. Park's class. c. For each data set, what percentage of the data is between the smallest value and the first quartile? the first quartile and the median? the median and the third quartile? the third quartile and the largest value? What percentage of the data is between the first quartile and the largest value? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 105 d. Create a box plot for each set of data. Use one number line for both box plots. e. Which box plot has the widest spread for the middle 50 percent of the data,the data between the first and third quartiles? What does this mean for that set of data in comparison to the other set of data? Solution 2.25 a. Min = 32 Q1 = 56 M = 74.5 Q3 = 82.5 Max = 99 b. Min = 25.5 Q1 = 78 M = 81 Q3 = 89 Max = 98 c. Mr. Ramirez's class: There are six data values ranging from 32 to 56: 30 percent. There are six data values ranging from 56 to 74.5: 30 percent. There are five data values ranging from 74.5 to 82.5: 25 percent. There are five data values ranging from 82.5 to 99: 25 percent. There are 16 data values between the first quartile, 56, and the largest value, 99: 75 percent. Ms. Park’s class: There are six data values ranging from 25.5 to 78: 27 percent. There are five data values ranging from 78 to the first instance of 81: 23 percent. There are six data values ranging from the second instance of 81 to 89: 27 percent. There are five data values ranging from 90 to 98: 23 percent. There are 17 values between the first quartile, 78, and the largest value, 98: 77 percent. d. Figure 2.16 e. The first data set has the wider spread for the middle 50 percent of the data. The IQR for the first data set is greater than the IQR for the second set. This means that there is more variability in the middle 50 percent of the first data set. 2.25 The following data set shows the heights in inches for the boys in a class of 40 students: 66, 66, 67, 67, 68, 68, 68, 68, 68, 69, 69, 69, 70, 71, 72, 72, 72, 73, 73, 74. The following data set shows the heights in inches for the girls in a class of 40 students: 61 61, 62, 62, 63, 63, 63, 65, 65, 65, 66, 66, 66, 67, 68, 68, 68, 69, 69, 69. Construct a box plot using a graphing calculator for each data set, and state which box plot has the wider spread for the middle 50 percent of the data. 106 Chapter 2 \| Descriptive Statistics Example 2.26 Graph a box-and-whisker plot for the following data values shown: 10, 10, 10, 15, 35, 75, 90, 95, 100, 175, 420, 490, 515, 515, 790 The five numbers used to create a box-and-whisker plot are as follows: Min: 10 Q1: 15 Med: 95 Q3: 490 Max: 790 The following graph shows the box-and-whisker plot. Figure 2.17 2.26 Follow the steps you used to graph a box-and-whisker plot for the data values shown: 0, 5, 5, 15, 30, 30, 45, 50, 50, 60, 75, 110, 140, 240, 330 2.5 \| Measures of the Center of the Data The center of a data set is also a way of describing location. The two most widely used measures of the center of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. NOTE The words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is arithmetic mean and average is technically a center location. However, in practice among non statisticians, average is commonly accepted for arithmetic mean. When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced “x bar”): x ¯ . The sample mean is a statistic. The Greek letter μ (pronounced "mew") represents the population mean. The population mean is a parameter. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. To see that both ways of calculating the mean are the same, consider the following sample: 1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 107 ¯ 11 3(1) + 2(2) + 1(3) + 5(4) 11 = 2.7. x¯ = = 2.7 In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4). You can quickly find the location of the median by using the expression n + 1 2 . The letter n is the total number of data values in the sample. As discussed earlier, if n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by two after the data have been ordered. For example, if the total number of data values is 97, then n + 1 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100, = 97 + 1 2 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and 2 then n + 1 2 = 100 + 1 2 the value of the median are not the same. The uppercase letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median. Example 2.27 Data indicating the number of months a patient with a specific disease lives after taking a new antibody drug are as follows (smallest to largest): 3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47 Calculate the mean and the median. Solution 2.27 The calculation for the mean is x¯ = [3 + 4 + (8)(2) + 10 + 11 + 12 + 13 + 14 + (15)(2) + (16)(2) + (17)(2) + 18 + 21 + (22)(2) + (24)(2) + 25 + (26)(2) + (27)(2) + (29)(2) + 31 + 32 + (33)(2) + (34)(2) + 35 + 37 + 40 + (44)(2) + 47] / 40 = 23.6. To find the median, M, first use the formula for the location. The location is n + 1 2 = 40 + 1 2 = 20.5. Start from the smallest value and count up; the median is located between the 20th and 21st values (the two 24s): 3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47 M = 24 + 24 2 = 24 To find the mean and the median: Clear list L1. Pres STAT 4:ClrList. Enter 2nd 1 for list L1. Press ENTER. Enter data into the list editor. Press STAT 1:EDIT. Put the data values into list L1. Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then ENTER. Press the down and up arrow keys to scroll. x¯ = 23.6, M = 24 108 Chapter 2 \| Descriptive Statistics 2.27 The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median. 3, 4, 5, 7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 17, 17, 18, 19, 19, 19, 21, 21, 22, 22, 23, 24, 24, 24, 24 Example 2.28 Suppose that in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn $30,000. Which is the better measure of the center: the mean or the median? Solution 2.28 x¯ = 5, 000, 000 + 49(30, 000) 50 = 129,400 M = 30,000 There are 49 people who earn $30,000 and one person who earns $5,000,000. The median is a better measure of the center than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data. 2.28 In a sample of 60 households, one house is worth $2,500,000. Half of the rest are worth $280,000, and all the others are worth $315,000. Which is the better measure of the center: the mean or the median? Another measure of the center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. Example 2.29 Statistics exam scores for 20 students are as follows: 50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93 Find the mode. Solution 2.29 The most frequent score is 72, which occurs five times. Mode = 72. 2.29 The number of books checked out from the library by 25 students are as follows: 0, 0, 0, 1, 2, 3, 3, 4, 4, 5, 5, 7, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 11, 12, 12 Find the mode. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 109 Example 2.30 Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice. When is the mode the best measure of the center? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing. NOTE The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is red, red, red, green, green,
yellow, purple, black, blue, the mode is red. Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software. 2.30 Five credit scores are 680, 680, 700, 720, 720. The data set is bimodal because the scores 680 and 720 each occur twice. Consider the annual earnings of workers at a factory. The mode is $25,000 and occurs 150 times out of 301. The median is $50,000, and the mean is $47,500. What would be the best measure of the center? The Law of Large Numbers and the Mean ¯ The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean x of the sample is very likely to get closer and closer to µ. This law is discussed in more detail later in the text. Sampling Distributions and Statistic of a Sampling Distribution You can think of a sampling distribution as a relative frequency distribution with a great many samples. See Chapter 1: Sampling and Data for a review of relative frequency. Suppose 30 randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below. Number of Movies Relative Frequency 0 1 2 3 4 Table 2.26 5 30 15 30 6 30 3 30 1 30 A relative frequency distribution includes the relative frequencies of a number of samples. Recall that a statistic is a number calculated from a sample. Statistic examples include the mean, the median, and the mode 110 Chapter 2 \| Descriptive Statistics ¯ as well as others. The sample mean x is an example of a statistic that estimates the population mean μ. Calculating the Mean of Grouped Frequency Tables When only grouped data is available, you do not know the individual data values (we know only intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table, we can apply the basic . We simply need to modify the definition to fit within the restrictions definition of mean: mean = data sum number o f data values of a frequency table. Since we do not know the individual data values, we can instead find the midpoint of each interval. The midpoint is lower boundary + upper boundary 2 . We can now modify the mean definition to be Mean o f Frequency Table = ∑ f m ∑ f , where f = the frequency of the interval, m = the midpoint of the interval, and sigma (∑) is read as "sigma" and means to sum up. So this formula says that we will sum the products of each midpoint and the corresponding frequency and divide by the sum of all of the frequencies. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 111 Example 2.31 A frequency table displaying Professor Blount’s last statistic test is shown. Find the best estimate of the class mean. Grade Interval Number of Students 50–56.5 56.5–62.5 62.5–68.5 68.5–74.5 74.5–80.5 80.5–86.5 86.5–92.5 92.5–98.5 Table 2.27 1 0 4 4 2 3 4 1 Solution 2.31 • Find the midpoints for all intervals. Grade Interval Midpoint 50–56.5 56.5–62.5 62.5–68.5 68.5–74.5 74.5–80.5 80.5–86.5 86.5–92.5 92.5–98.5 Table 2.28 53.25 59.5 65.5 71.5 77.5 83.5 89.5 95.5 • Calculate the sum of the product of each interval frequency and midpoint. ∑ f m 53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25 • μ = ∑ f m ∑ f = 1460.25 19 = 76.86 112 Chapter 2 \| Descriptive Statistics 2.31 Maris conducted a study on the effect that playing video games has on memory recall. As part of her study, she compiled the following data: Hours Teenagers Spend on Video Games Number of Teenagers 0–3.5 3.5–7.5 7.5–11.5 11.5–15.5 15.5–19.5 Table 2.29 3 7 12 7 9 What is the best estimate for the mean number of hours spent playing video games? 2.6 \| Skewness and the Mean, Median, and Mode Consider the following data set: 4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10 This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the middle of an interval. Figure 2.18 The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other. The mean, the median, and the mode are each seven for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode (unimodal), and the mode is the same as the mean and median. In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median. The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems chopped off compared to the left-hand side. A distribution of this type is called skewed to the left because it is pulled out to the left. A skewed left distribution has more high values. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 113 Figure 2.19 The mean is 6.3, the median is 6.5, and the mode is seven. Notice that the mean is less than the median, and they are both less than the mode. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution. The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is skewed to the right. A skewed right distribution has more low values. Figure 2.20 The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, the mean is the largest, while the mode is the smallest. Again, the mean reflects the skewing the most. To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean. Skewness and symmetry become important when we discuss probability distributions in later chapters. 114 Chapter 2 \| Descriptive Statistics Example 2.32 Statistics are used to compare and sometimes identify authors. The following lists show a simple random sample that compares the letter counts for three authors. Terry: 7, 9, 3, 3, 3, 4, 1, 3, 2, 2 Davis: 3, 3, 3, 4, 1, 4, 3, 2, 3, 1 Maris: 2, 3, 4, 4, 4, 6, 6, 6, 8, 3 a. Make a dot plot for the three authors and compare the shapes. b. Calculate the mean for each. c. Calculate the median for each. d. Describe any pattern you notice between the shape and the measures of center. Solution 2.32 a. Figure 2.21 Terry’s distribution has a right (positive) skew. Figure 2.22 Davis’s distribution has a left (negative) skew. Figure 2.23 Maris’s distribution is symmetrically shaped. b. Terry’s mean is 3.7, Davis’s mean is 2.7, and Maris’s mean is 4.6. c. Terry’s median is 3, Davis’s median is 3, and Maris’s median is four. It would be helpful to manually calculate these descriptive statistics, using the given data sets and then compare to the graphs. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 115 d. It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution. 2.32 Discuss the mean, median, and mode for each of the following problems. Is there a pattern between the shape and measure of the center? a. Figure 2.24 b. c. The Ages at Which Former U.S. Presidents Died Key: 8\|0 means 80. Table 2.30 116 Chapter 2 \| Descriptive Statistics Figure 2.25 2.7 \| Measures of the Spread of the Data An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The standard deviation is a number that measures how far data values are from their mean. The standard deviation • provides a numerical measure of the overall amount of variation in a data set and • can be used to determine whether a particular data value is close to or far from the mean. The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation. Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes. Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average. The standard deviation can be used to determine whether a data value is close to or far from the mean. Suppose that both Rosa and Binh shop at Supermarket A. Rosa waits at the checkout counter for seven minutes, and Binh waits for one minute. At Supermarket A, the mean waiting time is five minutes,
and the standard deviation is two minutes. The standard deviation can be used to determine whether a data value is close to or far from the mean. A z-score is a standardized score that lets us compare data sets. It tells us how many standard deviations a data value is from the mean and is calculated as the ratio of the difference in a particular score and the population mean to the population standard deviation. We can use the given information to create the table below. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 117 Supermarket Population Standard Deviation, σ Individual Score, x Population Mean, μ Supermarket A Supermarket B Table 2.31 2 minutes 4 minutes 7, 1 5 5 Since Rosa and Binh only shop at Supermarket A, we can ignore the row for Supermarket B. We need the values from the first row to determine the number of standard deviations above or below the mean each individual wait time is; we can do so by calculating two different z-scores. Rosa waited for seven minutes, so the z-score representing this deviation from the population mean may be calculated as The z-score of one tells us that Rosa’s wait time is one standard deviation above the mean wait time of five minutes. Binh waited for one minute, so the z-score representing this deviation from the population mean may be calculated as . The z-score of −2 tells us that Binh’s wait time is two standard deviations below the mean wait time of five minutes. A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if they are more than two standard deviations away is more of an approximate rule of thumb than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters. The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is one standard deviation to the right of five because 5 + (1)(2) = 7. If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (–2)(2) = 1. Figure 2.26 • In general, a value = mean + (#ofSTDEV)(standard deviation) • where #ofSTDEVs = the number of standard deviations • #ofSTDEV does not need to be an integer • One is two standard deviations less than the mean of five because 1 = 5 + (–2)(2). The equation value = mean + (#ofSTDEVs)(standard deviation) can be expressed for a sample and for a population as follows: • Sample: x = x¯ + ( # o f STDEV)(s) • Population: x = μ + ( # o f STDEV)(σ). The lowercase letter s represents the sample standard deviation and the Greek letter σ (lower case) represents the population standard deviation. ¯ The symbol x is the sample mean, and the Greek symbol μ is the population mean. Calculating the Standard Deviation If x is a number, then the difference x – mean is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is x – μ. For sample data, in symbols, a deviation is x – x¯ . The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data 118 Chapter 2 \| Descriptive Statistics from a sample. The calculations are similar but not identical. Therefore, the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lowercase letter s represents the sample standard deviation and the Greek letter σ (lowercase sigma) represents the population standard deviation. If the sample has the same characteristics as the population, then s should be a good estimate of σ. To calculate the standard deviation, we need to calculate the variance first. The variance is the average of the squares of the deviations (the x – x¯ values for a sample or the x – μ values for a population). The symbol σ2 represents the population variance; the population standard deviation σ is the square root of the population variance. The symbol s2 represents the sample variance; the sample standard deviation s is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations. If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N, the number of items in the population. If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n – 1, one less than the number of items in the sample. Formulas for the Sample Standard Deviation • s = 2 Σ(x − x¯ ) n − 1 or s = 2 Σ f (x − x¯ ) n − 1 • For the sample standard deviation, the denominator is n−; that is, the sample size minus 1. Formulas for the Population Standard Deviation • σ = Σ(x − μ)2 N or σ = Σ f (x – μ)2 N • For the population standard deviation, the denominator is N, the number of items in the population. In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three. Types of Variability in Samples When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. Variability is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following: • Observational or measurement variability • Natural variability • Induced variability • Sample variability Here are some examples to describe each type of variability: Example 1: Measurement variability Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability. Example 2: Natural variability Natural variability arises from the differences that naturally occur because members of a population differ from each other. For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability. Example 3: Induced variability Induced variability is the counterpart to natural variability. This occurs because we have artificially induced an element of variation that, by definition, was not present naturally. For example, we assign people to two different groups to study This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 119 memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may be affected by induced variability. Example 4: Sample variability Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability. Sampling Variability of a Statistic The statistic of a sampling distribution was discussed in Descriptive Statistics: Measures of the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is σ , where σ is the standard n deviation of the population and n is the size of the sample. NOTE In practice, use a calculator or computer software to calculate the standard deviation. If you are using a TI-83, 83+, or 84+ calculator, you need to select the appropriate standard deviation σx or sx from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However, you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. The calculator instructions appear at the end of this example. 120 Chapter 2 \| Descriptive Statistics Example 2.33 In a fifth-grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of n = 20 fifth-grade students. The ages are rounded to the nearest half year. 9, 9.5, 9.5, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5 x¯ = 9 + 9.5(2) + 10(4) + 10.5(4) + 11(6) + 11.5(3) 20 = 10.525 The average age is 10.53 years, rounded to two places. T
he variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating s. Data Frequency Deviations Deviations2 (Frequency)(Deviations2) x 9 9.5 10 10.5 11 11.5 f 1 2 4 4 6 3 Table 2.32 (x – x¯ ) 9 – 10.525 = –1.525 (x – x¯ )2 (–1.525)2 = 2.325625 1 × 2.325625 = 2.325625 (f)(x – x¯ )2 9.5 – 10.525 = –1.025 (–1.025)2 = 1.050625 2 × 1.050625 = 2.101250 10 – 10.525 = –.525 (–.525)2 = .275625 4 × .275625 = 1.1025 10.5 – 10.525 = –.025 (–.025)2 = .000625 4 × .000625 = .0025 11 – 10.525 = .475 (.475)2 = .225625 6 × .225625 = 1.35375 11.5 – 10.525 = .975 (.975)2 = .950625 3 × .950625 = 2.851875 The total is 9.7375. The last column simply multiplies each squared deviation by the frequency for the corresponding data value. The sample variance, s2, is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1): s2 = 9.7375 20 − 1 = .5125 The sample standard deviation s is equal to the square root of the sample variance: s = .5125 = .715891, which is rounded to two decimal places, s = .72. Typically, you do the calculation for the standard deviation on your calculator or computer. The intermediate results are not rounded. This is done for accuracy. • For the following problems, recall that value = mean + (#ofSTDEVs)(standard deviation). Verify the mean and standard deviation on a calculator or computer. Note that these formulas are derived by algebraically manipulating the z-score formulas, given either parameters or statistics. • For a sample: x = x¯ + (#ofSTDEVs)(s) • For a population: x = μ + (#ofSTDEVs)(σ) • For this example, use x = x¯ + (#ofSTDEVs)(s) because the data is from a sample a. Verify the mean and standard deviation on your calculator or computer. b. Find the value that is one standard deviation above the mean. Find ( x¯ + 1s). c. Find the value that is two standard deviations below the mean. Find ( x¯ – 2s). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 121 d. Find the values that are 1.5 standard deviations from (below and above) the mean. Solution 2.33 a. ◦ Clear lists L1 and L2. Press STAT 4:ClrList. Enter 2nd 1 for L1, the comma (,), and 2nd 2 for L2. ◦ Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down. ◦ Put the data values (9, 9.5, 10, 10.5, 11, 11.5) into list L1 and the frequencies (1, 2, 4, 4, 6, 3) into list L2. Use the arrow keys to move around. ◦ Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER. ◦ x¯ = 10.525. ◦ Use Sx because this is sample data (not a population): Sx=.715891. b. c. d. ( x¯ + 1s) = 10.53 + (1)(.72) = 11.25 ( x¯ – 2s) = 10.53 – (2)(.72) = 9.09 ◦ ◦ ( x¯ – 1.5s) = 10.53 – (1.5)(.72) = 9.45 ( x¯ + 1.5s) = 10.53 + (1.5)(.72) = 11.61 2.33 On a baseball team, the ages of each of the players are as follows: 21, 21, 22, 23, 24, 24, 25, 25, 28, 29, 29, 31, 32, 33, 33, 34, 35, 36, 36, 36, 36, 38, 38, 38, 40 Use your calculator or computer to find the mean and standard deviation. Then find the value that is two standard deviations above the mean. Explanation of the standard deviation calculation shown in the table The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11, which is indicated by the deviations .97 and .47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is –1.525 for the data value nine. If you add the deviations, the sum is always zero. We can sum the always products of zero. to show that the deviations 1(−1.525) + 2(−1.025) + 4(−.525) + 4(−.025) + 6(.475) + 3(.975) = 0 For Example 2.33, there are n = 20 deviations. So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation. and deviations frequencies sum of the the is The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data. Notice that instead of dividing by n = 20, the calculation divided by n – 1 = 20 – 1 = 19 because the data is a sample. For the sample variance, we divide by the sample size minus one (n – 1). Why not divide by n? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by (n – 1) gives a better estimate of the population variance. 122 NOTE Chapter 2 \| Descriptive Statistics Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number that measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic. The standard deviation, s or σ, is either zero or larger than zero. Describing the data with reference to the spread is called variability. The variability in data depends on the method by which the outcomes are obtained, for example, by measuring or by random sampling. When the standard deviation is zero, there is no spread; that is, all the data values are equal to each other. The standard deviation is small when all the data are concentrated close to the mean and larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make s or σ very large. The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better feel for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful, but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data. Display your data in a histogram or a box plot. Example 2.34 Use the following data (first exam scores) from Susan Dean's spring precalculus class: 33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100 a. Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places. b. Calculate the following to one decimal place using a TI-83+ or TI-84 calculator: i. The sample mean ii. The sample standard deviation iii. The median iv. The first quartile v. The third quartile vi. IQR c. Construct a box plot and a histogram on the same set of axes. Make comments about the box plot, the histogram, and the chart. Solution 2.34 a. See Table 2.33. b. Entering the data values into a list in your graphing calculator and then selecting Stat, Calc, and 1-Var Stats will produce the one-variable statistics you need. c. The x-axis goes from 32.5 to 100.5; the y-axis goes from –2.4 to 15 for the histogram. The number of intervals is 5, so the width of an interval is (100.5 – 32.5) divided by 5, equal to 13.6. Endpoints of the intervals are as follows: the starting point is 32.5, 32.5 + 13.6 = 46.1, 46.1 + 13.6 = 59.7, 59.7 + 13.6 = 73.3, 73.3 + 13.6 = 86.9, 86.9 + 13.6 = 100.5 = the ending value; no data values fall on an interval boundary. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 123 Figure 2.27 The long left whisker in the box plot is reflected in the left side of the histogram. The spread of the exam scores in the lower 50 percent is greater (73 – 33 = 40) than the spread in the upper 50 percent (100 – 73 = 27). The histogram, box plot, and chart all reflect this. There are a substantial number of A and B grades (80s, 90s, and 100). The histogram clearly shows this. The box plot shows us that the middle 50 percent of the exam scores (IQR = 29) are Ds, Cs, and Bs. The box plot also shows us that the lower 25 percent of the exam scores are Ds and Fs. Data Frequency Relative Frequency Cumulative Relative Frequency 33 42 49 53 55 61 63 67 68 69 72 73 74 78 80 83 88 90 Table 2.33 .032 .032 .065 .032 .065 .032 .032 .032 .065 .065 .032 .032 .032 .032 .032 .032 .097 .032 .032 .064 .129 .161 .226 .258 .290 .322 .387 .452 .484 .516 .548 .580 .612 .644 .741 .773 124 Chapter 2 \| Descriptive Statistics Data Frequency Relative Frequency Cumulative Relative Frequency 92 94 96 100 1 4 1 1 Table 2.33 .032 .129 .032 .032 .805 .934 .966 .998 (Why isn't this value 1?) 2.34 The following data show the different types of pet food that stores in the area carry: 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12 Calculate the sample mean and the sample standard deviation to one decimal place using a TI-83+ or TI-84 calculator. Standard deviation of Grouped Frequency Tables Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula Mean o f Frequency Table = ∑ f m ∑ f , where f = interval frequencies and m = interval midpoints. Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes
numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how unusual individual data are when compared to the mean. Example 2.35 Find the standard deviation for the data in Table 2.34. Class Frequency, f Midpoint, m m2 0–2 3–5 6–8 9–11 12–14 15–17 1 6 10 7 0 2 Table 2.34 1 4 7 10 13 16 1 16 49 100 169 256 ¯ 2 x 7.58 7.58 7.58 7.58 7.58 7.58 fm2 Standard Deviation 1 96 490 700 0 512 3.5 3.5 3.5 3.5 3.5 3.5 For this data set, we have the mean, x¯ = 7.58, and the standard deviation, sx = 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 – 3.5 – 3.5 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 125 = .58. While the formula for calculating the standard deviation is not complicated, s x = 2 f (m − x¯ ) n − 1 , where sx = sample standard deviation, x¯ = sample mean; the calculations are tedious. It is usually best to use technology when performing the calculations. 2.35 Find the standard deviation for the data from the previous example: Class Frequency, f 0–2 3–5 6–8 9–11 12–14 15–17 1 6 10 7 0 2 Table 2.35 First, press the STAT key and select 1:Edit. Figure 2.28 Input the midpoint values into L1 and the frequencies into L2. 126 Chapter 2 \| Descriptive Statistics Figure 2.29 Select STAT, CALC, and 1: 1-Var Stats. Figure 2.30 Select 2nd, then 1, then, 2nd, then 2 Enter. Figure 2.31 You will see displayed both a population standard deviation, σx, and the sample standard deviation, sx. Comparing Values from Different Data Sets As explained before, a z-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading. • For each data value, calculate how many standard deviations away from its mean the value is. • In symbols, the formulas for calculating z-scores become the following. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 127 z = x − x¯ s z = x − μ σ Sample Population Table 2.36 As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used. Example 2.36 Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school? Student GPA School Mean GPA School Standard Deviation John Ali 2.85 77 3.0 80 Table 2.37 .7 10 Solution 2.36 For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer. z = # of STDEVs = value – mean standard deviation = x + μ σ For John, z = # o f STDEVs = 2.85 – 3.0 .7 = − 0.21 For Ali, z = # o f STDEVs = 77 − 80 10 = − 0.3 John has the better GPA when compared to his school because his GPA is 0.21 standard deviations below his school's mean, while Ali's GPA is .3 standard deviations below his school's mean. John's z-score of –.21 is higher than Ali's z-score of –.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school. The z-score representing John's score does not fall as far below the mean as the z-score representing Ali's score. 128 Chapter 2 \| Descriptive Statistics 2.36 Two swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the 50-meter freestyle when compared to her team. Which swimmer had the fastest time when compared to her team? Swimmer Time (seconds) Team Mean Time Team Standard Deviation Angie Beth 26.2 27.3 Table 2.38 27.2 30.1 .8 1.4 The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data. For any data set, no matter what the distribution of the data is, the following are true: • At least 75 percent of the data is within two standard deviations of the mean. • At least 89 percent of the data is within three standard deviations of the mean. • At least 95 percent of the data is within 4.5 standard deviations of the mean. • This is known as Chebyshev's Rule. A bell-shaped distribution is one that is normal and symmetric, meaning the curve can be folded along a line of symmetry drawn through the median, and the left and right sides of the curve would fold on each other symmetrically.. With a bellshaped distribution, the mean, median, and mode are all located at the same place. For data having a distribution that is bell-shaped and symmetric, the following are true: • Approximately 68 percent of the data is within one standard deviation of the mean. • Approximately 95 percent of the data is within two standard deviations of the mean. • More than 99 percent of the data is within three standard deviations of the mean. • This is known as the Empirical Rule. • It is important to note that this rule applies only when the shape of the distribution of the data is bell-shaped and symmetric; we will learn more about this when studying the Normal or Gaussian probability distribution in later chapters. 2.8 \| Descriptive Statistics This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 129 2.1 Descriptive Statistics Student Learning Outcomes • The student will construct a histogram and a box plot. • The student will calculate univariate statistics. • The student will examine the graphs to interpret what the data imply. Collect the Data Record the number of pairs of shoes you own. 1. Randomly survey 30 classmates about the number of pairs of shoes they own. Record their values. _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ Table 2.39 Survey Results 2. Construct a histogram. Make five to six intervals. Sketch the graph using a ruler and pencil and scale the axes. Figure 2.32 3. Calculate the following values: a. b. x¯ = _____ s = _____ 4. Are the data discrete or continuous? How do you know? 5. In complete sentences, describe the shape of the histogram. 6. Are there any potential outliers? List the value(s) that could be outliers. Use a formula to check the end values to determine if they are potential outliers. 130 Chapter 2 \| Descriptive Statistics Analyze the Data 1. Determine the following values: a. Min = _____ b. M = _____ c. Max = _____ d. Q1 = _____ e. Q3 = _____ f. IQR = _____ 2. Construct a box plot of data. 3. What does the shape of the box plot imply about the concentration of data? Use complete sentences. 4. Using the box plot, how can you determine if there are potential outliers? 5. How does the standard deviation help you to determine concentration of the data and whether there are potential outliers? 6. What does the IQR represent in this problem? 7. Show your work to find the value that is 1.5 standard deviations a. above the mean. b. below the mean. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 131 KEY TERMS box plot a graph that gives a quick picture of the middle 50 percent of the data first quartile the value that is the median of the lower half of the ordered data set frequency the number of times a value of the data occurs frequency polygon a data display that looks like a line graph but uses intervals to display ranges of large amounts of data frequency table a data representation in which grouped data are displayed along with the corresponding frequencies histogram a graphical representation in x-y form of the distribution of data in a data set; x represents the data and y represents the frequency, or relative frequency; the graph consists of contiguous rectangles interquartile range or IQR, is the range of the middle 50 percent of the data values; the IQR is found by subtracting the first quartile from the third quartile interval also called a class interval; an interval represents a range of data and is used when displaying large data sets mean a number that measures the central tendency of the data; a common name for mean is average. The term mean is a shortened form of arithmetic mean. By definition, the mean for a sample (denoted by x x¯ = the mean population (denoted Sum of all values in the sample Number of values in the sample and for by μ) a , ¯ ) is is μ = Sum of all values in the population Number of values in the population median a number that separates ordered data into halves; half the values are the same number or smaller than the median, and half the values are the same number or larger than the median The median may or may not be part of the data. midpoint the mean of an interval in a frequency table mode the value that appears most frequently in a set of data outlier an observation that does not fit the rest of the data paired data set two data sets that have a one-to-one relationship so that • both data sets are the same size, and • each data point in one data set is matched with exactly one point from the other set percentile a number that divides ordered data into hundredths; percentiles may or may not be part of the data. The median of the data is the second quartile and the 50th percentile The first and third quartiles are the 25th and the 75th percentiles, respectively. quartiles the numbers that separate the data into quarters; quartiles may or may not be part of the data; the second quartile is the median of the data relative frequency of all outcomes the ratio of the number of times
a value of the data occurs in the set of all outcomes to the number skewed used to describe data that is not symmetrical; when the right side of a graph looks chopped off compared to the left side, we say it is skewed to the left. When the left side of the graph looks chopped off compared to the right side, we say the data are skewed to the right. Alternatively, when the lower values of the data are more spread out, we say the data are skewed to the left. When the greater values are more spread out, the data are skewed to the right. standard deviation a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: s for sample standard deviation and σ for population standard deviation 132 Chapter 2 \| Descriptive Statistics variance mean of the squared deviations from the mean, or the square of the standard deviation; for a set of data, a deviation can be represented as x – x¯ where x is a value of the data and x¯ variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and 1 is the sample mean; the sample CHAPTER REVIEW 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends, that is, finding a general pattern in data sets, including temperature, sales, employment, company profit, or cost, over a period of time. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Bar graphs are especially useful when categorical data are being used. 2.2 Histograms, Frequency Polygons, and Time Series Graphs A histogram is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets. A frequency polygon can also be used when graphing large data sets with data points that repeat. The data usually go on the y-axis with the frequency being graphed on the x-axis. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time. 2.3 Measures of the Location of the Data The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50th percentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quartile (Q1) is the 25th percentile, the second quartile (Q2 or median) is the 50th percentile, and the third quartile (Q3) is the 75th percentile. The interquartile range, or IQR, is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q1 from Q3 and can help determine outliers by using the following two expressions. • Q3 + IQR(1.5) • Q1 – IQR(1.5) 2.4 Box Plots Box plots are a type of graph that can help visually organize data. Before a box plot can be graphed, the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data. 2.5 Measures of the Center of the Data The mean and the median can be calculated to help you find the center of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your data set. The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges that lack specific values, the mean may seem impossible to calculate. However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval. Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set. 2.6 Skewness and the Mean, Median, and Mode Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There are three types of distributions. A right (or positive) skewed distribution has a shape like Figure 2.19. A left (or negative) skewed distribution has a shape like Figure 2.20. A symmetrical distribution looks like Figure 2.18. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 133 2.7 Measures of the Spread of the Data The standard deviation can help you calculate the spread of data. There are different equations to use if you are calculating the standard deviation of a sample or of a population. • The standard deviation allows us to compare individual data or classes to the data set mean numerically. • s = ∑ (x − x¯ ) n − 1 2 or s = ∑ f (x − x¯ ) n − 1 2 is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ, and the formula σ = ∑ (x − μ)2 N or σ = ∑ f (x − μ)2 N . FORMULA REVIEW 2.3 Measures of the Location of the Data 2.5 Measures of the Center of the Data i = ⎛ ⎝ k 100 ⎞ ⎠(n + 1) where i = the ranking or position of a data value, k = the kth percentile, n = total number of data. Expression for finding the percentile of a data value ⎛ ⎝ x + 0.5y n (100) ⎞ ⎠ where x = the number of values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile, y = the number of data values equal to the data value for which you want to find the percentile, n = total number of data. PRACTICE where f = interval frequencies and m = μ = ∑ f m ∑ f interval midpoints. 2.7 Measures of the Spread of the Data s x = ∑ f m2 n − x¯ 2 s x = sample standard deviation x¯ = sample mean ⎛ ⎝x - x¯ ) s z = where and z = (x - μ) θ . 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs For each of the following data sets, create a stemplot and identify any outliers. 1. The miles-per-gallon ratings for 30 cars are shown below (lowest to highest): 19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43. 2. The height in feet of 25 trees is shown below (lowest to highest): 25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54. 3. The data are the prices of different laptops at an electronics store. Round each value to the nearest 10. 249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610 4. The following data are daily high temperatures in a town for one month: 61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95. For the next three exercises, use the data to construct a line graph. 134 Chapter 2 \| Descriptive Statistics 5. In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in Table 2.40. Number of Times in Store Frequency 1 2 3 4 5 Table 2.40 4 10 16 6 4 6. In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table 2.41. Years Since Last Purchase Frequency 0 1 2 3 4 5 Table 2.41 2 8 13 22 16 9 7. Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.42. Number of TV Shows Frequency 0 1 2 3 4 Table 2.42 12 18 36 7 2 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 135 8. The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table 2.43 shows the four seasons, the number of students who have birthdays in each season, and the percentage of students in each group. Construct a bar graph showing the number of students. Seasons Number of Students Proportion of Population Spring Summer Autumn Winter 8 9 11 6 Table 2.43 24% 26% 32% 18% 9. Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.8, construct a bar graph showing the percentages. 10. David County has six high schools. Each school sent students to participate in a county-wide science competition. Table 2.44 shows the percentage breakdown of competitors from each school and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school. High School Science Competition Population Overall Student Population Alabaster 28.9% Concordia 7.6% Genoa 12.1% Mocksville 18.5% Tynneson 24.2% West End 8.7% Table 2.44 8.6% 23.2% 15.0% 14.3% 10.1% 28.8% 11. Use the data from the David County science competition supplied in Exercise 2.10. Construct a bar graph that shows the county-wide population percentage of students at each school. 2.2 Histograms, Frequency Polygons, and Time Series Graphs 12. 65 randomly selected car salespersons were asked the number of cars they generally sell in one week. 14 people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five
cars, nine generally sell six cars, and 11 generally sell seven cars. Complete the table. Data Value (Number of Cars) Frequency Relative Frequency Cumulative Relative Frequency Table 2.45 13. What does the frequency column in Table 2.45 sum to? Why? 136 Chapter 2 \| Descriptive Statistics 14. What does the relative frequency column in Table 2.45 sum to? Why? 15. What is the difference between relative frequency and frequency for each data value in Table 2.45? 16. What is the difference between cumulative relative frequency and relative frequency for each data value? 17. To construct the histogram for the data in Table 2.45, determine appropriate minimum and maximum x- and y-values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling. Figure 2.33 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 137 18. Construct a frequency polygon for the following. a. b. c. Pulse Rates for Women Frequency 60–69 70–79 80–89 90–99 100–109 110–119 120–129 Table 2.46 12 14 11 1 1 0 1 Actual Speed in a 30-MPH Zone Frequency 42–45 46–49 50–53 54–57 58–61 Table 2.47 25 14 7 3 1 Tar (mg) in Nonfiltered Cigarettes Frequency 10–13 14–17 18–21 22–25 26–29 Table 2.48 1 0 15 7 2 138 Chapter 2 \| Descriptive Statistics 19. Construct a frequency polygon from the frequency distribution for the 50 highest-ranked countries for depth of hunger. Depth of Hunger Frequency 230–259 260–289 290–319 320–349 350–379 380–409 410–439 Table 2.49 21 13 5 7 1 1 1 20. Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlaid frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men? Life Expectancy at Birth – Women Frequency 49–55 56–62 63–69 70–76 77–83 84–90 3 3 1 3 8 2 Table 2.50 Life Expectancy at Birth – Men Frequency 49–55 56–62 63–69 70–76 77–83 84–90 Table 2.51 3 3 1 1 7 5 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 139 21. Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births. Sex/Year 1855 1856 1857 1858 1859 1860 1861 Female 45,545 49,582 50,257 50,324 51,915 51,220 52,403 47,804 52,239 53,158 53,694 54,628 54,409 54,606 93,349 101,821 103,415 104,018 106,543 105,629 107,009 Male Total Table 2.52 Sex/Year 1862 1863 1864 1865 1866 1867 1868 1869 Female 51,812 53,115 54,959 54,850 55,307 55,527 56,292 55,033 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354 Male Total Table 2.53 Sex/Year 1871 1870 1872 1871 1872 1827 1874 1875 Female 56,099 56,431 57,472 56,099 57,472 58,233 60,109 60,146 60,029 58,959 61,293 60,029 61,293 61,467 63,602 63,432 116,128 115,390 118,765 116,128 118,765 119,700 123,711 123,578 Male Total Table 2.54 22. The following data sets list full-time police per 100,000 citizens along with incidents of a certain crime per 100,000 citizens for the city of Detroit, Michigan, during the period from 1961 to 1973. Year Police 1961 1962 1963 1964 1965 1966 1967 260.35 269.8 272.04 272.96 272.51 261.34 268.89 Incidents 8.6 8.9 8.52 8.89 13.07 14.57 21.36 Table 2.55 Year Police 1968 1969 1970 1971 1972 1973 295.99 319.87 341.43 356.59 376.69 390.19 Incidents 28.03 31.49 37.39 46.26 47.24 52.33 Table 2.56 a. Construct a double time series graph using a common x-axis for both sets of data. b. Which variable increased the fastest? Explain. c. Did Detroit’s increase in police officers have an impact on the incident rate? Explain. 140 Chapter 2 \| Descriptive Statistics 2.3 Measures of the Location of the Data 23. Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest: 18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the 40th percentile. b. Find the 78th percentile. 24. Listed are 32 ages for Academy Award-winning best actors in order from smallest to largest: 18, 18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77 a. Find the percentile of 37. b. Find the percentile of 72. 25. Jesse was ranked 37th in his graduating class of 180 students. At what percentile is Jesse’s ranking? 26. 27. a. For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race? b. The 20th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20th percentile in the context of the situation. c. A bicyclist in the 90th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90th percentile in the context of the situation. a. For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race? b. The 40th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40th percentile in the context of the situation. 28. On an exam, would it be more desirable to earn a grade with a high or a low percentile? Explain. 29. Mina is waiting in line at the Department of Motor Vehicles. Her wait time of 32 minutes is the 85th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85th percentile in the context of this situation. 30. In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78th percentile. Should Li be pleased or upset by this result? Explain. 31. In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90th percentile in the context of this problem. 32. The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system: a. Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an admissions index score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12 percent of high school students in the state. In this context, what percentile does the top 12 percent represent? b. Students whose GPAs are at or above the 96th percentile of all students at their high school are eligible, called eligible in the local context, even if they are not in the top 12 percent of all students in the state. What percentage of students from each high school are eligible in the local context? 33. Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34th percentile. The 34th percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34 percent of the houses or 66 percent of the houses? Use Exercise 2.25 to calculate the following values. 34. First quartile = ________ 35. Second quartile = median = 50th percentile = ________ 36. Third quartile = ________ 37. Interquartile range (IQR) = ________ – ________ = ________ This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 141 38. 10th percentile = ________ 39. 70th percentile = ________ 2.4 Box Plots Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. 40. Construct a box plot below. Use a ruler to measure and scale accurately. 41. Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas but not in others? How can you tell? 2.5 Measures of the Center of the Data 42. Find the mean for the following frequency tables: a. b. c. Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 Table 2.57 Daily Low Temperature Frequency 49.5–59.5 59.5–69.5 69.5–79.5 79.5–89.5 89.5–99.5 Table 2.58 53 32 15 1 0 Points per Game Frequency 49.5–59.5 59.5–69.5 69.5–79.5 79.5–89.5 89.5–99.5 Table 2.59 14 32 15 23 2 Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16, 17, 19, 20, 20, 21, 23, 24, 25, 25, 25, 26, 26, 27, 27, 27, 28, 29, 30, 32, 33, 33, 34, 35, 37, 39, 40 43. Calculate the mean. 142 Chapter 2 \| Descriptive Statistics 44. Identify the median. 45. Identify the mode. Use the following information to answer the next three exercises: Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars, 19 generally sell four cars, 12 generally sell five cars, nine generally sell six cars, and 11 generally sell seven cars. Calculate the following. 46. sample mean = x ¯ = ________ 47. median = ________ 48. mode = ________ 2.6 Skewness and the Mean, Median, and Mode Use the following information to answer the next three exercises. State whether the data are symmetrical, skewed to the left, or skewed to the right. 49. 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5 50. 16, 17, 19, 22, 22, 22
, 22, 22, 23 51. 87, 87, 87, 87, 87, 88, 89, 89, 90, 91 52. When the data are skewed left, what is the typical relationship between the mean and median? 53. When the data are symmetrical, what is the typical relationship between the mean and median? 54. What word describes a distribution that has two modes? 55. Describe the shape of this distribution. Figure 2.34 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 143 56. Describe the relationship between the mode and the median of this distribution. Figure 2.35 57. Describe the relationship between the mean and the median of this distribution. Figure 2.36 58. Describe the shape of this distribution. Figure 2.37 144 Chapter 2 \| Descriptive Statistics 59. Describe the relationship between the mode and the median of this distribution. Figure 2.38 60. Are the mean and the median the exact same in this distribution? Why or why not? Figure 2.39 61. Describe the shape of this distribution. Figure 2.40 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 145 62. Describe the relationship between the mode and the median of this distribution. Figure 2.41 63. Describe the relationship between the mean and the median of this distribution. Figure 2.42 64. The mean and median for the data are the same. 3, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7 Is the data perfectly symmetrical? Why or why not? 65. Which is the greatest, the mean, the mode, or the median of the data set? 11, 11, 12, 12, 12, 12, 13, 15, 17, 22, 22, 22 66. Which is the least, the mean, the mode, and the median of the data set? 56, 56, 56, 58, 59, 60, 62, 64, 64, 65, 67 67. Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why? 68. In a perfectly symmetrical distribution, when would the mode be different from the mean and median? 2.7 Measures of the Spread of the Data For each of the examples given below, tell whether the differences in outcomes may be explained by measurement variability, natural variability, induced variability, or sampling variability. 146 Chapter 2 \| Descriptive Statistics 69. Scientists randomly select five groups of 10 women from a population of 1,000 women to record their body fat percentage. The scientists compute the mean body fat percentage from each group. The differences in outcomes may be attributed to which type of variability? 70. A pharmaceutical company randomly assigns participants to one of two groups: one is a control group receiving a placebo, and another is a treatment group receiving a new drug to lower blood pressure. The differences in outcomes may be attributed to which type of variability? 71. Jaiqua and Harold are trying to determine how ramp steepness affects the speed of a ball rolling down the ramp. They measure the time it takes for the ball to roll down ramps of differing slopes. When Jaiqua rolls the ball and Harold works the stopwatch, they get different results than when Harold rolls the ball and Jaiqua works the stopwatch. The differences in outcomes may be attributed to which type of variability? 72. Twenty people begin the same workout program on the same day and continue for three months. During that time, all participants worked out for the same amount of time and did the same number of exercises and repetitions. Each person was weighed at both the beginning and the end of the program. The differences in outcomes regarding the amount of weight lost may be attributed to which type of variability? Use the following information to answer the next two exercises. The following data are the distances between 20 retail stores and a large distribution center. The distances are in miles. 29, 37, 38, 40, 58, 67, 68, 69, 76, 86, 87, 95, 96, 96, 99, 106, 112, 127, 145, 150 73. Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth. 74. Find the value that is one standard deviation below the mean. 75. Two baseball players, Fredo and Karl, on different teams wanted to find out who had the higher batting average when compared to his team. Which baseball player had the higher batting average when compared to his team? Baseball Player Batting Average Team Batting Average Team Standard Deviation Fredo Karl Table 2.60 .158 .177 .166 .189 .012 .015 76. Use Table 2.60 to find the value that is three standard deviations a. above the mean, and b. below the mean This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 147 77. Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/ 84. a. b. c. Grade Frequency 49.5–59.5 2 59.5–69.5 3 69.5–79.5 8 79.5–89.5 12 89.5–99.5 5 Table 2.61 Daily Low Temperature Frequency 49.5–59.5 59.5–69.5 69.5–79.5 79.5–89.5 89.5–99.5 Table 2.62 53 32 15 1 0 Points per Game Frequency 49.5–59.5 59.5–69.5 69.5–79.5 79.5–89.5 89.5–99.5 Table 2.63 14 32 15 23 2 HOMEWORK 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs 78. Student grades on a chemistry exam were 77, 78, 76, 81, 86, 51, 79, 82, 84, and 99. a. Construct a stem-and-leaf plot of the data. b. Are there any potential outliers? If so, which scores are they? Why do you consider them outliers? 148 Chapter 2 \| Descriptive Statistics 79. Table 2.64 contains the 2010 rates for a specific disease in U.S. states and Washington, DC. Percent (%) State Percent (%) State Percent (%) State Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware 32.2 24.5 24.3 30.1 24.0 21.0 22.5 28.0 Washington, DC 22.2 Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Table 2.64 26.6 29.6 22.7 26.5 28.2 29.6 28.4 29.4 Kentucky Louisiana Maine Maryland 31.3 31.0 26.8 27.1 North Dakota 27.2 Ohio Oklahoma Oregon 29.2 30.4 26.8 Massachusetts 23.0 Pennsylvania 28.6 Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada 30.9 24.8 34.0 30.5 23.0 26.9 22.4 New Hampshire 25.0 New Jersey New Mexico New York 23.8 25.1 23.9 North Carolina 27.8 Rhode Island 25.5 South Carolina 31.5 South Dakota 27.3 Tennessee Texas Utah Vermont Virginia Washington 30.8 31.0 22.5 23.2 26.0 25.5 West Virginia 32.5 Wisconsin Wyoming 26.3 25.1 a. Use a random number generator to randomly pick eight states. Construct a bar graph of the rates of a specific disease of those eight states. b. Construct a bar graph for all the states beginning with the letter A. c. Construct a bar graph for all the states beginning with the letter M. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 149 2.2 Histograms, Frequency Polygons, and Time Series Graphs 150 Chapter 2 \| Descriptive Statistics 80. Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows: Number of Books Frequency Relative Frequency 0 1 2 3 4 5 6 8 10 12 16 12 8 6 2 2 Table 2.65 Publisher A Number of Books Frequency Relative Frequency 0 1 2 3 4 5 7 9 18 24 24 22 15 10 5 1 Table 2.66 Publisher B Number of Books Frequency Relative Frequency 0–1 2–3 4–5 6–7 8–9 20 35 12 2 1 Table 2.67 Publisher C a. Find the relative frequencies for each survey. Write them in the charts. b. Using either a graphing calculator or computer or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of 1. For Publisher C, make bar widths of 2. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical. c. d. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not? e. Make new histograms for Publisher A and Publisher B. This time, make bar widths of 2. f. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 151 81. Often, cruise ships conduct all onboard transactions, with the exception of souvenirs, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their onboard bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group: Amount ($) Frequency Relative Frequency 51–100 101–150 151–200 201–250 251–300 301–350 5 10 15 15 10 5 Table 2.68 Singles Amount ($) Frequency Relative Frequency 100–150 201–250 251–300 301–350 351–400 401–450 451–500 501–550 551–600 601–650 5 5 5 5 10 10 10 10 5 5 Table 2.69 Couples a. Fill in the relative frequency for each group. b. Construct a histogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis. c. Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis. d. Compare the two graphs: i. List two similarities between the graphs. ii. List two differences between the graphs. iii. Overall, are the graphs more similar or different? e. Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by $50, scale it by $100. Use relative frequency on the y-axis. f. Compare the graph for the singles with the new graph for the couples: i. List two similarities between the graphs. ii. Overall, are the graphs more similar or different? g. How did scaling the couples graph differently change the way you compared it to the singles graph? h. Based on the graphs, do you think that individuals spend the same amount, more or less, as sin
gles as they do person by person as a couple? Explain why in one or two complete sentences. 152 Chapter 2 \| Descriptive Statistics 82. 25 randomly selected students were asked the number of movies they watched the previous week. The results are as follows: Number of Movies Frequency Relative Frequency Cumulative Relative Frequency 0 1 2 3 4 Table 2.70 5 9 6 4 1 a. Construct a histogram of the data. b. Complete the columns of the chart. Use the following information to answer the next two exercises: Suppose 111 people who shopped in a special T-shirt store were asked the number of T-shirts they own costing more than $19 each. 83. The percentage of people who own at most three T-shirts costing more than $19 each is approximately ________. a. 21 b. 59 c. 41 d. cannot be determined 84. If the data were collected by asking the first 111 people who entered the store, then the type of sampling is ________. a. cluster simple random b. c. stratified d. convenience This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 153 85. Following are the 2010 obesity rates by U.S. states and Washington, DC. Percent (%) State Percent (%) State Percent (%) State Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware 32.2 24.5 24.3 30.1 24.0 21.0 22.5 28.0 Washington, DC 22.2 Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Table 2.71 26.6 29.6 22.7 26.5 28.2 29.6 28.4 29.4 Kentucky Louisiana Maine Maryland 31.3 31.0 26.8 27.1 North Dakota 27.2 Ohio Oklahoma Oregon 29.2 30.4 26.8 Massachusetts 23.0 Pennsylvania 28.6 Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada 30.9 24.8 34.0 30.5 23.0 26.9 22.4 New Hampshire 25.0 New Jersey New Mexico New York 23.8 25.1 23.9 North Carolina 27.8 Rhode Island 25.5 South Carolina 31.5 South Dakota 27.3 Tennessee Texas Utah Vermont Virginia Washington 30.8 31.0 22.5 23.2 26.0 25.5 West Virginia 32.5 Wisconsin Wyoming 26.3 25.1 Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint—Label the x-axis with the states. 2.3 Measures of the Location of the Data 86. The median age for U.S. ethnicity A currently is 30.9 years; for U.S. ethnicity B, it is 42.3 years. a. Based on this information, give two reasons why ethnicity A median age could be lower than the ethnicity B median age. b. Does the lower median age for ethnicity A necessarily mean that ethnicity A die younger than ethnicity B? Why or why not? c. How might it be possible for ethnicity A and ethnicity B to die at approximately the same age but for the median age for ethnicity B to be higher? 154 Chapter 2 \| Descriptive Statistics 87. Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in Table 2.72. Also, include the left endpoint but not the right endpoint. Salary ($) Relative Frequency < 20,000 .02 20,000–25,000 .09 25,000–30,000 .19 30,000–40,000 .26 40,000–50,000 .18 50,000–75,000 .17 75,000–99,999 .02 100,000+ .01 Table 2.72 a. What percentage of the survey answered "not sure"? b. What percentage think that middle class is from $25,000 to $50,000? c. Construct a histogram of the data. i. Should all bars have the same width, based on the data? Why or why not? ii. How should the < 20,000 and the 100,000+ intervals be handled? Why? d. Find the 40th and 80th percentiles. e. Construct a bar graph of the data. 88. Given the following box plot, answer the questions. Figure 2.43 a. Which quarter has the smallest spread of data? What is that spread? b. Which quarter has the largest spread of data? What is that spread? c. Find the interquartile range (IQR). d. Are there more data in the interval 5–10 or in the interval 10–13? How do you know this? e. Which interval has the fewest data in it? How do you know this? i. 0–2 ii. 2–4 iii. 10–12 iv. 12–13 v. need more information This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 155 89. The following box plot shows the ages of the U.S. population for 1990, the latest available year: Figure 2.44 a. Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know? b. 12.6 percent are age 65 and over. Approximately what percentage of the population are working-age adults (above age 17 to age 65)? 2.4 Box Plots 90. In a survey of 20-year-olds in China, Germany, and the United States, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results: Figure 2.45 a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected. b. Have more Americans or more Germans surveyed been to more than eight foreign countries? c. Compare the three box plots. What do they imply about the foreign travel of 20-year-old residents of the three countries when compared to each other? 91. Given the following box plot, answer the questions. Figure 2.46 a. Think of an example (in words) where the data might fit into the above box plot. In two to five sentences, write down the example. b. What does it mean to have the first and second quartiles so close together, while the second to third quartiles are far apart? 156 Chapter 2 \| Descriptive Statistics 92. Given the following box plots, answer the questions. Figure 2.47 a. In complete sentences, explain why each statement is false. i. Data 1 has more data values above two than Data 2 has above two. ii. The data sets cannot have the same mode. iii. For Data 1, there are more data values below four than there are above four. b. For which group, Data 1 or Data 2, is the value of 7 more likely to be an outlier? Explain why in complete sentences. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 157 93. A survey was conducted of 130 purchasers of new black sports cars, 130 purchasers of new red sports cars, and 130 purchasers of new white sports cars. In it, people were asked the age they were when they purchased their car. The following box plots display the results: Figure 2.48 a. In complete sentences, describe what the shape of each box plot implies about the distribution of the data collected for that car series. b. Which group is most likely to have an outlier? Explain how you determined that. c. Compare the three box plots. What do they imply about the age of purchasing a sports car from the series when compared to each other? d. Look at the red sports cars. Which quarter has the smallest spread of data? What is the spread? e. Look at the red sports cars. Which quarter has the largest spread of data? What is the spread? f. Look at the red sports cars. Estimate the interquartile range (IQR). g. Look at the red sports cars. Are there more data in the interval 31–38 or in the interval 45–55? How do you know this? h. Look at the red sports cars. Which interval has the fewest data in it? How do you know this? i. 31–35 ii. 38–41 iii. 41–64 94. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows: Number of Movies Frequency 0 1 2 3 4 Table 2.73 5 9 6 4 1 Construct a box plot of the data. 158 Chapter 2 \| Descriptive Statistics 2.5 Measures of the Center of the Data 95. Scientists are studying a particular disease. They found that countries that have the highest rates of people who have ever been diagnosed with this disease range from 11.4 percent to 74.6 percent. Percentage of Population Diagnosed Number of Countries 11.4–20.45 20.45–29.45 29.45–38.45 38.45–47.45 47.45–56.45 56.45–65.45 65.45–74.45 74.45–83.45 Table 2.74 29 13 4 0 2 1 0 1 a. What is the best estimate of the average percentage affected by the disease for these countries? b. The United States has an average disease rate of 33.9 percent. Is this rate above average or below? c. How does the United States compare to other countries? 96. Table 2.75 gives the percentage of children under age five have been diagnosed with a medical condition. What is the best estimate for the mean percentage of children with the condition? Percentage of Children with the Condition Number of Countries 16–21.45 21.45–26.9 26.9–32.35 32.35–37.8 37.8–43.25 43.25–48.7 Table 2.75 23 4 9 7 6 1 2.6 Skewness and the Mean, Median, and Mode 97. The median age of the U.S. population in 1980 was 30.0 years. In 1991, the median age was 33.1 years. a. What does it mean for the median age to rise? b. Give two reasons why the median age could rise. c. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not? 2.7 Measures of the Spread of the Data Use the following information to answer the next nine exercises: The population parameters below describe the full-time equivalent number of students (FTES) each year at Lake Tahoe Community College from 1976–1977 through 2004–2005. • μ = 1,000 FTES • median = 1,014 FTES • σ = 474 FTES This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 159 • • first quartile = 528.5 FTES third quartile = 1,447.5 FTES • n = 29 years 98. A sample of 11 years is taken. About how many are expected to have an FTES of 1,014 or above? Explain how you determined your answer. 99. Seventy-five percent of all years have an FTES a. at or below ______. b. at or above ______. 100. The population standard deviation = ______. 101. What percentage of the FTES were from 528.5 to 1,447.5? How do you know? 102. What is the IQR? What does the IQR represent? 103. How many standard deviations away from the mean is the median? Additional Information: The population FTES for 2005–2006 through 2010–2011 was given in an updated report. The data are reported here. Year 2005–2006 2006–2007 2007–2008 2008–2009 2009–2010 2010–2011 Total FTES 1,585 1,690 1,735 1,935 2,021
1,890 Table 2.76 104. Calculate the mean, median, standard deviation, the first quartile, the third quartile, and the IQR. Round to one decimal place. 105. Construct a box plot for the FTES for 2005–2006 through 2010–2011 and a box plot for the FTES for 1976–1977 through 2004–2005. 106. Compare the IQR for the FTES for 1976–1977 through 2004–2005 with the IQR for the FTES for 2005-2006 through 2010–2011. Why do you suppose the IQRs are so different? 107. Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best GPA when compared to other students at his school? Explain how you determined your answer. Student GPA School Average GPA School Standard Deviation Thuy Vichet Kamala 2.7 87 8.6 3.2 75 8 Table 2.77 .8 20 .4 108. A music school has budgeted to purchase three musical instruments. The school plans to purchase a piano costing $3,000, a guitar costing $550, and a drum set costing $600. The mean cost for a piano is $4,000 with a standard deviation of $2,500. The mean cost for a guitar is $500 with a standard deviation of $200. The mean cost for drums is $700 with a standard deviation of $100. Which cost is the lowest when compared to other instruments of the same type? Which cost is the highest when compared to other instruments of the same type? Justify your answer. 109. An elementary school class ran one mile with a mean of 11 minutes and a standard deviation of three minutes. Rachel, a student in the class, ran one mile in eight minutes. A junior high school class ran one mile with a mean of nine minutes and a standard deviation of two minutes. Kenji, a student in the class, ran one mile in 8.5 minutes. A high school class ran one mile with a mean of seven minutes and a standard deviation of four minutes. Nedda, a student in the class, ran one mile in eight minutes. a. Why is Kenji considered a better runner than Nedda even though Nedda ran faster than he? b. Who is the fastest runner with respect to his or her class? Explain why. 160 Chapter 2 \| Descriptive Statistics 110. Scientists are studying a particular disease. They found that countries that have the highest rates of people who have ever been diagnosed with this disease range from 11.4 percent to 74.6 percent. Percentage of Population with Disease Number of Countries 11.4–20.45 20.45–29.45 29.45–38.45 38.45–47.45 47.45–56.45 56.45–65.45 65.45–74.45 74.45–83.45 Table 2.78 29 13 4 0 2 1 0 1 What is the best estimate of the average percentage of people with the disease for these countries? What is the standard deviation for the listed rates? The United States has an average disease rate of 33.9 percent. Is this rate above average or below? How unusual is the U.S. obesity rate compared to the average rate? Explain. 111. Table 2.79 gives the percentage of children under age five diagnosed with a specific medical condition. Percentage of Children with the Condition Number of Countries 16–21.45 21.45–26.9 26.9–32.35 32.35–37.8 37.8–43.25 43.25–48.7 Table 2.79 23 4 9 7 6 1 What is the best estimate for the mean percentage of children with the condition? What is the standard deviation? Which interval(s) could be considered unusual? Explain. BRINGING IT TOGETHER: HOMEWORK This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 161 112. Santa Clara County, California, has approximately 27,873 Japanese Americans. Table 2.80 shows their ages by group and each age-group's percentage of the Japanese American community. Age-Group Percentage of Community 0–17 18–24 25–34 35–44 45–54 55–64 65+ Table 2.80 18.9 8.0 22.8 15.0 13.1 11.9 10.3 a. Construct a histogram of the Japanese American community in Santa Clara County. The bars will not be the same width for this example. Why not? What impact does this have on the reliability of the graph? b. What percentage of the community is under age 35? c. Which box plot most resembles the information above? Figure 2.49 162 Chapter 2 \| Descriptive Statistics 113. Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed 100 shoppers. The samples yielded the following information. Javier Ercilia ¯ 6.0 miles 6.0 miles x s 4.0 miles 7.0 miles Table 2.81 a. How can you determine which survey was correct? b. Explain what the difference in the results of the surveys implies about the data. c. If the two histograms depict the distribution of values for each supervisor, which one depicts Ercilia’s sample? How do you know? d. Figure 2.50 If the two box plots depict the distribution of values for each supervisor, which one depicts Ercilia’s sample? How do you know? Figure 2.51 Use the following information to answer the next three exercises: We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section. Number of Years Frequency Number of Years Frequency 1 3 1 1 4 22 23 26 40 42 7 14 15 18 19 Table 2.82 1 1 1 2 2 Total = 20 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 163 Number of Years Frequency Number of Years Frequency 20 3 Table 2.82 Total = 20 114. What is the IQR? a. 8 b. 11 c. 15 d. 35 115. What is the mode? a. 19 b. 19.5 c. 14 and 20 d. 22.65 116. Is this a sample or the entire population? sample a. b. entire population c. neither 117. Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows: Number of Movies Frequency 0 1 2 3 4 Table 2.83 5 9 6 4 1 ¯ . a. Find the sample mean x b. Find the approximate sample standard deviation, s. 164 Chapter 2 \| Descriptive Statistics 118. Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. The results are as follows: X Frequency 12 12 0 1 Table 2.84 ¯ a. Find the sample mean, x b. Find the sample standard deviation, s. c. Construct a histogram of the data. d. Complete the columns of the chart. e. Find the first quartile. f. Find the median. g. Find the third quartile. h. Construct a box plot of the data. i. What percentage of the students owned at least five pairs? j. Find the 40th percentile. k. Find the 90th percentile. l. Construct a line graph of the data. m. Construct a stemplot of the data. 119. Following are the published weights (in pounds) of all of the football team members of the San Francisco 49ers from a previous year: 177, 205, 210, 210, 232, 205, 185, 185, 178, 210, 206, 212, 184, 174, 185, 242, 188, 212, 215, 247, 241, 223, 220, 260, 245, 259, 278, 270, 280, 295, 275, 285, 290, 272, 273, 280, 285, 286, 200, 215, 185, 230, 250, 241, 190, 260, 250, 302, 265, 290, 276, 228, 265 a. Organize the data from smallest to largest value. b. Find the median. c. Find the first quartile. d. Find the third quartile. e. Construct a box plot of the data. f. The middle 50 percent of the weights are from ________ to ________. g. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why? If our population included every team member who ever played for a California-based football team, would the above data be a sample of weights or the population of weights? Why? h. i. Assume the population was a California-based football team. Find i. ii. iii. iv. the population mean, μ, the population standard deviation, σ, and the weight that is two standard deviations below the mean. In addition, when the team's most famous quarterback, played football, he weighed 205 pounds. Also find how many standard deviations above or below the mean was he? j. That same year, the mean weight for a player from a Texas football team was 240.08 pounds with a standard deviation of 44.38 pounds. One player weighed in at 209 pounds. With respect to his team, who was lighter, the California quarterback or the Texas player? How did you determine your answer? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 165 120. One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The 12 change scores are as follows: 3, 8, –1, 2, 0, 5, –3, 1, –1, 6, 5, –2 a. What is the mean change score? b. What is the standard deviation for this population? c. What is the median change score? d. Find the change score that is 2.2 standard deviations below the mean. 121. Refer to Figure 2.52 to determine which of the following are true and which are false. Explain your solution to each part in complete sentences. Figure 2.52 a. The medians for all three graphs are the same. b. We cannot determine if any of the means for the three graphs are different. c. The standard deviation for Graph b is larger than the standard deviation for Graph a. d. We cannot determine if any of the third quartiles for the three graphs are different. 122. In a recent issue of the IEEE Spectrum, 84 engineering conferences were announced. Four conferences lasted two days. Thirty-six lasted three days. Eighteen lasted four days. Nineteen lasted five days. Four lasted six days. One lasted seven days. One lasted eight days. One lasted nine days. Let X = the length (in days) of an engineering conference. a. Organize the data in a chart. b. Find the median, the first quartile, and the third quartile. c. Find the 65th percentile. d. Find the 10th percentile. e. Construct a box plot of the data. f. The middle 50 percent of the conferences last from ________ days to ________ days. g. Calculate the sample mean of days of eng
ineering conferences. h. Calculate the sample standard deviation of days of engineering conferences. i. Find the mode. j. If you were planning an engineering conference, which would you choose as the length of the conference, mean, median, or mode? Explain why you made that choice. k. Give two reasons why you think that three to five days seem to be popular lengths of engineering conferences. 123. A survey of enrollment at 35 community colleges across the United States yielded the following figures: 6,414; 1,550; 2,109; 9,350; 21,828; 4,300; 5,944; 5,722; 2,825; 2,044; 5,481; 5,200; 5,853; 2,750; 10,012; 6,357; 27,000; 9,414; 7,681; 3,200; 17,500; 9,200; 7,380; 18,314; 6,557; 13,713; 17,768; 7,493; 2,771; 2,861; 1,263; 7,285; 28,165; 5,080; 11,622 a. Organize the data into a chart with five intervals of equal width. Label the two columns Enrollment and Frequency. b. Construct a histogram of the data. c. If you were to build a new community college, which piece of information would be more valuable: the mode or the mean? d. Calculate the sample mean. e. Calculate the sample standard deviation. f. A school with an enrollment of 8,000 would be how many standard deviations away from the mean? 166 Chapter 2 \| Descriptive Statistics Use the following information to answer the next two exercises. X = the number of days per week that 100 clients use a particular exercise facility. X Frequency 0 1 2 3 4 5 6 3 12 33 28 11 9 4 Table 2.85 124. The 80th percentile is ________. a. 5 b. 80 c. 3 d. 4 125. The number that is 1.5 standard deviations below the mean is approximately ________. a. 0.7 b. 4.8 c. –2.8 d. cannot be determined This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 167 126. Suppose that a publisher conducted a survey asking adult consumers the number of fiction paperback books they had purchased in the previous month. The results are summarized in Table 2.86. Number of Books Frequency Relative Frequency 0 1 2 3 4 5 7 9 Table 2.86 18 24 24 22 15 10 5 1 a. Are there any outliers in the data? Use an appropriate numerical test involving the IQR to identify outliers, if any, and clearly state your conclusion. If a data value is identified as an outlier, what should be done about it? b. c. Are any data values farther than two standard deviations away from the mean? In some situations, statisticians may use this criterion to identify data values that are unusual, compared to the other data values. Note that this criterion is most appropriate to use for data that is mound shaped and symmetric rather than for skewed data. d. Do Parts a and c of this problem give the same answer? e. Examine the shape of the data. Which part, a or c, of this question gives a more appropriate result for this data? f. Based on the shape of the data, which is the most appropriate measure of center for this data, mean, median, or mode? REFERENCES 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs Burbary, K. (2011, March 7). Facebook demographics revisited – 2001 statistics. Social Media Today. Retrieved from http://www.kenburbary.com/2011/03/facebook-demographics-revisited-2011-statistics-2/ Centers for Disease Control and Prevention. (n.d.). Overweight and obesity: Adult obesity facts. Available online http://www.cdc.gov/obesity/data/adult.html CollegeBoard. (2013). The 9th annual AP report to the nation. Retrieved from http://apreport.collegeboard.org/goalsandfindings/promoting-equity 2.2 Histograms, Frequency Polygons, and Time Series Graphs Bureau of Labor Statistics, U.S. Department of Labor. (n.d.). Consumer price index. Retrieved from https://www.bls.gov/ cpi/ CIA World Factbook. http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (n.d.). Demographics: Children under the age of 5 years underweight. Available at Centers for Disease Control and Prevention. (n.d.). Overweight and obesity: Adult obesity facts. Available online http://www.cdc.gov/obesity/data/adult.html Food and Agriculture Organization of http://www.fao.org/economic/ess/ess-fs/en/ the United Nations. (n.d.). Food security statistics. Retrieved from General Register Office for Scotland. Births time series data. (2013). Retrieved from http://www.gro-scotland.gov.uk/ statistics/theme/vital-events/births/time-series.html 168 Chapter 2 \| Descriptive Statistics Gunst, R., and Mason, R. (1980). Regression analysis and its application: A data-oriented approach. Boca Raton, FL: CRC Press. Sandbox Networks. (2007). Presidents. Available online at http://www.factmonster.com/ipka/A0194030.html Scholastic. (2013). Timeline: Guide to the U.S. presidents. Retrieved from http://www.scholastic.com/teachers/article/ timeline-guide-us-presidents World Bank Group. (2013). DataBank: CO2 emissions (kt). Retrieved from http://databank.worldbank.org/data/home.aspx 2.3 Measures of the Location of the Data Cauchon, D., and Overberg, P. (2012). Census data shows minorities now a majority of U.S. births. USA Today. Retrieved from http://usatoday30.usatoday.com/news/nation/story/2012-05-17/minority-birthscensus/55029100/1 The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/ Time. (n.d.). Survey by Yankelovich Partners, Inc. U.S. Census Bureau. (1990). 1990 census. Retrieved from http://www.census.gov/main/www/cen1990.html U.S. Census Bureau. (n.d.). Data. Retrieved from http://www.census.gov/ 2.4 Box Plots West Magazine. (n.d.). Retrieved from https://westmagazine.net/ 2.5 Measures of the Center of the Data CIA World Factbook. r.aspx?t=50&v=2228&l=en (n.d.). Obesity – adult prevalence rate. Available at http://www.indexmundi.com/g/ World Bank Group. (n.d.). Retrieved from http://www.worldbank.org 2.7 Measures of the Spread of the Data King, B. (2005, Dec.). Graphically Speaking. Retrieved from http://www.ltcc.edu/web/about/institutional-research Microsoft Bookshelf. (n.d.). SOLUTIONS 1 Stem Leaf Table 2.87 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 169 Stem Leaf Table 2.88 3 5 Figure 2.53 7 Figure 2.54 Chapter 2 \| Descriptive Statistics 170 9 Figure 2.55 11 Figure 2.56 13 65 15 The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value. 17 Answers will vary. One possible histogram is shown below. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 171 Figure 2.57 19 Find the midpoint for each class. These will be graphed on the x-axis. The frequency values will be graphed on the y-axis values. Figure 2.58 172 21 Chapter 2 \| Descriptive Statistics Figure 2.59 23 a. The 40th percentile is 37 years. b. The 78th percentile is 70 years. 25 Jesse graduated 37th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37. x = 143 and y = 1. x + .5y (100) = 79.72. Jesse’s rank of 37 puts him at the 80th percentile. (100) = 143 + .5(1) n 180 27 a. For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster. b. 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster). 29 When waiting in line at the DMV, the 85th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer. 31 The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of $1,700 or less; only 10 percent had damage repair costs of $1,700 or more. 33 You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost $240,000 or less; 66 percent of houses cost $240,000 or more. 35 4 37 6 – 4 = 2 39 6 41 More than 25 percent of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25 percent and the bottom 25 percent are spread out evenly; the whiskers have the same length. 43 Mean: 16 + 17 + 19 + 20 + 20 + 21 + 23 + 24 + 25 + 25 + 25 + 26 + 26 + 27 + 27 + 27 + 28 + 29 + 30 + 32 + 33 + 33 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 173 + 34 + 35 + 37 + 39 + 40 = 738; 738 27 = 27.33 45 The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27 47 4 49 The data are symmetrical. The median is 3, and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical. 51 The data are skewed right. The median is 87.5, and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right. 53 When the data are symmetrical, the mean and median are close or the same. 55 The distribution is skewed right because it looks pulled out to the right. 57 The mean is 4.1 and is slightly greater than the median, which is 4. 59 The mode and the median are the same. In this case, both 5. 61 The distribution is skewed left because it looks pulled out to the left. 63 Both the mean and the median are 6. 65 The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest. 67 The mean tends to reflect skewing the most because it is affected the most by outliers. 69 sampling variability 70 induced variability 71 measurement variability 72 natural variability 73 s = 34.5 75 For
Fredo: z = .158 – .166 .012 = –0.67. For Karl: z = .177 – .189 .015 = –.8. Fredo’s z score of –.67 is higher than Karl’s z score of –.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team. 77 a. s x = b. s x = c. s x = ∑ f m2 n ∑ f m2 n ∑ f m2 n 79 − x¯ 2 = 193,157.45 30 − 79.52 = 10.88 − x¯ 2 = 380,945.3 101 − 60.942 = 7.62 − x¯ 2 = 440,051.5 86 − 70.662 = 11.14 a. Example solution for using the random number generator for the TI-84+ to generate a simple random sample of eight states. Instructions are as follows. Number the entries in the table 1–51 (includes Washington, DC; numbered vertically) Press MATH Arrow over to PRB Press 5:randInt( Enter 51,1,8) Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}. 174 Chapter 2 \| Descriptive Statistics Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}. Figure 2.60 b. Figure 2.61 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 175 Figure 2.62 c. 81 Amount($) Frequency Relative Frequency 51–100 101–150 151–200 201–250 251–300 301–350 5 10 15 15 10 5 Table 2.89 Singles .08 .17 .25 .25 .17 .08 Amount ($) Frequency Relative Frequency 100–150 201–250 251–300 301–350 351–400 401–450 451–500 501–550 551–600 601–650 5 5 5 5 10 10 10 10 5 5 Table 2.90 Couples .07 .07 .07 .07 .14 .14 .14 .14 .07 .07 176 Chapter 2 \| Descriptive Statistics a. See Table 2.69 and Table 2.69. b. In the following histogram, data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted, with the exception of the first interval, where both boundary values are included. Figure 2.63 c. In the following histogram, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted, with the exception of the first interval, where values on both boundaries are included. Figure 2.64 d. Compare the two graphs. i. Answers may vary. Possible answers include the following: ▪ Both graphs have a single peak. ▪ Both graphs use class intervals with width equal to $50 ii. Answers may vary. Possible answers include the following: ▪ The couples graph has a class interval with no values ▪ It takes almost twice as many class intervals to display the data for couples iii. Answers may vary. Possible answers include the following. The graphs are more similar than different because This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 177 the overall patterns for the graphs are the same. e. Check student's solution. f. Compare the graph for the singles with the new graph for the couples: i. ▪ Both graphs have a single peak ▪ Both graphs display six class intervals ▪ Both graphs show the same general pattern ii. Answers may vary. Possible answers include the following. Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different. g. Answers may vary. Possible answers include the following. You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison. h. Answers may vary. Possible answers include the following. Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals. 83 c 85 Answers will vary. 87 a. 1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06 b. .19+.26+.18 = .63 c. Check student’s solution. d. 40th percentile will fall between 30,000 and 40,000 80th percentile will fall between 50,000 and 75,000 e. Check student’s solution. 89 a. more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent b. 62.4 percent 91 a. Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year. b. Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile. 93 a. Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50 percent of buyers are more variable than the ages of the lower 50 percent. b. The black sports car is most likely to have an outlier. It has the longest whisker. c. Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a rule, because there is so much variability in each data set. d. The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median. e. The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile. 178 Chapter 2 \| Descriptive Statistics f. IQR ~ 17 years g. There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is are concentrated. h. The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25 percent fall between 41 and 64. Since 25 percent of values fall between 31 and 38, we know that fewer than 25 percent fall between 31 and 35. 96 the mean percentage, x¯ = 1,328.65 50 = 26.75 98 The median value is the middle value in the ordered list of data values. The median value of a set of 11 will be the sixth number in order. Six years will have totals at or below the median. 100 474 FTES 102 919 104 • mean = 1,809.3 • median = 1,812.5 • • • • standard deviation = 151.2 first quartile = 1,690 third quartile = 1,935 IQR = 245 106 Hint: think about the number of years covered by each time period and what happened to higher education during those periods. 108 For pianos, the cost of the piano is .4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type. 110 • x¯ = 23.32 • Using the TI 83/84, we obtain a standard deviation of: s x = 12.95. • The obesity rate of the United States is 10.58 percent higher than the average obesity rate. • Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the disease percentage that is one standard deviation from the mean. The U.S. disease rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, although 34 percent have the disease, does not have an unusually high percentage of people with the disease. 112 a. For graph, check student's solution. b. 49.7 percent of the community is under the age of 35 c. Based on the information in the table, graph (a) most closely represents the data. 114 a 116 b 117 a. 1.48 b. 1.12 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 2 \| Descriptive Statistics 179 119 a. 174, 177, 178, 184, 185, 185, 185, 185, 188, 190, 200, 205, 205, 206, 210, 210, 210, 212, 212, 215, 215, 220, 223, 228, 230, 232, 241, 241, 242, 245, 247, 250, 250, 259, 260, 260, 265, 265, 270, 272, 273, 275, 276, 278, 280, 280, 285, 285, 286, 290, 290, 295, 302 b. 241 c. 205.5 d. 272.5 e. f. 205.5, 272.5 g. sample h. population i. i. 236.34 ii. 37.50 iii. 161.34 iv. .84 standard deviations below the mean j. young true true true false 121 a. b. c. d. 123 a. b. Check student’s solution. c. mode d. 8,628.74 e. 6,943.88 f. –0.09 Enrollment Frequency 1,000–5,000 5,000–10,000 10 16 10,000–15,000 3 15,000–20,000 3 20,000–25,000 1 25,000–30,000 2 Table 2.91 180 125 a Chapter 2 \| Descriptive Statistics This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 181 3 \| PROBABILITY TOPICS Figure 3.1 Meteor showers are rare, but the probability of them occurring can be calculated. (credit: Navicore/flickr) Introduction Chapter Objectives By the end of this chapter, the student should be able to do the following: • Understand and use the terminology of probability • Determine whether two events are mutually exclusive and whether two events are independent • Calculate probabilities using the addition rules and multiplication rules • Construct and interpret contingency tables • Construct and interpret Venn diagrams • Construct and interpret tree diagrams It is often necessary to guess about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their asses
sment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs. You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach. 182 Chapter 3 \| Probability Topics How likely is it that a randomly chosen person in your class has change in his or her pocket? Would you say that it is very likely? Somewhat likely? Not likely? How likely is it that a randomly chosen person in your class has ridden a bus in the past month? If a person is chosen at random from your classroom and you know that he or she has ridden a bus in the past month, do you think that person is more likely or less likely to have change? Probability theory allows us to measure how likely—or unlikely—a given result is. Your instructor will survey your class. Count the number of students in the class today. • Raise your hand if you have any change in your pocket or purse. Record the number of raised hands. • Raise your hand if you rode a bus within the past month. Record the number of raised hands. • Raise your hand if you answered yes to BOTH of the first two questions. Record the number of raised hands. Use the class data as estimates of the following probabilities. P(change) means the probability that a randomly chosen person in your class has change in his/her pocket or purse. P(bus) means the probability that a randomly chosen person in your class rode a bus within the last month and so on. Discuss your answers. • Find P(change). • Find P(bus). • Find P(change AND bus). Find the probability that a randomly chosen student in your class has change in his/her pocket or purse and rode a bus within the last month. • Find P(change\|bus). Find the probability that a randomly chosen student has change given that he or she rode a bus within the last month. Count all the students who rode a bus. From the group of students who rode a bus, count those who have change. The probability is equal to those who have change and rode a bus divided by those who rode a bus. 3.1 \| Terminology Probability is a measure that is associated with how certain we are of results, or outcomes, of a particular activity. When the activity is a planned operation carried out under controlled conditions, it is called an experiment. If the result is not predetermined, then the experiment is said to be a chance experiment. Each time the experiment is attempted is called a trial. Examples of chance experiments include the following: • • flipping a fair coin, spinning a spinner, • drawing a marble at random from a bag, and • rolling a pair of dice. A result of an experiment is called an outcome. The sample space of an experiment is the set, or collection, of all possible outcomes. There are four main ways to represent a sample space: Systematic List of Outcomes heads (H) HH Flipping a Fair Coin Flipping Two Fair Coins Table 3.1 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 183 Flipping a Fair Coin Flipping Two Fair Coins tails (T) HT TH TT Figure 3.2 Figure 3.3 Tree Diagram* Venn Diagram* Figure 3.4 Figure 3.5 Set Notation S = {H, T} S = {HH, HT, TH, TT} Table 3.1 *We will investigate tree diagrams and Venn diagrams in Section 3.5. Note—when represented as a set, the sample space is denoted with an uppercase S. An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like A and B are commonly used to represent events. For example, if the experiment is to flip three fair coins, event A might be getting at most one head. The probability of an event A is written P(A), and 0 ≤ P(A) ≤ 1.P(A) = 0 means the event A can never happen. P(A) = 1 means the event A always happens. P(A) = 0.5 means the event A is equally likely to occur or not to occur. Figure 3.6 If two outcomes or events are equally likely, then they have equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer. To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. This is known as the theoretical probability of A. 184 Chapter 3 \| Probability Topics Theoretical Probability of Event A P(A) = Number of outcomes in event A Total number of possible outcomes. For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. Let A represent the outcome getting one head. There are two outcomes that meet this condition {HT, TH}, so P(A) = 2 4 = 1 2 = .5. Theoretical probability is not sufficient in all situations, however. Suppose we want to calculate the probability that a randomly selected car will run a red light at a given intersection. In this case, we need to look at events that have occurred, not theoretical possibilities. We could install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a period of time. These data will allow us to calculate the experimental, or empirical, probability that a car runs the red light. Experimental Probability of Event A P(A) = Number of times event A occurs. Total number of trials While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two possible outcomes. So, the theoretical probability of heads is 1 2 . Probability does not predict short-term results, however. If an experiment involves flipping a coin 10 times, you should not expect exactly five heads and five tails. The probability of any outcome measures the long-term relative frequency of that outcome. If you continue to flip the coin (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches .5 (the probability of heads).This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed, or empirical, relative frequency will approach the theoretical probability. Suppose you roll one fair, six-sided die with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five. There are two outcomes {5, 6}. P(E) = 2 . If you were to roll the die only a few times, you would not be 6 surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, 2 6 of the rolls would result in an outcome of at least five. You would not expect exactly 2 6 , but the long-term relative frequency of obtaining this result would approach the theoretical probability of 2 6 as the number of repetitions grows larger and larger. It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair, or biased. Two math professors in Europe had their statistics students test the Belgian one-euro coin and discovered that in 250 trials, a head was obtained 56 percent of the time and a tail was obtained 44 percent of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely. OR Event An outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B. For example, let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. A OR B = {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are not listed twice. AND Event An outcome is in the event A AND B if the outcome is in both A and B at the same time. For example, let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then A AND B = {4, 5}. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 185 The complement of event A is denoted A′ (read "A prime"). A′ consists of all outcomes that are not in A. Notice that P(A) + P(A′) = 1. For example, let S = {1, 2, 3, 4, 5, 6} and let A = {1, 2, 3, 4}. Then, A′ = {5, 6}. P(A) = 4 6 , P(A′) = 2 6 , and P(A) + P(A′) = 4 6 + 2 6 = 1. The conditional probability of A given B is written P(A\|B), r
ead "the probability of A, given B." P(A\|B) is the probability that event A will occur given that the event B has already occurred. A conditional probability reduces the sample space. We calculate the probability of A from the reduced sample space B. The formula to calculate P(A\|B) is P(A\|B) = P(A AND B) P(B) where P(B) is greater than zero. For example, suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = {2, 3} and B = {2, 4, 6}. P(A\|B) represents the probability that a randomly selected outcome is in A given that it is in B. We know that the outcome must lie in B, so there are three possible outcomes. There is only one outcome in B that also lies in A, so P(A\|B) = 1 3 . We get the same result by using the formula. Remember that S has six outcomes. P(A\|B) = P(A AND B) P(B) = (the number of outcomes that are 2 or 3 and even in S) 6 (the number of outcomes that are even in S Understanding Terminology and Symbols It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any. Example 3.1 The sample space S is the whole numbers starting at one and less than 20. a. S = ________ Let event A = the even numbers and event B = numbers greater than 13. b. A = ________, B = ________ c. P(A) = ________, P(B) = ________ d. A AND B = ________, A OR B = ________ e. P(A AND B) = ________, P(A OR B) = ________ f. A′ = ________, P(A′) = ________ g. P(A) + P(A′) = ________ h. P(A\|B) = ________, P(B\|A) = ________; are the probabilities equal? Solution 3.1 a. S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} b. A = {2, 4, 6, 8, 10, 12, 14, 16, 18}, B = {14, 15, 16, 17, 18, 19} c. P(A) = number of outcomes in A number of outcomes in S = 9 19 , P(B) = P(A) = number of outcomes in B number of outcomes in S = 6 19 d. The set A AND B contains all outcomes that lie in both sets A and B, so A AND B = {14,16,18}, The set A OR B contains all outcomes that lie either of the sets A or B, so A OR B = {2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19}. e. P(A AND B) = 3 19 , P(A OR B) = 12 19 186 Chapter 3 \| Probability Topics f. A' consists of all outcomes in the sample space, S, that DO NOT lie in A, so A′ = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19; P(A′) = 10 19 . g. P(A) + P(A′) = 9 19 + 10 19 = 1 h. P(A\|B) = P(AANDB) P(B) = 3 19 6 19 = 3 6 , P(B\|A) = P(AANDB) P(A) = 3 19 9 19 = 3 9 , No, the probabilities are not equal. 3.1 The sample space S is all the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (1, 4)). a. S = ________ Let event A = the sum is even and event B = the first number is prime. b. A = ________, B = ________ c. P(A) = ________, P(B) = ________ d. A AND B = ________, A OR B = ________ e. P(A AND B) = ________, P(A OR B) = ________ f. B′ = ________, P(B′) = ________ g. P(A) + P(A′) = ________ h. P(A\|B) = ________, P(B\|A) = ________; are the probabilities equal? Example 3.2 A fair, six-sided die is rolled. The sample space, S, is {1, 2, 3, 4, 5, 6}. Describe each event and calculate its probability. a. Event T = the outcome is two. b. Event A = the outcome is an even number. c. Event B = the outcome is less than four. d. The complement of A e. A GIVEN B f. B GIVEN A g. A AND B h. A OR B i. A OR B′ j. Event N = the outcome is a prime number. k. Event I = the outcome is seven. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 187 Solution 3.2 a. T = {2}, P(T) = number of outcomes in T number of outcomes in S = 1 6 b. A = {2, 4, 6}, P(A) = 3 6 = 1 2 c. B = {1, 2, 3}, P(B) = 3 6 = 1 2 d. A′ = {1, 3, 5}, P(A′) = 3 6 = 1 2 e. A\|B = {2}, There are three outcomes in B, and only 1 of these lies in A, so P(A\|B) = 1 3 f. B\|A = {2}, There are three outcomes in A, and only 1 of these lies in B, so P(B\|A) = 1 3 g. A AND B = {2}, P(A AND B) = 1 6 h. A OR B = {1, 2, 3, 4, 6}, P(A OR B) = 5 6 i. A OR B′ = {2, 4, 5, 6}, P(A OR B′) = 4 6 = 2 3 j. N = {2, 3, 5}, P(N) = 1 2 k. It is impossible to roll a die and get an outcome of 7, so P(7) = 0. Example 3.3 Table 3.2 describes the distribution of a random sample S of 100 individuals, organized by gender and whether they are right or left-handed. Right-Handed Left-Handed Males 43 Females 44 Table 3.2 9 4 Let’s denote the events M = the subject is male, F = the subject is female, R = the subject is right-handed, L = the subject is left-handed. Compute the following probabilities: a. P(M) b. P(F) c. P(R) d. P(L) e. P(M AND R) f. P(F AND L) g. P(M OR F) h. P(M OR R) 188 Chapter 3 \| Probability Topics i. P(F OR L) j. P(M') k. P(R\|M) l. P(F\|L) m. P(L\|F) Solution 3.3 a. P(M) = number of males total number of subjects = 43 + 9 43 + 9 + 44 + 4 = 52 100 = .52 b. P(F) = number of females total number of subjects = 44 + 4 43 + 9 + 44 + 4 = 48 100 = .48 c. P(R) = number of right-handed subjects total number of subjects = 43 + 44 43 + 9 + 44 + 4 = 87 100 = .87 d. P(L) = number of left-handed subjects total number of subjects = 9 + 4 43 + 9 + 44 + 4 = 13 100 = .13 e. P(MandR) = number of male, right-handed subjects total number of subjects = 43 100 = .43 f. P(FandL) = number of female, left-handed subjects total number of subjects = 4 100 = .04 g. P(MorF) = number of subjects that are male or female total number of subjects = 52 + 48 100 = 100 100 = 1 h. P(MorR) = number of subjects that are male or right-handed total number of subjects = 43 + 9 + 44 100 = 96 100 = .96 i. P(ForL) = number of subjects that are female or left-handed total number of subjects = 44 + 4 + 9 100 = 57 100 = .57 j. P(M') = number of subjects who are not male total number of subjects = 44 + 4 43 + 9 + 44 + 4 = 48 100 = .48 k. P(R\|M) = l. P(F\|L) = m. P(L\|F) = P(RandM) P(M) = 0.43 0.52 =.8269 (rounded to four decimal places) P(FandL) P(L) = 0.04 0.13 P(LandF) P(F) = 0.04 0.48 =.3077 (rounded to four decimal places) =.0833 (rounded to four decimal places) 3.2 \| Independent and Mutually Exclusive Events Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: • P(A\|B) = P(A) • P(B\|A) = P(B) • P(A AND B) = P(A)P(B) This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 189 Two events A and B are independent events if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show only one of the above conditions. If two events are not independent, then we say that they are dependent events. Sampling may be done with replacement or without replacement. • With replacement: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick. A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is 4 7 . When James draws a marble from the bag a second time, the probability of drawing blue is still 4 7 white marbles. . James replaced the marble after the first draw, so there are still four blue and three Figure 3.7 • Without replacement: When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent. The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is 4 7 . Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now three blue and three white marbles. So, the probability of drawing blue is now 3 6 . Removing the first marble without replacing it influences the probabilities = 1 2 on the second draw. 190 Chapter 3 \| Probability Topics Figure 3.8 If it is not known whether A and B are independent or dependent, assume they are dependent until you can show otherwise. Example 3.4 You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. Clubs and spades are black, while diamonds and hearts are red cards. There are 13 cards in each suit consisting of A (ace), 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit. Figure 3.9 a. Sampling with replacement Suppose you pick three cards with replacement. The first card you pick out of the 52 cards is the Q of spades. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 191 You put this card back, reshuffle the cards and pick a second card from the 52-card deck. It is the 10 of clubs. You put this card back, reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the Q of spades again. Your picks are {Q of spades, 10 of clubs, Q of spades}. You have picked the Q of spades twice. You pick each card from the 52-card deck. b. Sampling without replacement Suppose you pick three cards without replacement. The first card you pick out of the 52 cards is the K of hearts. You put this card aside and pick
the second card from the 51 cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining 50 cards in the deck. The third card is the J of spades. Your picks are {K of hearts, three of diamonds, J of spades}. Because you have picked the cards without replacement, you cannot pick the same card twice. 3.4 You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), K (king) of that suit. Three cards are picked at random. a. Suppose you know that the picked cards are Q of spades, K of hearts and Q of spades. Can you decide if the sampling was with or without replacement? b. Suppose you know that the picked cards are Q of spades, K of hearts, and J of spades. Can you decide if the sampling was with or without replacement? Example 3.5 You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), and K (king) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs. a. Suppose you pick four cards, but do not put any cards back into the deck. Your cards are QS, 1D, 1C, QD. b. Suppose you pick four cards and put each card back before you pick the next card. Your cards are KH, 7D, 6D, KH. Which of a. or b. did you sample with replacement and which did you sample without replacement? Solution 3.5 a. Because you do not put any cards back, the deck changes after each draw. These events are dependent, and this is sampling without replacement; b. Because you put each card back before picking the next one, the deck never changes. These events are independent, so this is sampling with replacement. 3.5 You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (jack), Q (queen), and K (king) of that suit. S = spades, H = Hearts, D = Diamonds, C = Clubs. Suppose that you sample four cards without replacement. Which of the following outcomes are possible? Answer the same question for sampling with replacement. a. QS, 1D, 1C, QD b. KH, 7D, 6D, KH c. QS, 7D, 6D, KS 192 Chapter 3 \| Probability Topics Mutually Exclusive Events A and B are mutually exclusive events if they cannot occur at the same time. This means that A and B do not share any outcomes and P(A AND B) = 0. For example, suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, and C = {7, 9}. A AND B = {4, 5}. P(A AND B) = 2 10 and is not equal to zero. Therefore, A and B are not mutually exclusive. A and C do not have any numbers in common so P(A AND C) = 0. Therefore, A and C are mutually exclusive. If it is not known whether A and B are mutually exclusive, assume they are not until you can show otherwise. The following examples illustrate these definitions and terms. Example 3.6 Flip two fair coins. This is an experiment. The sample space is {HH, HT, TH, TT}, where T = tails and H = heads. The outcomes are HH, HT, TH, and TT. The outcomes HT and TH are different. The HT means that the first coin showed heads and the second coin showed tails. The TH means that the first coin showed tails and the second coin showed heads. • Let A = the event of getting at most one tail. At most one tail means zero or one tail. Then A can be written as {HH, HT, TH}. The outcome HH shows zero tails. HT and TH each show one tail. • Let B = the event of getting all tails. B can be written as {TT}. B is the complement event of A, so B = A′. Also, P(A) + P(B) = P(A) + P(A′) = 1. • The probabilities for A and for B are P(A) = 3 4 and P(B) = 1 4 . • Let C = the event of getting all heads. C = {HH}. Since B = {TT}, P(B AND C) = 0. B and C are mutually exclusive. (B and C have no members in common because you cannot have all tails and all heads at the same time.) • Let D = event of getting more than one tail. D = {TT}. P(D) = 1 4 . • Let E = event of getting a head on the first roll. This implies you can get either a head or tail on the second roll. E = {HT, HH}. P(E) = 2 4 . • Find the probability of getting at least one (one or two) tail in two flips. Let F = event of getting at least one tail in two flips. F = {HT, TH, TT}. P(F) = 3 4 . 3.6 Draw two cards from a standard 52-card deck with replacement. Find the probability of getting at least one black card. Example 3.7 Flip two fair coins. Find the probabilities of the events. a. Let F = the event of getting at most one tail (zero or one tail). b. Let G = the event of getting two faces that are the same. c. Let H = the event of getting a head on the first flip followed by a head or tail on the second flip. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 193 d. Are F and G mutually exclusive? e. Let J = the event of getting all tails. Are J and H mutually exclusive? Solution 3.7 Look at the sample space in Example 3.6. a. Zero (0) or one (1) tails occur when the outcomes HH, TH, HT show up. P(F) = 3 4 . b. Two faces are the same if HH or TT show up. P(G) = 2 4 . c. A head on the first flip followed by a head or tail on the second flip occurs when HH or HT show up. P(H) = 2 4 . d. F and G share HH so P(F AND G) is not equal to zero (0). F and G are not mutually exclusive. e. Getting all tails occurs when tails shows up on both coins (TT). H’s outcomes are HH and HT. J and H have nothing in common so P(J AND H) = 0. J and H are mutually exclusive. 3.7 A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Find the probability of the following events: a. Let F = the event of getting the white ball twice. b. Let G = the event of getting two balls of different colors. c. Let H = the event of getting white on the first pick. d. Are F and G mutually exclusive? e. Are G and H mutually exclusive? Example 3.8 Roll one fair, six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. Let event A = a face is odd. Then A = {1, 3, 5}. Let event B = a face is even. Then B = {2, 4, 6}. • Find the complement of A, A′. The complement of A, A′, is B because A and B together make up the sample space. P(A) + P(B) = P(A) + P(A′) = 1. Also, P(A) = 3 6 and P(B) = 3 6 . • Let event C = odd faces larger than two. Then C = {3, 5}. Let event D = all even faces smaller than five. Then D = {2, 4}. P(C AND D) = 0 because you cannot have an odd and even face at the same time. Therefore, C and D are mutually exclusive events. • Let event E = all faces less than five. E = {1, 2, 3, 4}. Are C and E mutually exclusive events? Answer yes or no. Why or why not? Solution 3.8 No. C = {3, 5} and E = {1, 2, 3, 4}. P(C AND E) = 1 6 . To be mutually exclusive, P(C AND E) must be zero. • Find P(C\|A). This is a conditional probability. Recall that event C is {3, 5} and event A is {1, 3, 5}. To find P(C\|A), find the probability of C using the sample space A. You have reduced the sample space from the 194 Chapter 3 \| Probability Topics original sample space {1, 2, 3, 4, 5, 6} to {1, 3, 5}. So, P(C\|A) = 2 3 . 3.8 Let event A = learning Spanish. Let event B = learning German. Then A AND B = learning Spanish and German. Suppose P(A) = 0.4 and P(B) = .2. P(A AND B) = .08. Are events A and B independent? Hint—You must show one of the following: • P(A\|B) = P(A) • P(B\|A) • P(A AND B) = P(A)P(B) Example 3.9 Let event G = taking a math class. Let event H = taking a science class. Then, G AND H = taking a math class and a science class. Suppose P(G) = .6, P(H) = .5, and P(G AND H) = .3. Are G and H independent? If G and H are independent, then you must show ONE of the following: • P(G\|H) = P(G) • P(H\|G) = P(H) • P(G AND H) = P(G)P(H) NOTE The choice you make depends on the information you have. You could choose any of the methods here because you have the necessary information. a. Show that P(G\|H) = P(G). Solution 3.9 P(G\|H) = P(G AND H) P(H) = .3 .5 = .6 = P(G) b. Show P(G AND H) = P(G)P(H). Solution 3.9 P(G)P(H) = (.6)(.5) = .3 = P(G AND H) Since G and H are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent, that is, they are dependent, then knowing that a person is taking a science class would change the chance he or she is taking math. For practice, show that P(H\|G) = P(H) to show that G and H are independent events. 3.9 In a bag, there are six red marbles and four green marbles. The red marbles are marked with the numbers 1, 2, 3, 4, 5, and 6. The green marbles are marked with the numbers 1, 2, 3, and 4. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 195 • R = a red marble • G = a green marble • O = an odd-numbered marble • The sample space is S = {R1, R2, R3, R4, R5, R6, G1, G2, G3, G4}. S has 10 outcomes. What is P(G AND O)? Example 3.10 Let event C = taking an English class. Let event D = taking a speech class. Suppose P(C) = .75, P(D) = .3, P(C\|D) = .75 and P(C AND D) = .225. Justify your answers to the following questions numerically. a. Are C and D independent? b. Are C and D mutually exclusive? c. What is P(D\|C)? Solution 3.10 a. Yes, because P(C\|D) = .75 = P(C). b. No, because P(C AND D) is not equal to zero. c. P(D\|C) = P(C AND D) P(C) = 0.225 .75 = .3 3.10 A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = .40, P(D) = .30 and P(B AND D) = .20. a. Find P(B\|D). b. Find P(D\|B). c. Are B and D independent? d. Are B and D mutually exclusive? Example 3.11 In a box there are three red cards and five blue c
ards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card. Let R = red card is drawn, B = blue card is drawn, E = even-numbered card is drawn. The sample space S = R1, R2, R3, B1, B2, B3, B4, B5. S has eight outcomes. • P(R) = 3 8 . P(B) = 5 8 . P(R AND B) = 0. You cannot draw one card that is both red and blue. • P(E) = 3 8 . There are three even-numbered cards, R2, B2, and B4. 196 Chapter 3 \| Probability Topics • P(E\|B = 2 5 . There are five blue cards: B1, B2, B3, B4, and B5. Out of the blue cards, there are two even cards; B2 and B4. • P(B\|E) = 2 3 are blue; B2 and B4. . There are three even-numbered cards: R2, B2, and B4. Out of the even-numbered cards, two • The events R and B are mutually exclusive because P(R AND B) = 0. • Let G = card with a number greater than 3. G = {B4, B5}. P(G) = 2 8 . Let H = blue card numbered between one and four, inclusive. H = {B1, B2, B3, B4}. P(G\|H) = 1 4 . The only card in H that has a number greater than three is B4. Since 2 8 = 1 4 , P(G) = P(G\|H), which means that G and H are independent. 3.11 In a basketball arena, • 70 percent of the fans are rooting for the home team, • 25 percent of the fans are wearing blue, • 20 percent of the fans are wearing blue and are rooting for the away team, and • Of the fans rooting for the away team, 67 percent are wearing blue. Let A be the event that a fan is rooting for the away team. Let B be the event that a fan is wearing blue. Are the events of rooting for the away team and wearing blue independent? Are they mutually exclusive? Example 3.12 In a particular class, 60 percent of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75 percent have long hair. Let F be the event that a student is female. Let L be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent? The following probabilities are given in this example: • P(F) = 0.60; P(L) = 0.50 • P(F AND L) = 0.45 • P(L\|F) = 0.75 NOTE The choice you make depends on the information you have. You could use the first or last condition on the list for this example. You do not know P(F\|L) yet, so you cannot use the second condition. Solution 1 Check whether P(F AND L) = P(F)P(L). We are given that P(F AND L) = 0.45, but P(F)P(L) = (.60)(.50) = .30. The events of being female and having long hair are not independent because P(F AND L) does not equal P(F)P(L). Solution 2 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 197 Check whether P(L\|F) equals P(L). We are given that P(L\|F) = .75, but P(L) = .50; they are not equal. The events of being female and having long hair are not independent. Interpretation of Results The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair. 3.12 Mark is deciding which route to take to work. His choices are I = the Interstate and F = Fifth Street. • P(I) = .44 and P(F) = .55 • P(I AND F) = 0 because Mark will take only one route to work. What is the probability of P(I OR F)? Example 3.13 a. Toss one fair coin (the coin has two sides, H and T). The outcomes are ________. Count the outcomes. There are ________ outcomes. b. Toss one fair, six-sided die (the die has 1, 2, 3, 4, 5, or 6 dots on a side). The outcomes are ________. Count the outcomes. There are ________ outcomes. c. Multiply the two numbers of outcomes. The answer is ________. d. If you flip one fair coin and follow it with the toss of one fair, six-sided die, the answer in Part c is the number of outcomes (size of the sample space). List the outcomes. Hint—Two of the outcomes are H1 and T6. e. Event A = heads (H) on the coin followed by an even number (2, 4, 6) on the die. A = {________}. Find P(A). f. Event B = heads on the coin followed by a three on the die. B = {________}. Find P(B). g. Are A and B mutually exclusive? Hint—What is P(A AND B)? If P(A AND B) = 0, then A and B are mutually exclusive. h. Are A and B independent? Hint—Is P(A AND B) = P(A)P(B)? If P(A AND B) = P(A)P(B), then A and B are independent. If not, then they are dependent. Solution 3.13 a. H and T; 2 b. 1, 2, 3, 4, 5, 6; 6 c. 2(6) = 12 d. Make a systematic list of possible outcomes. Start by listing all possible outcomes when the coin shows tails (T). Then list the outcomes that are possible when the coin shows heads (H): T1, T2, T3, T4, T5, T6, H1, H2, H3, H4, H5, H6 e. A = {H2, H4, H6}; P(A) = number of outcomes inA number of possible outcomes = 3 12 f. B = {H3}; P(B) = 1 12 g. Yes, because P(A AND B) = 0 198 Chapter 3 \| Probability Topics h. P(A AND B) = 0. P(A)P(B) = ⎛ ⎝ ⎞ ⎠ 3 12 ⎛ ⎝ 1 12 ⎞ ⎠ . P(A AND B) does not equal P(A)P(B), so A and B are dependent. 3.13 A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, and S the event of picking the white ball in the second drawing. a. Compute P(T). b. Compute P(T\|F). c. Are T and F independent? d. Are F and S mutually exclusive? e. Are F and S independent? 3.3 \| Two Basic Rules of Probability In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not. The Multiplication Rule If A and B are two events defined on a sample space, then P(A AND B) = P(B)P(A\|B). This equation can be rewritten as P(A AND B) = P(B)P(A\|B), the multiplication rule. If A and B are independent, then P(A\|B) = P(A). In this special case, P(A AND B) = P(A\|B)P(B) becomes P(A AND B) = P(A)P(B). A bag contains four green marbles, three red marbles, and two yellow marbles. Mark draws two marbles from the bag without replacement. The probability that he draws a yellow marble and then a green marble is P⎛ ⎝yellow and green⎞ ⎠ ⋅ P⎛ ⎝green \| yellow⎞ ⎠ ⎝yellow⎞ ⋅ 4 8 ⎠ = P⎛ = 2 9 = 1 9 Notice that P⎛ ⎝green \| yellow⎞ ⎠ = 4 8 . After the yellow marble is drawn, there are four green marbles in the bag and eight marbles in all. The Addition Rule If A and B are defined on a sample space, then P(A OR B) = P(A) + P(B) − P(A AND B). Draw one card from a standard deck of playing cards. Let H = the card is a heart, and let J = the card is a jack. These events are not mutually exclusive because a card can be both a heart and a jack. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 199 = 13 52 P(H or J) = P(H) + P(J) − P(H and J) + 4 52 = 16 52 = 4 13 − 1 52 ≈ .3077 If A and B are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) − P(A AND B) becomes P(A OR B) = P(A) + P(B). Draw one card from a standard deck of playing cards. Let H = the card is a heart and S = the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade is P(H or S) = P(H) + P(S) + 13 52 = 13 52 = 26 52 = 1 2 = .5 Example 3.14 Klaus is trying to choose where to go on vacation. His two choices are: A = New Zealand and B = Alaska. • Klaus can only afford one vacation. The probability that he chooses A is P(A) = .6 and the probability that he chooses B is P(B) = .35. • P(A AND B) = 0 because Klaus can only afford to take one vacation. • Therefore, the probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) = .6 + .35 = .95. Note that the probability that he does not choose to go anywhere on vacation must be .05. Example 3.15 Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game. A = the event Carlos is successful on his first attempt. P(A) = .65. B = the event Carlos is successful on his second attempt. P(B) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal given that he made the first goal is .90. a. What is the probability that he makes both goals? Solution 3.15 a. The problem is asking you to find P(A AND B) = P(B AND A). Since P(B\|A) = .90: P(B AND A) = P(B\|A) P(A) = (.90)(.65) = .585. Carlos makes the first and second goals with probability .585. b. What is the probability that Carlos makes either the first goal or the second goal? Solution 3.15 b. The problem is asking you to find P(A OR B). 200 Chapter 3 \| Probability Topics P(A OR B) = P(A) + P(B) − P(A AND B) = .65 + .65 − .585 = .715 Carlos makes either the first goal or the second goal with probability .715. c. Are A and B independent? Solution 3.15 c. No, they are not, because P(B AND A) = .585. P(B)P(A) = (.65)(.65) = .423 .423 ≠ .585 = P(B AND A) So, P(B AND A) is not equal to P(B)P(A). d. Are A and B mutually exclusive? Solution 3.15 d. No, they are not because P(A and B) = .585. To be mutually exclusive, P(A AND B) must equal zero. 3.15 Helen plays basketball. For free throws, she makes the shot 75 percent of the time. Helen must now attempt two free throws. C = the event that Helen makes the first shot. P(C) = .75. D = the event Helen makes the second shot. P(D) = .75. The probability that Helen makes the second free throw given that she made the first is .85. What is the probability that Helen makes both free throws? Example 3.16 A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Fortyseven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice four times a week. Thirty of the intermediate swimmers practice four times a week. Ten of the novice s
wimmers practice four times a week. Suppose one member of the swim team is chosen randomly. a. What is the probability that the member is a novice swimmer? Solution 3.16 a. There are 150 members; 75 of these are advanced, and 47 of these are intermediate swimmers. So there are 150 − 75 − 47 = 28 novice swimmers. The probability that a randomly selected swimmer is a novice is 28 150 . b. What is the probability that the member practices four times a week? Solution 3.16 b. 40 + 30 + 10 150 = 80 150 c. What is the probability that the member is an advanced swimmer and practices four times a week? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 201 Solution 3.16 c. There are 40 advanced swimmers who practice four times per week, so the probability is 40 150 . d. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and being an intermediate swimmer mutually exclusive? Why or why not? Solution 3.16 d. P(advanced AND intermediate) = 0, so these are mutually exclusive events. A swimmer cannot be an advanced swimmer and an intermediate swimmer at the same time. e. Are being a novice swimmer and practicing four times a week independent events? Why or why not? Solution 3.16 e. No, these are not independent events. P(novice AND practices four times per week) = .0667 P(novice)P(practices four times per week) = .0996 .0667 ≠ .0996 3.16 A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. The remainder are taking a gap year. Fifty of the seniors going to college are on their school's sports teams. Thirty of the seniors going directly to work are on their school's sports teams. Five of the seniors taking a gap year are on their schools sports teams. What is the probability that a senior is taking a gap year? Example 3.17 Felicity attends a school in Modesto, CA. The probability that Felicity enrolls in a math class is .2 and the probability that she enrolls in a speech class is .65. The probability that she enrolls in a math class GIVEN that she enrolls in speech class is .25. Let M = math class, S = speech class, and M\|S = math given speech. a. What is the probability that Felicity enrolls in math and speech? Find P(M AND S) = P(M\|S)P(S). b. What is the probability that Felicity enrolls in math or speech classes? Find P(M OR S) = P(M) + P(S) − P(M AND S). c. Are M and S independent? Is P(M\|S) = P(M)? d. Are M and S mutually exclusive? Is P(M AND S) = 0? Solution 3.17 a. P(M AND S) = P(M\|S)P(S) = .25(.65) = .1625 b. P(M OR S) = P(M) + P(S) − P(M AND S) = .2 + .65 − .1625 = .6875 c. No, P(M\|S) = .25 and P(M) = .2. d. No, P(M AND S) = .1625. 202 Chapter 3 \| Probability Topics 3.17 A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = .40, P(D) = .30, and P(D\|B) = .5. a. Find P(B AND D). b. Find P(B OR D). Example 3.18 Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative about 85 percent of the time. Let B = woman develops the disease and let N = tests negative. Suppose one woman is selected at random. a. What is the probability that the woman develops the disease? What is the probability that woman tests negative? Solution 3.18 a. P(B) = .143; P(N) = .85 b. Given that the woman develops the disease, what is the probability that she tests negative? Solution 3.18 b. Among women who develop the disease, the test is negative 2 percent of the time, so P(N\|B) = .02 c. What is the probability that the woman has the disease AND tests negative? Solution 3.18 c. P(B AND N) = P(B)P(N\|B) = (.143)(.02) = .0029 d. What is the probability that the woman has the disease OR tests negative? Solution 3.18 d. P(B OR N) = P(B) + P(N) − P(B AND N) = .143 + .85 − .0029 = .9901 e. Are having the disease and testing negative independent events? Solution 3.18 e. No. P(N) = .85; P(N\|B) = .02. So, P(N\|B) does not equal P(N). f. Are having the disease and testing negative mutually exclusive? Solution 3.18 f. No. P(B AND N) = .0029. For B and N to be mutually exclusive, P(B AND N) must be zero. 3.18 A school has 200 seniors of whom 140 will be going to college next year. Forty will be going directly to work. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 203 The remainder are taking a gap year. Fifty of the seniors going to college are on their school's sports teams. Thirty of the seniors going directly to work are on their school's sports teams. Five of the seniors taking a gap year are on their school's sports teams. What is the probability that a senior is going to college and plays sports? Example 3.19 Refer to the information in Example 3.18. P = tests positive. a. Given that a woman develops the disease, what is the probability that she tests positive? Find P(P\|B) = 1 − P(N\|B). b. What is the probability that a woman develops the disease and tests positive? Find P(B AND P) = P(P\|B)P(B). c. What is the probability that a woman does not develop the disease? Find P(B′) = 1 − P(B). d. What is the probability that a woman tests positive for the disease? Find P(P) = 1 − P(N). Solution 3.19 a. P(P\|B) = 1 − P(N\|B) = 1 − .02 = .98 b. P(B AND P) = P(P\|B)P(B) = .98(.143) = .1401 c. P(B') = 1 − P(B) = 1 − .143 = .857 d. P(P) = 1 − P(N) = 1 − .85 = .15 3.19 A student goes to the library. Let events B = the student checks out a book and D = the student checks out a DVD. Suppose that P(B) = .40, P(D) = .30, and P(D\|B) = .5. a. Find P(B′). b. Find P(D AND B). c. Find P(B\|D). d. Find P(D AND B′). e. Find P(D\|B′). 3.4 \| Contingency Tables A two-way table provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a contingency table. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. We used two-way tables in Chapters 1 and 2 to calculate marginal and conditional distributions. These tables organize data in a way that supports the calculation of relative frequency and, therefore, experimental (empirical) probability. Later on, we will use contingency tables again, but in another manner. Example 3.20 Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data: 204 Chapter 3 \| Probability Topics Speeding Violation in the Last Year No Speeding Violation in the Last Year Total Uses a cell phone while driving Does not use a cell phone while driving Total Table 3.3 25 45 70 280 405 685 305 450 755 The total number of people in the sample is 755. The row totals are 305 and 450. The column totals are 70 and 685. Notice that 305 + 450 = 755 and 70 + 685 = 755. Using the table, calculate the following probabilities: a. Find P(Person uses a cell phone while driving). b. Find P(Person had no violation in the last year). c. Find P(Person had no violation in the last year and uses a cell phone while driving). d. Find P(Person uses a cell phone while driving or person had no violation in the last year). e. Find P(Person uses a cell phone while driving given person had a violation in the last year). f. Find P(Person had no violation last year given person does not use a cell phone while driving). Solution 3.20 a. This is the same as the marginal distribution (Section 1.2). P⎛ ⎝Person uses a cell phone while driving⎞ ⎠ = number who use cell phones while driving number in study = 305 755 ≈ .4040 b. The marginal distribution is P⎛ ⎝Person had no violation in the last year⎞ ⎠ = number who had no violation number in study = 685 755 ≈ .9073. c. Find the number of participants who satisfy both conditions. P(Person had no violation in the last year AND uses a cell phone while driving) = number who had no violation AND uses cell phone while driving number in study = 280 755 ≈ .3709 d. To find this probability, you need to identify how many participants use a cell phone while driving OR have no violation in the past year OR both. P⎛ ⎝Person uses a cell phone while driving OR had no violation in the last year⎞ ⎠ = 25 + 405 + 280 755 = 710 755 ≈ .9404 e. This is a conditional probability. You are given that the person had no violation in the last year, so you need only consider the values in that column of data. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 205 (Person uses a cell phone while driving GIVEN the person had a violation in the last year) = number who used cell phone AND had a violation number in study who had a violation in the last year = 25 70 ≈ .3571 f. For this conditional probability, consider only values in the row labeled “Does not use a cell phone while driving.” P⎛ ⎝Person had no violation last year GIVEN person does not use cell phone while driving⎞ ⎠ = 405 450 = .9 3.20 Table 3.4 shows the number of athletes who stretch before exercising and how many had injuries within the past year. Injury in Past Year No Injury in Past Year Total Stretches 55 Does not stretch 231 Total 286 Table 3.4 295 219 514 350 450 800 a. What is P(Athlete stretches before exercising)? b. What is P(Athlete stretches before exercising\|no injury in the last year)? Example 3.21 Table 3.5 shows a random sample of 100 hikers and the areas of hiking they prefer. Sex The Coastline Near Lakes and Streams On Mountain Peaks Total Female 18 Male Total ___ ___ 16 ___ 41 Table 3.5 Hiking Area Preference
___ 14 ___ 45 55 ___ a. Complete the table. Solution 3.21 a. There are 45 females in the sample; 18 prefer the coastline and 16 prefer hiking near lakes and streams. So, we know there are 45 − 18 − 16 = 11 female students who prefer hiking on mountain peaks. Continue reasoning in this way to complete the table. 206 Chapter 3 \| Probability Topics Sex The Coastline Near Lakes and Streams On Mountain Peaks Total Female 18 Male Total 16 34 16 25 41 Table 3.6 Hiking Area Preference 11 14 25 45 55 100 b. Are the events being female and preferring the coastline independent events? Let F = being female and let C = preferring the coastline. 1. Find P(F AND C). 2. Find P(F)P(C). Are these two numbers the same? If they are, then F and C are independent. If they are not, then F and C are not independent. Solution 3.21 b. 1. P(F AND C) = 18 100 = .18 2. P(F)P(C) = ⎛ ⎝ 45 100 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 34 100 = (.45)(.34) = .153 P(F AND C) ≠ P(F)P(C), so the events F and C are not independent. c. Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let M = being male, and let L = prefers hiking near lakes and streams. 1. What word tells you this is a conditional? 2. Is the sample space for this problem all 100 hikers? If not, what is it? 3. Fill in the blanks and calculate the probability: P(_____\|_____) = _____. Solution 3.21 c. 1. The word given tells you that this is a conditional. 2. No, the sample space for this problem is the 41 hikers who prefer lakes and streams. 3. Find the conditional probability P(M\|L). Because it is given that the person prefers hiking near lakes and streams, you need only consider the values in the column labeled "Near Lakes and Streams." P(M\|L) = 25 41 d. Find the probability that a person is female or prefers hiking on mountain peaks. Let F = being female, and let P = prefers mountain peaks. 1. Find P(F). 2. Find P(P). 3. Find P(F AND P). 4. Find P(F OR P). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 207 Solution 3.21 d. 1. P(F) = 45 100 2. P(P) = 25 100 3. P(F AND P) = number of hikers that are both female AND prefers mountain peaks number of hikers in study = 11 100 4. P(F OR P) = P(F) + P(P) − P(F AND P) = 45 100 + 25 100 - 11 100 = 59 100 3.21 Table 3.7 shows a random sample of 200 cyclists and the routes they prefer. Let M = males and H = hilly path. Gender Lake Path Hilly Path Wooded Path Total Female Male Total 45 26 71 Table 3.7 38 52 90 27 12 39 110 90 200 a. Out of the males, what is the probability that the cyclist prefers a hilly path? b. Are the events being male and preferring the hilly path independent events? 208 Chapter 3 \| Probability Topics Example 3.22 Muddy Mouse lives in a cage with three doors. If Muddy goes out the first door, the probability that he gets caught by Alissa the cat is 1 5 and the probability he is not caught is 4 5 . If he goes out the second door, the probability he gets caught by Alissa is 1 4 and the probability he is not caught is 3 4 . The probability that Alissa catches Muddy coming out of the third door is 1 2 and the probability she does not catch Muddy is 1 2 Muddy will choose any of the three doors, so the probability of choosing each door is 1 3 . It is equally likely that . Caught or Not Door One Door Two Door Three Total Caught Not Caught 1 15 4 15 1 12 3 12 1 6 1 6 Total ____ ____ ____ ____ ____ 1 Table 3.8 Door Choice • The first entry 1 15 = ⎛ ⎝ is P(Door One AND Caught). • The entry 4 15 = ⎛ ⎝ Verify the remaining entries. is P(Door One AND Not Caught). a. Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1. Solution 3.22 a. Caught or Not Door One Door Two Door Three Total Caught Not Caught Total 1 15 4 15 5 15 Table 3.9 Door Choice 1 12 3 12 4 12 1 6 1 6 2 6 19 60 41 60 1 b. What is the probability that Alissa does not catch Muddy? Solution 3.22 b. 41 60 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 209 c. What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa? Solution 3.22 c. This is a conditional probability, so consider only probabilities in the row labeled "Caught." Choosing Door One and choosing Door Two are mutually exclusive, so P⎛ ⎝Choosing Door One OR Choosing Door Two AND Caught ⎞ ⎠ = 1 15 + 1 12 = 9 60 . Use the formula for conditional probability P(A\|B) = P(AANDB) P(B) . P⎛ ⎝Door One OR Door Two\|Caught ⎞ ⎠ = P⎛ ⎝Door One OR Door Two AND Caught ⎞ ⎝Caught⎞ P⎛ ⎠ ⎠ = 9 60 19 60 = 9 19. . Example 3.23 Table 3.10 contains the number of crimes per 100,000 inhabitants from 2008 to 2011 in the United States. Year Crime A Crime B Crime C Crime D Total 2008 145.7 2009 133.1 2010 119.3 2011 113.7 Total 732.1 717.7 701 702.2 29.7 29.1 27.7 26.8 314.7 259.2 239.1 229.6 Table 3.10 U.S. Crime Index Rates Per 100,000 Inhabitants 2008–2011 TOTAL each column and each row. Total data = 4,520.7. a. Find P(2009 AND Crime A). b. Find P(2010 AND Crime B). c. Find P(2010 OR Crime B). d. Find P(2011\|Crime A). e. Find P(Crime D\|2008). Solution 3.23 a. 133.1 4,520.7 B) = 1,087.1 4,520.7 = .0294, b. 701 4,520.7 = .1551, c. P(2010 OR Crime B) = P(2010) + P(Crime B) – P(2010 AND Crime + 2,852.9 4,520.7 − 701 4,520.7 = .7165, d. 113.7 511.8 = .2222, e. 314.7 1,222.2 = .2575 3.23 Table 3.11 relates the weights and heights of a group of individuals participating in an observational study. 210 Chapter 3 \| Probability Topics Ages Tall Medium Short Totals 28 51 25 14 28 9 Under 18 18 20 12 18–50 51+ Totals Table 3.11 a. Find the total for each row and column. b. Find the probability that a randomly chosen individual from this group is tall. c. Find the probability that a randomly chosen individual from this group is Under 18 and tall. d. Find the probability that a randomly chosen individual from this group is tall given that the individual is Under 18. e. Find the probability that a randomly chosen individual from this group is Under 18 given that the individual is tall. f. Find the probability a randomly chosen individual from this group is tall and age 51+. g. Are the events under 18 and tall independent? 3.5 \| Tree and Venn Diagrams Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities. Tree Diagrams A tree diagram is a special type of graph used to determine the outcomes of an experiment. It consists of branches that are labeled with either frequencies or probabilities. Tree diagrams can make some probability problems easier to visualize and solve. The following example illustrates how to use a tree diagram: Example 3.24 In an urn, there are 11 balls. Three balls are red (R) and eight balls are blue (B). Draw two balls, one at a time, with replacement. With replacement means that you put the first ball back in the urn before you select the second ball. Therefore, you are selecting from exactly the same group each time, so each draw is independent. The tree diagram shows all the possible outcomes. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 211 Figure 3.10 Total = 64 + 24 + 24 + 9 = 121. The first set of branches represents the first draw. There are 8 ways to draw a blue marble and 3 ways to draw a red one. The second set of branches represents the second draw. Regardless of the choice on the first draw, there are again eight ways to draw a blue marble and 3 ways to draw a red one. Read down each branch to see the total number of possible outcomes. For example, there are 8 ways to get a blue marble on the first draw, and eight ways to get one on the second draw, so there are 8 × 8 = 64 different ways to draw two blue marbles in succession. Each of the outcomes is distinct. In fact, we can list each red ball as R1, R2, and R3 and each blue ball as B1, B2, B3, B4, B5, B6, B7, and B8. Then the nine RR outcomes can be written as follows: R1R1, R1R2, R1R3, R2R1, R2R2, R2R3, R3R1, R3R2, R3R3. The other outcomes are similar. There are a total of 11 balls in the urn. Draw two balls, one at a time, with replacement. There are 11(11) = 121 outcomes, the size of the sample space. a. List the 24 BR outcomes: B1R1, B1R2, B1R3, . . . Solution 3.24 a. We know that there will be 24 different possible outcomes because there are eight ways to draw blue and three ways to draw red. Make a systematic list of possible outcomes that consist of a blue marble on the first draw and a red marble on the second draw. B1R1, B1R2, B1R3 B2R1, B2R2, B2R3 B3R1, B3R2, B3R3 B4R1, B4R2, B4R3 B5R1, B5R2, B5R3 B6R1, B6R2, B6R3 B7R1, B7R2, B7R3 B8R1, B8R2, B8R3 b. Calculate P(RR). Solution 3.24 b. You can use the tree diagram. There are nine ways to draw two reds and 121 possible outcomes. So, P(RR) = 9 121. . Each draw is independent, so you can also use the formula: P(RR) = P(R)P(R) = ⎛ ⎝ 3 11 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 3 11 = 9 121. 212 Chapter 3 \| Probability Topics c. Calculate P(RB OR BR). Solution 3.24 c. The tree diagram shows that there are 24 ways to draw RB and 24 ways to draw BR. There are 121 possible outcomes, so P(RB or BR) = 24 + 24 . = 48 121 121 The events RB and BR are mutually exclusive, so P(RB OR BR) = P(RB) + P(BR) = P(R)P(B) + P(B)P(R 11 8 11 8 11 3 11 = 48 121. d. Using the tree diagram, calculate P(R on 1st draw AND B on 2nd draw). Solution 3.24 d. Follow the path on the tree. There are three ways to get a red marble on the first draw and eight ways to get a blue on the second draw. There are 3 × 8 = 24 ways to draw red then blue, so P(RB) = 24 121 . Can you think of another way to find this probability? P(R on 1st draw AND B on 2nd draw) = P(RB) = ⎛ ⎝ 3 11 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 8 11 = 24 121 e. Using the tree diagram, calculate P(R on 2nd draw GIVEN B on 1st draw). Solution 3.24 e. Given tha
t a blue marble is selected first, we need only follow the left set of branches on the tree diagram. In this case, there are three ways to obtain red on the second draw and 11 possible outcomes. Figure 3.11 P(R on 2nd draw GIVEN B on 1st) = P(R on 2nd \| B on 1st) = 3 11 You can also use the formula This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 213 P(R on 2nd \| B on 1st) = P(R on 2nd AND B on 1st) P(B on 1st) = 24 121 64 + 24 121 = 24 88 = 3 11. f. Using the tree diagram, calculate P(BB). Solution 3.24 f. P(BB) = 64 121 g. Using the tree diagram, calculate P(B on the 2nd draw GIVEN R on the first draw). Solution 3.24 g. P(B on 2nd draw\|R on 1st draw) = 8 11 There are 9 + 24 outcomes that have R on the first draw (9 RR and 24 RB). The sample space is then 9 + 24 = 33. Twenty-four of the 33 outcomes have B on the second draw. The probability is then 24 33 . 3.24 In a standard deck, there are 52 cards. Twelve cards are face cards (event F) and 40 cards are not face cards (event N). Draw two cards, one at a time, with replacement. All possible outcomes are shown in the tree diagram as frequencies. Using the tree diagram, calculate P(FF). Figure 3.12 Example 3.25 An urn has three red marbles and eight blue marbles in it. Draw two marbles, one at a time, this time without replacement, from the urn. Without replacement means that you do not put the first ball back before you select the second marble. Following is a tree diagram for this situation. The branches are labeled with probabilities instead of frequencies. The numbers at the ends of the branches are calculated by multiplying the numbers on the two ⎞ corresponding branches, for example, P(RR) = ⎛ ⎠ = 6 ⎝ 110 3 11 2 10 ⎛ ⎞ ⎠ ⎝ . 214 Chapter 3 \| Probability Topics Figure 3.13 Total = 56 + 24 + 24 + 6 110 = 110 110 = 1. NOTE If you draw a red on the first draw from the three red possibilities, there are two red marbles left to draw on the second draw. You do not put back or replace the first marble after you have drawn it. You draw without replacement, so that on the second draw there are 10 marbles left in the urn. Calculate the following probabilities using the tree diagram: a. P(RR) = ________ Solution 3.25 a. P(RR) = ⎛ 3 ⎝ 11 ⎛ ⎞ ⎠ ⎝ 2 10 ⎞ ⎠ = 6 110 b. Fill in the blanks. P(RB OR BR) = ⎛ ⎝ 3 11 ⎛ ⎞ ⎠ ⎝ 8 10 ⎞ ⎠ + (________)(________) = 48 110 Solution 3.25 b. P(RB OR BR) = P(RB) + P(BR) = P(R on 1st) P(B on 2nd) + P(B on 1st) P(R on 2nd) = ⎛ ⎝ 3 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 8 10 + ⎛ ⎝ 8 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 3 10 = 48 110 c. Because this is a conditional probability, we restrict the sample space to consider only those outcomes that have a blue marble in the first draw. Look at the second level of the tree to see that P(R on 2nd\|B on 1st) = 3 10 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 215 Solution 3.25 c. P(R on 2nd\|B on 1st) = 3 10 d. Fill in the blanks. P(R on 1st AND B on 2nd) = P(RB) = (________)(________) = 24 100 Solution 3.25 d. P(R on 1st AND B on 2nd) = P(RB) = ⎛ ⎝ 3 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 8 10 = 24 100 e. Find P(BB). Solution 3.25 e. P(BB) = ⎛ 8 ⎝ 11 ⎛ ⎞ ⎠ ⎝ ⎞ ⎠ 7 10 f. Find P(B on 2nd\|R on 1st). Solution 3.25 f. Using the tree diagram, P(B on 2nd\|R on 1st) = P(R\|B) = 8 10 . If we are using probabilities, we can label the tree in the following general way: • P(R\|R) here means P(R on 2nd\|R on 1st) • P(B\|R) here means P(B on 2nd\|R on 1st) • P(R\|B) here means P(R on 2nd\|B on 1st) • P(B\|B) here means P(B on 2nd\|B on 1st) 3.25 In a standard deck, there are 52 cards. Twelve cards are face cards (F) and 40 cards are not face cards (N). Draw two cards, one at a time, without replacement. The tree diagram is labeled with all possible probabilities. 216 Chapter 3 \| Probability Topics Figure 3.14 a. Find P(FN OR NF). b. Find P(N\|F). c. Find P(at most one face card). Hint: At most one face card means zero or one face card. d. Find P(at least one face card). Hint: At least one face card means one or two face cards. Example 3.26 A litter of kittens available for adoption at the Humane Society has four tabby kittens and five black kittens. A family comes in and randomly selects two kittens (without replacement) for adoption. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 217 a. Which shows the probability that both kittens are tabby? ⎛ a. What is the probability that one kitten of each coloring is selected? ⎛ a. What is the probability that a tabby is chosen as the second kitten when a black kitten was chosen as the first? d. What is the probability of choosing two kittens of the same color? Solution 3.26 ⎠ , b. ⎛ a. 4 8 , d. 32 72 3.26 Suppose there are four red balls and three yellow balls in a box. Three balls are drawn from the box without replacement. What is the probability that one ball of each coloring is selected? Venn Diagram A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. Example 3.27 Suppose an experiment has the outcomes 1, 2, 3, . . . , 12 where each outcome has an equal chance of occurring. Let event A = {1, 2, 3, 4, 5, 6} and event B = {6, 7, 8, 9}. Then A AND B = {6} and A OR B = {1, 2, 3, 4, 5, 6, 7, 218 Chapter 3 \| Probability Topics 8, 9}. The Venn diagram is as follows: Figure 3.15 3.27 Suppose an experiment has outcomes black, white, red, orange, yellow, green, blue, and purple, where each outcome has an equal chance of occurring. Let event C = {green, blue, purple} and event P = {red, yellow, blue}. Then C AND P = {blue} and C OR P = {green, blue, purple, red, yellow}. Draw a Venn diagram representing this situation. Example 3.28 Flip two fair coins. Let A = tails on the first coin. Let B = tails on the second coin. Then A = {TT, TH} and B = {TT, HT}. Therefore, A AND B = {TT}. A OR B = {TH, TT, HT}. The sample space when you flip two fair coins is X = {HH, HT, TH, TT}. The outcome HH is in NEITHER A NOR B. The Venn diagram is as follows: Figure 3.16 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 219 3.28 Roll a fair, six-sided die. Let A = a prime number of dots is rolled. Let B = an odd number of dots is rolled. Then A = {2, 3, 5} and B = {1, 3, 5}. Therefore, A AND B = {3, 5}. A OR B = {1, 2, 3, 5}. The sample space for rolling a fair die is S = {1, 2, 3, 4, 5, 6}. Draw a Venn diagram representing this situation. Example 3.29 Forty percent of the students at a local college belong to a club and 50 percent work part time. Five percent of the students work part time and belong to a club. Draw a Venn diagram showing the relationships. Let C = student belongs to a club and PT = student works part time. Start by drawing a rectangle to represent the sample space. Then draw two circles or ovals inside the rectangle to represent the events of interest: belonging to a club (C) and working part time (PT). Always draw overlapping shapes to represent outcomes that are in both events. Figure 3.17 Label each piece of the diagram clearly and note the probability or frequency of each part. Start by labeling the overlapping section first. Note that the probabilities in C total 0.40 and the sum of the probabilities in PT is 0.50. The total of all probabilities displayed must be 1, representing 100 percent of the sample space. If a student is selected at random, find the following: a. b. c. d. e. the probability that the student belongs to a club. the probability that the student works part time. the probability that the student belongs to a club AND works part time. the probability that the student belongs to a club given that the student works part time. the probability that the student belongs to a club OR works part time. Solution 3.29 P(C) = .40 220 Chapter 3 \| Probability Topics P(PT) = .50 P(C AND PT) = .05 P(C\|PT) = P(C AND PT) P(PT) = .05 .50 = .1 P(C OR PT) = P(C) + P(PT) − P(C AND PT) = .40 + .50 − .05 = .85 3.29 Fifty percent of the workers at a factory work a second job, 25 percent have a spouse who also works, and 5 percent work a second job and have a spouse who also works. Draw a Venn diagram showing the relationships. Let W = works a second job and S = spouse also works. Example 3.30 A person with type O blood and a negative Rh factor (Rh–) can donate blood to any person with any blood type. Four percent of African Americans have type O blood and a negative Rh factor, 5−10 percent of African Americans have the Rh– factor, and 51 percent have type O blood. Figure 3.18 The “O” circle represents the African Americans with type O blood. The “Rh––" oval represents the African Americans with the Rh– –factor. We will use the average of 5 percent and 10 percent, 7.5 percent, as the percentage of African Americans who have the Rh–– factor. Let O = African American with Type O blood and R = African American with Rh– –factor. a. P(O) = ___________ b. P(R) = ___________ c. P(O AND R) = ___________ This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 221 d. P(O OR R) = ____________ e. f. In the Venn Diagram, describe the overlapping area using a complete sentence. In the Venn Diagram, describe the area in the rectangle but outside both the circle and the oval using a complete sentence. Solution 3.30 a. P(O) = .51 b. P(R) = .075 because an average of 7.5 percent of African Americans have the Rh– –factor. c. P(O AND R) = 0.04 because 4 percent of African Americans have both Type O blood and the Rh– –factor. d. P(O OR R) = P(O) + P(R) - P(O AND R) = .51 + .075 − .04 = .545 e. The area represents the African Americans that have type O blood and the Rh–– factor. f. The area represents the African Americans that have neither type O blood nor the Rh–– factor. 3.30 In a bookstore, the probability that the custome
r buys a novel is .6, and the probability that the customer buys a nonfiction book is .4. Suppose that the probability that the customer buys both is .2. a. Draw a Venn diagram representing the situation. b. Find the probability that the customer buys either a novel or a nonfiction book. c. In the Venn diagram, describe the overlapping area using a complete sentence. d. Suppose that some customers buy only compact disks. Draw an oval in your Venn diagram representing this event. 3.6 \| Probability Topics 222 Chapter 3 \| Probability Topics 3.1 Probability Topics Student Learning Outcomes • The student will use theoretical and empirical methods to estimate probabilities. • The student will appraise the differences between the two estimates. • The student will demonstrate an understanding of long-term relative frequencies. Do the Experiment Count out 40 mixed-color candies, which is approximately one small bag’s worth. Record the number of each color in Table 3.12. Use the information from this table to complete Table 3.13. Next, put the candies in a cup. The experiment is to pick two candies, one at a time. Do not look into the cup as you pick them. The first time through, replace the first candy before picking the second one. Record the results in the With Replacement column of Table 3.14. Do this 24 times. The second time through, after picking the first candy, do not replace it before picking the second one. Then, pick the second one. Record the results in the Without Replacement column section of Table 3.15. After you record the pick, put both candies back. Do this a total of 24 times, also. Use the data from Table 3.15 to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do not multiply out any fractions. Color Quantity Yellow (Y) Green (G) Blue (BL) Brown (B) Orange (O) Red (R) Table 3.12 Population With Replacement Without Replacement P(2 reds) P(R1B2 OR B1R2) P(R1 AND G2) P(G2\|R1) P(no yellows) P(doubles) P(no doubles) Table 3.13 Theoretical Probabilities This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 223 NOTE G2 = green on second pick, R1 = red on first pick, B1 = brown on first pick, B2 = brown on second pick, doubles = both picks are the same color. With Replacement Without Replacement ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ ) Table 3.14 Empirical Results With Replacement Without Replacement P(2 reds) P(R1B2 OR B1R2) P(R1 AND G2) P(G2\|R1) P(no yellows) P(doubles) P(no doubles) Table 3.15 Empirical Probabilities Discussion Questions 1. Why are the With Replacement and Without Replacement probabilities different? 2. Convert P(no yellows) to decimal format for both Theoretical With Replacement and for Empirical With Replacement. Round to four decimal places. a. Theoretical With Replacement: P(no yellows) = _______ b. Empirical With Replacement: P(no yellows) = _______ c. Are the decimal values close? Did you expect them to be closer together or farther apart? Why? 3. If you increased the number of times you picked two candies to 240 times, why would empirical probability values change? 224 Chapter 3 \| Probability Topics 4. Would this change (see Question 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know? 5. Explain the differences in what P(G1 AND R2) and P(R1\|G2) represent. Hint: Think about the sample space for each probability. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 225 KEY TERMS conditional probability the likelihood that an event will occur given that another event has already occurred contingency table the method of displaying a frequency distribution as a table with rows and columns to show how two variables may be dependent (contingent) upon each other; the table provides an easy way to calculate conditional probabilities dependent events if two events are NOT independent, then we say that they are dependent equally likely each outcome of an experiment has the same probability event a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by S. An event is an arbitrary subset in S. It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as A, B, C, and so on experiment a planned activity carried out under controlled conditions independent events The occurrence of one event has no effect on the probability of the occurrence of another event; events A and B are independent if one of the following is true: 1. P(A\|B) = P(A) 2. P(B\|A) = P(B) 3. P(A AND B) = P(A)P(B) mutually exclusive two events are mutually exclusive if the probability that they both happen at the same time is zero; if events A and B are mutually exclusive, then P(A AND B) = 0 outcome a particular result of an experiment probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following three axioms (by A.N. Kolmogorov, 1930s): Let S denote the sample space and A and B are two events in S; then • 0 ≤ P(A) ≤ 1, • If A and B are any two mutually exclusive events, then P(A OR B) = P(A) + P(B), and • P(S) = 1 sample space the set of all possible outcomes of an experiment sampling with replacement if each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once sampling without replacement when sampling is done without replacement, each member of a population may be chosen only once the AND event an outcome is in the event A AND B if the outcome is in both A AND B at the same time the complement event the complement of event A consists of all outcomes that are NOT in A the conditional probability of one event GIVEN another event P(A\|B) is the probability that event A will occur given that the event B has already occurred the OR event an outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B the OR of two events an outcome is in the event A OR B if the outcome is in A, is in B, or is in both A and B tree diagram the useful visual representation of a sample space and events in the form of a tree with branches marked by possible outcomes together with associated probabilities (frequencies, relative frequencies) Venn diagram the visual representation of a sample space and events in the form of circles or ovals showing their 226 Chapter 3 \| Probability Topics intersections CHAPTER REVIEW 3.1 Terminology In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive. 3.2 Independent and Mutually Exclusive Events Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other. 3.3 Two Basic Rules of Probability The multiplication rule and the addition rule are used for computing the probability of A and B, as well as the probability of A or B for two given events A, B defined on the sample space. In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered to be not independent. The events A and B are mutually exclusive events when they do not have any outcomes in common. 3.4 Contingency Tables There are several tools you can use to help organize and sort data when calculating probabilities. Contingency tables, also known as two-way tables, help display data and are particularly useful when calculating probabilites that have multiple dependent variables. 3.5 Tree and Venn Diagrams A tree diagram uses branches to show the different outcomes of experiments and makes complex probability questions easy to visualize. A Venn diagram is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events. A Venn diagram is especially helpful for visualizing the OR event, the AND event, and the complement of an event and for understanding conditional probabilities. FORMULA REVIEW 3.1 Terminology A and B are events P(S) = 1 where S is the sample space 0 ≤ P(A) ≤ 1 P(A\|B) = P(AANDB) P(B) 3.2 Independent and Mutually Exclusive Events If A and B are independent, P(A AND B) = P(A)P(B), This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 P(A\|B) = P(A), and P(B\|
A) = P(B). If A and B are mutually exclusive, P(A OR B) = P(A) + P(B) and P(A AND B) = 0. 3.3 Two Basic Rules of Probability The multiplication rule—P(A AND B) = P(A\|B)P(B) The addition rule—P(A OR B) = P(A) + P(B) − P(A AND B) Chapter 3 \| Probability Topics 227 PRACTICE 3.1 Terminology 1. In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts A through J of this question. Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols. • Let F be the event that a student is female. • Let M be the event that a student is male. • Let S be the event that a student has short hair. • Let L be the event that a student has long hair. a. The probability that a student does not have long hair. b. The probability that a student is male or has short hair. c. The probability that a student is female and has long hair. d. The probability that a student is male, given that the student has long hair. e. The probability that a student has long hair, given that the student is male. f. Of all female students, the probability that a student has short hair. g. Of all students with long hair, the probability that a student is female. h. The probability that a student is female or has long hair. i. The probability that a randomly selected student is a male student with short hair. j. The probability that a student is female. Use the following information to answer the next four exercises. A box is filled with several party favors. It contains 12 hats, 15 noisemakers, 10 finger traps, and five bags of confetti. Let H = the event of getting a hat. Let N = the event of getting a noisemaker. Let F = the event of getting a finger trap. Let C = the event of getting a bag of confetti. 2. Find P(H). 3. Find P(N). 4. Find P(F). 5. Find P(C). Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange. Let B = the event of getting a blue jelly bean Let G = the event of getting a green jelly bean. Let O = the event of getting an orange jelly bean. Let P = the event of getting a purple jelly bean. Let R = the event of getting a red jelly bean. Let Y = the event of getting a yellow jelly bean. 6. Find P(B). 7. Find P(G). 8. Find P(P). 9. Find P(R). 10. Find P(Y). 11. Find P(O). Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 countries in Oceania (Pacific Ocean region). Let A = the event that a country is in Asia. 228 Chapter 3 \| Probability Topics Let E = the event that a country is in Europe. Let F = the event that a country is in Africa. Let N = the event that a country is in North America. Let O = the event that a country is in Oceania. Let S = the event that a country is in South America. 12. Find P(A). 13. Find P(E). 14. Find P(F). 15. Find P(N). 16. Find P(O). 17. Find P(S). 18. What is the probability of drawing a red card in a standard deck of 52 cards? 19. What is the probability of drawing a club in a standard deck of 52 cards? 20. What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six? 21. What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six? Use the following information to answer the next two exercises. You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area. Figure 3.19 Let B = the event of landing on blue. Let R = the event of landing on red. Let G = the event of landing on green. Let Y = the event of landing on yellow. 22. If you land on Y, you get the biggest prize. Find P(Y). 23. If you land on red, you don’t get a prize. What is P(R)? Use the following information to answer the next 10 exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters. Let I = the event that a player in an infielder. Let O = the event that a player is an outfielder. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 229 Let H = the event that a player is a great hitter. Let N = the event that a player is not a great hitter. 24. Write the symbols for the probability that a player is not an outfielder. 25. Write the symbols for the probability that a player is an outfielder or is a great hitter. 26. Write the symbols for the probability that a player is an infielder and is not a great hitter. 27. Write the symbols for the probability that a player is a great hitter, given that the player is an infielder. 28. Write the symbols for the probability that a player is an infielder, given that the player is a great hitter. 29. Write the symbols for the probability that of all the outfielders, a player is not a great hitter. 30. Write the symbols for the probability that of all the great hitters, a player is an outfielder. 31. Write the symbols for the probability that a player is an infielder or is not a great hitter. 32. Write the symbols for the probability that a player is an outfielder and is a great hitter. 33. Write the symbols for the probability that a player is an infielder. 34. What is the word for the set of all possible outcomes? 35. What is conditional probability? 36. A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book Let F = event that book is fiction Let N = event that book is nonfiction What is the sample space? 37. What is the sum of the probabilities of an event and its complement? Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let E = the event that it lands on an even number. Let M = the event that it lands on a multiple of three. 38. What does P(E\|M) mean in words? 39. What does P(E OR M) mean in words? 3.2 Independent and Mutually Exclusive Events 40. E and F are mutually exclusive events. P(E) = .4; P(F) = .5. Find P(E∣F). 41. J and K are independent events. P(J\|K) = .3. Find P(J). 42. U and V are mutually exclusive events. P(U) = .26; P(V) = .37. Find the following: a. P(U AND V) = b. P(U\|V) = c. P(U OR V) = 43. Q and R are independent events. P(Q) = .4 and P(Q AND R) = .1. Find P(R). 3.3 Two Basic Rules of Probability Use the following information to answer the next 10 exercises. Forty-eight percent of all voters of a certain state prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Among Latino registered voters in this state, 55 percent prefer life in prison without parole over the death penalty for a person convicted of first-degree murder. Of all citizens in this state, 37.6 percent are Latino. In this problem, let • C = citizens of a certain state (registered voters) preferring life in prison without parole over the death penalty for a person convicted of first-degree murder. • L = registered voters of the state who are Latino. Suppose that one citizen is randomly selected. 44. Find P(C). Chapter 3 \| Probability Topics 230 45. Find P(L). 46. Find P(C\|L). 47. In words, what is C\|L? 48. Find P(L AND C). 49. In words, what is L AND C? 50. Are L and C independent events? Show why or why not. 51. Find P(L OR C). 52. In words, what is L OR C? 53. Are L and C mutually exclusive events? Show why or why not. 3.4 Contingency Tables Use the following information to answer the next four exercises. Table 3.16 shows a random sample of musicians and how they learned to play their instruments. Gender Self-Taught Studied in School Private Instruction Total Female Male Total 12 19 31 Table 3.16 38 24 62 22 15 37 72 58 130 54. Find P(musician is a female). 55. Find P(musician is a male AND had private instruction). 56. Find P(musician is a female OR is self taught). 57. Are the events being a female musician and learning music in school mutually exclusive events? 3.5 Tree and Venn Diagrams 58. The probability that a man develops some form of cancer in his lifetime is 0.4567. The probability that a man has at least one false-positive test result, meaning the test comes back for cancer when the man does not have it, is .51. Let C = a man develops cancer in his lifetime; P = a man has at least one false-positive test. Construct a tree diagram of the situation. BRINGING IT TOGETHER: PRACTICE Use the following information to answer the next seven exercises. An article in the New England Journal of Medicine, reported about a study of people who use a product in California and Hawaii. In one part of the report, the self-reported ethnicity and using the product levels per day were given. Of the people using the product at most 10 times a day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans, and 7,650 whites. Of the people using the product 11 to 20 times per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people using the product 21 to 30 times per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 Whites. Of the people using the product at least 31 times per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 231 59. Complete the table using the data
provided. Suppose that one person from the study is randomly selected. Find the probability that person used the product 11 to 20 times a day. Product Use (times per day) African Americans Native Hawaiians Latinos Japanese Americans Whites TOTALS 1–10 11–20 21–30 31+ TOTALS Table 3.17 Product Use by Ethnicity 60. Suppose that one person from the study is randomly selected. Find the probability that the person used the product 11 to 20 times per day. 61. Find the probability that the person was Latino. 62. In words, explain what it means to pick one person from the study who is Japanese American AND uses the product 21 to 30 times per day. Also, find the probability. 63. In words, explain what it means to pick one person from the study who is Japanese American OR uses the product 21 to 30 times per day. Also, find the probability. 64. In words, explain what it means to pick one person from the study who is Japanese American GIVEN that the person uses the product 21 to 30 times per day. Also, find the probability. 65. Prove that product use/day and ethnicity are dependent events. HOMEWORK 232 Chapter 3 \| Probability Topics 3.1 Terminology 66. Figure 3.20 The graph in Figure 3.20 displays the sample sizes and percentages of people in different age and gender groups who were polled concerning their approval of Mayor Ford’s actions in office. The total number in the sample of all the age groups is 1,045. a. Define three events in the graph. b. Describe in words what the entry 40 means. c. Describe in words the complement of the entry in the previous question. d. Describe in words what the entry 30 means. e. Out of the males and females, what percent are males? f. Out of the females, what percent disapprove of Mayor Ford? g. Out of all the age groups, what percent approve of Mayor Ford? h. Find P(Approve\|Male). i. Out of the age groups, what percent are more than 44 years old? j. Find P(Approve\|Age < 35). 67. Explain what is wrong with the following statements. Use complete sentences. a. If there is a 60 percent chance of rain on Saturday and a 70 percent chance of rain on Sunday, then there is a 130 percent chance of rain over the weekend. b. The probability that a baseball player hits a home run is greater than the probability that he gets a successful hit. 3.2 Independent and Mutually Exclusive Events Use the following information to answer the next 12 exercises. The graph shown is based on more than 170,000 interviews that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Health Index Scores are the sample space. We randomly sample one type of Health Index Score, the emotional wellbeing score. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 233 Figure 3.21 68. Find the probability that a Health Index Score is 82.7. 69. Find the probability that a Health Index Score is 81.0. 70. Find the probability that a Health Index Score is more than 81. 71. Find the probability that a Health Index Score is between 80.5 and 82. 72. If we know a Health Index Score is 81.5 or more, what is the probability that it is 82.7? 73. What is the probability that a Health Index Score is 80.7 or 82.7? 74. What is the probability that a Health Index Score is less than 80.2 given that it is already less than 81? 75. What occupation has the highest Health Index Score? 76. What occupation has the lowest emotional index score? 77. What is the range of the data? 78. Compute the average Health Index Score. 79. If all occupations are equally likely for a certain individual, what is the probability that he or she will have an occupation with lower than average Health Index Score? 234 Chapter 3 \| Probability Topics 3.3 Two Basic Rules of Probability 80. On February 28, 2013, a Field Poll Survey reported that 61 percent of California registered voters approved of a law that was about to be passed. Among 18- to 39-year olds (California registered voters), the approval rating was 78 percent. Six in 10 California registered voters said that the upcoming Supreme Court’s ruling about the constitutionality of the law was either very or somewhat important to them. Out of those registered voters who supported the law, 75 percent say the ruling is important to them. In this problem, let • C = California registered voters who supported the law, • B = California registered voters who say the Supreme Court’s ruling about the law is very or somewhat important to them, and • A = California registered voters who are 18 to 39 years old. a. Find P(C). b. Find P(B). c. Find P(C\|A). d. Find P(B\|C). e. f. g. Find P(C AND B). h. i. Find P(C OR B). j. Are C and B mutually exclusive events? Show why or why not. In words, what is C\|A? In words, what is B\|C? In words, what is C AND B? 81. After a mayor of a major Canadian city announced his plans to cut budget costs in late 2011, researchers polled 1,046 people to measure the mayor’s popularity. Everyone polled expressed either approval or disapproval. These are the results their poll produced: • • • In early 2011, 60 percent of the population approved of the mayor's actions in office. In mid-2011, 57 percent of the population approved of his actions. In late 2011, the percentage of popular approval was measured at 42 percent. a. What is the sample size for this study? b. What proportion in the poll disapproved of the mayor, according to the results from late 2011? c. How many people polled responded that they approved of the mayor in late 2011? d. What is the probability that a person supported the mayor, based on the data collected in mid-2011? e. What is the probability that a person supported the mayor, based on the data collected in early 2011? Use the following information to answer the next three exercises. A local restaurant sells pork chops and chicken breasts. The given values below are the weights (in ounces) of pork chops and chicken breasts listed on the menu. Your server will randomly select one piece of meat (pork chop or chicken breast) that you will be served. 17 20 21 18 20 20 20 18 19 19 20 19 21 20 18 20 20 19 18 19 17 19 17 21 17 21 18 21 19 21 20 17 20 18 19 20 20 17 21 20 Pork Chops Chicken Breasts Table 3.18 82. a. List the sample space of the possible items that are on the menu. b. Find P(you will get a 17-oz. piece of meat). c. Find P(you will get a pork chop). d. Find P(you will get a 17-oz. pork chop). e. f. Find two mutually exclusive events. g. Are the events getting 17 oz. of meat and getting a pork chop independent? Is getting a pork chop the complement of getting a chicken breast? Why? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 235 83. Compute the probabilities. a. P(you will get a chicken breast) b. P(you will get a 17-oz. chicken breast) c. P(you will get a chicken breast or you will not get a 17-oz. pork chop) d. P(you will not get a chicken breast and you will get an 18-oz. pork chop) e. P(you will get a piece of meat that is not 21 oz.) f. P(you will get a piece of chicken that is not 21 oz.) g. P(you will not get a chicken breast and you will not get a pork chop) 84. Compute the probabilities: a. P(you will not get a pork chop) b. P(you will get a 20-oz. pork chop) c. P(you will not get a chicken breast or you will not get an 18-oz. pork chop) d. P(you will not get a chicken breast and you will not get an 18-oz. pork chop) e. P(you will get a pork chop that is not 21 oz.) f. P(you will not get a chicken breast or you will not get a pork chop) 85. Suppose that you have eight cards. Five are green and three are yellow. The five green cards are numbered 1, 2, 3, 4, and 5. The three yellow cards are numbered 1, 2, and 3. The cards are well shuffled. You randomly draw one card. • G = card drawn is green • E = card drawn is even-numbered a. List the sample space. b. P(G) = ________ c. P(G\|E) = ________ d. P(G AND E) = ________ e. P(G OR E) = ________ f. Are G and E mutually exclusive? Justify your answer numerically. 86. Roll two fair dice separately. Each die has six faces. a. List the sample space. b. Let A be the event that either a three or four is rolled first, followed by an even number. Find P(A). c. Let B be the event that the sum of the two rolls is at most seven. Find P(B). d. e. Are A and B mutually exclusive events? Explain your answer in one to three complete sentences, including In words, explain what P(A\|B) represents. Find P(A\|B). numerical justification. f. Are A and B independent events? Explain your answer in one to three complete sentences, including numerical justification. 87. A special deck of cards has 10 cards. Four are green, three are blue, and three are red. When a card is picked, its color is recorded. An experiment consists of first picking a card and then tossing a coin. a. List the sample space. b. Let A be the event that a blue card is picked first, followed by landing a head on the coin toss. Find P(A). c. Let B be the event that a red or green is picked, followed by landing a head on the coin toss. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification. d. Let C be the event that a red or blue is picked, followed by landing a head on the coin toss. Are the events A and C mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification. 88. An experiment consists of first rolling a die and then tossing a coin. a. List the sample space. b. Let A be the event that either a three or a four is rolled first, followed by landing a head on the coin toss. Find P(A). c. Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including numerical justification. 89. An experiment consists of tossing a nickel, a dime, and a quarter. Of interest
is the side the coin lands on. a. List the sample space. b. Let A be the event that there are at least two tails. Find P(A). c. Let B be the event that the first and second tosses land on heads. Are the events A and B mutually exclusive? Explain your answer in one to three complete sentences, including justification. 236 Chapter 3 \| Probability Topics 90. Consider the following scenario: Let P(C) = .4. Let P(D) = .5. Let P(C\|D) = .6. a. Find P(C AND D). b. Are C and D mutually exclusive? Why or why not? c. Are C and D independent events? Why or why not? d. Find P(C OR D). e. Find P(D\|C). 91. Y and Z are independent events. a. Rewrite the basic Addition Rule P(Y OR Z) = P(Y) + P(Z) - P(Y AND Z) using the information that Y and Z are independent events. b. Use the rewritten rule to find P(Z) if P(Y OR Z) = .71 and P(Y) = .42. 92. G and H are mutually exclusive events. P(G) = .5 P(H) = .3 a. Explain why the following statement MUST be false: P(H\|G) = .4. b. Find P(H OR G). c. Are G and H independent or dependent events? Explain in a complete sentence. 93. Approximately 281,000,000 people over age five live in the United States. Of these people, 55,000,000 speak a language other than English at home. Of those who speak another language at home, 62.3 percent speak Spanish. Let E = speaks English at home; E′ = speaks another language at home; and S = speaks Spanish. Finish each probability statement by matching the correct answer. Probability Statements Answers a. P(E′) = b. P(E) = c. P(S and E′) = d. P(S\|E′) = Table 3.19 i. .8043 ii. .623 iii. .1957 iv. .1219 94. In 1994, the U.S. government held a lottery to issue 55,000 licenses of a certain type. Renate Deutsch, from Germany, was one of approximately 6.5 million people who entered this lottery. Let G = won license. a. What was Renate’s chance of winning one of the licenses? Write your answer as a probability statement. b. In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen. Once the finalists were chosen, assuming that each finalist had an equal chance to win, what was Renate’s chance of winning one of the licenses? Write your answer as a conditional probability statement. Let F = was a finalist. c. Are G and F independent or dependent events? Justify your answer numerically and also explain why. d. Are G and F mutually exclusive events? Justify your answer numerically and explain why. 95. Three professors at George Washington University did an experiment to determine if economists are more likely to return found money than other people. They dropped 64 stamped, addressed envelopes with $10 cash in different classrooms on the George Washington campus. Forty-four percent were returned overall. From the economics classes 56 percent of the envelopes were returned. From the business, psychology, and history classes 31 percent were returned. Let R = money returned; E = economics classes; and O = other classes. a. Write a probability statement for the overall percentage of money returned. b. Write a probability statement for the percentage of money returned out of the economics classes. c. Write a probability statement for the percentage of money returned out of the other classes. d. e. Based upon this study, do you think that economists are more selfish than other people? Explain why or why not. Is money being returned independent of the class? Justify your answer numerically and explain it. Include numbers to justify your answer. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 237 96. The following table of data obtained from www.baseball-almanac.com shows hit information for four players. Suppose that one hit from the table is randomly selected. Name Single Double Triple Home Run Total Hits Babe Ruth 1,517 Jackie Robinson 1,054 Ty Cobb Hank Aaron Total Table 3.20 3,603 2,294 8,471 506 273 174 624 1,577 136 54 295 98 583 714 137 114 755 2,873 1,518 4,189 3,771 1,720 12,351 Are the hit being made by Hank Aaron and the hit being a double independent events? a. Yes, because P(hit by Hank Aaron\|hit is a double) = P(hit by Hank Aaron) b. No, because P(hit by Hank Aaron\|hit is a double) ≠ P(hit is a double) c. No, because P(hit is by Hank Aaron\|hit is a double) ≠ P(hit by Hank Aaron) d. Yes, because P(hit is by Hank Aaron\|hit is a double) = P(hit is a double) 97. United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. According to their website, a person with type O blood and a negative Rh factor (Rh–) can donate blood to any person with any bloodtype. Their data show that 43 percent of people have type O blood and 15 percent of people have Rh– factor; 52 percent of people have type O or Rh– factor. a. Find the probability that a person has both type O blood and the Rh– factor. b. Find the probability that a person does not have both type O blood and the Rh– factor. 98. At a college, 72 percent of courses have final exams and 46 percent of courses require research papers. Suppose that 32 percent of courses have a research paper and a final exam. Let F be the event that a course has a final exam. Let R be the event that a course requires a research paper. a. Find the probability that a course has a final exam or a research project. b. Find the probability that a course has neither of these two requirements. 99. In a box of assorted cookies, 36 percent contain chocolate and 12 percent contain nuts. Of those, 8 percent contain both chocolate and nuts. Sean is allergic to both chocolate and nuts. a. Find the probability that a cookie contains chocolate or nuts (he can't eat it). b. Find the probability that a cookie does not contain chocolate or nuts (he can eat it). 100. A college finds that 10 percent of students have taken a distance learning class and that 40 percent of students are part-time students. Of the part-time students, 20 percent have taken a distance learning class. Let D = event that a student takes a distance learning class and E = event that a student is a part-time student. a. Find P(D AND E). b. Find P(E\|D). c. Find P(D OR E). d. Using an appropriate test, show whether D and E are independent. e. Using an appropriate test, show whether D and E are mutually exclusive. 3.4 Contingency Tables Use the information in the Table 3.21 to answer the next eight exercises. The table shows the political party affiliation of each of 67 members of the U.S. Senate in June 2012, and when they would next be up for reelection. Up for Reelection: Democratic Party Republican Party Other Total November 2014 November 2016 20 10 Table 3.21 13 24 0 0 238 Chapter 3 \| Probability Topics Up for Reelection: Democratic Party Republican Party Other Total Total Table 3.21 101. What is the probability that a randomly selected senator had an Other affiliation? 102. What is the probability that a randomly selected senator would be up for reelection in November 2016? 103. What is the probability that a randomly selected senator was a Democrat and was up for reelection in November 2016? 104. What is the probability that a randomly selected senator was a Republican or was up for reelection in November 2014? 105. Suppose that a member of the U.S. Senate is randomly selected. Given that the randomly selected senator was up for reelection in November 2016, what is the probability that this senator was a Democrat? 106. Suppose that a member of the U.S. Senate is randomly selected. What is the probability that the senator was up for reelection in November 2014, knowing that this senator was a Republican? 107. The events Republican and Up for reelection in 2016 are ________. independent a. mutually exclusive b. c. both mutually exclusive and independent d. neither mutually exclusive nor independent 108. The events Other and Up for reelection in November 2016 are ________. independent a. mutually exclusive b. c. both mutually exclusive and independent d. neither mutually exclusive nor independent Use the following information to answer the next two exercises. The table of data obtained from www.baseball-almanac.com shows hit information for four well-known baseball players. Suppose that one hit from the table is randomly selected. Name Single Double Triple Home Run Total Hits Babe Ruth 1,517 Jackie Robinson 1,054 Ty Cobb Hank Aaron TOTAL Table 3.22 3,603 2,294 8,471 506 273 174 624 1,577 136 54 295 98 583 714 137 114 755 2,873 1,518 4,189 3,771 1,720 12,351 109. Find P(Hit was made by Babe Ruth). a. b. c. d. 1,518 2,873 2,873 12,351 583 12,351 4,189 12,351 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 239 110. Find P(Hit was made by Ty Cobb\|The hit was a Home Run). a. b. c. d. 4,189 12,351 114 1,720 1,720 4,189 114 12,351 111. Table 3.23 identifies a group of children by one of four hair colors, and by type of hair. Hair Type Brown Blond Black Red Totals 20 80 Wavy Straight Totals Table 3.23 15 3 12 43 215 15 20 a. Complete the table. b. What is the probability that a randomly selected child will have wavy hair? c. What is the probability that a randomly selected child will have either brown or blond hair? d. What is the probability that a randomly selected child will have wavy brown hair? e. What is the probability that a randomly selected child will have red hair, given that he or she has straight hair? f. g. If B is the event of a child having brown hair, find the probability of the complement of B. In words, what does the complement of B represent? 112. In a previous year, the weights of the members of a California football team and a Texas football team were published in a newspaper. The factual data were compiled into the following table. The weights in the column headings are in pounds. Shirt # ≤ 210 211–250 251–290 > 290 1–33 34–66 66–99 21 6 6 Table 3.24 5 18 12 0 7 22 0 4 5 For the following, suppose that you randomly select one player from the California team or the Texas team. a. Find
the probability that his shirt number is from 1 to 33. b. Find the probability that he weighs at most 210 pounds. c. Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds. d. Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds. e. Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210 pounds. 3.5 Tree and Venn Diagrams Use the following information to answer the next two exercises. This tree diagram shows the tossing of an unfair coin followed by drawing one bead from a cup containing three red (R), four yellow (Y), and five blue (B) beads. For the coin, P(H) = 2 3 where H is heads and T is tails. and P(T) = 1 3 240 Chapter 3 \| Probability Topics Figure 3.22 113. Find P(tossing a head on the coin AND a red bead). a. b. c. d. 2 3 5 15 6 36 5 36 a. b. 114. Find P(blue bead). 15 36 10 36 10 12 6 36 d. c. 115. A box of cookies contains three chocolate and seven butter cookies. Miguel randomly selects a cookie and eats it. Then he randomly selects another cookie and eats it. How many cookies did he take? a. Draw the tree that represents the possibilities for the cookie selections. Write the probabilities along each branch of the tree. b. Are the probabilities for the flavor of the second cookie that Miguel selects independent of his first selection? Explain. c. For each complete path through the tree, write the event it represents and find the probabilities. d. Let S be the event that both cookies selected were the same flavor. Find P(S). e. Let T be the event that the cookies selected were different flavors. Find P(T) by two different methods by using the complement rule and by using the branches of the tree. Your answers should be the same with both methods. f. Let U be the event that the second cookie selected is a butter cookie. Find P(U). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 241 BRINGING IT TOGETHER: HOMEWORK 116. A previous year, the weights of the members of a California football team and a Texas football team were published in a newspaper The factual data are compiled into Table 3.25. Shirt# ≤ 210 211–250 251–290 290≤ 1–33 21 34–66 66–99 6 6 Table 3.25 5 18 12 0 7 22 0 4 5 For the following, suppose that you randomly select one player from the California team or the Texas team. If having a shirt number from one to 33 and weighing at most 210 pounds were independent events, then what should be true about P(Shirt# 1–33\|≤ 210 pounds)? 117. The probability that a male develops some form of cancer in his lifetime is .4567. The probability that a male has at least one false-positive test result, meaning the test comes back for cancer when the man does not have it, is .51. Some of the following questions do not have enough information for you to answer them. Write not enough information for those answers. Let C = a man develops cancer in his lifetime and P = a man has at least one false-positive. a. P(C) = ______ b. P(P\|C) = ______ c. P(P\|C') = ______ d. If a test comes up positive, based upon numerical values, can you assume that man has cancer? Justify numerically and explain why or why not. 118. Given events G and H: P(G) = .43; P(H) = .26; P(H AND G) = .14 a. Find P(H OR G). b. Find the probability of the complement of event (H AND G). c. Find the probability of the complement of event (H OR G). 119. Given events J and K: P(J) = .18; P(K) = .37; P(J OR K) = .45 a. Find P(J AND K). b. Find the probability of the complement of event (J AND K). c. Find the probability of the complement of event (J OR K). Use the following information to answer the next two exercises. Suppose that you have eight cards. Five are green and three are yellow. The cards are well shuffled. 120. Suppose that you randomly draw two cards, one at a time, with replacement. Let G1 = first card is green Let G2 = second card is green a. Draw a tree diagram of the situation. b. Find P(G1 AND G2). c. Find P(at least one green). d. Find P(G2\|G1). e. Are G2 and G1 independent events? Explain why or why not. 121. Suppose that you randomly draw two cards, one at a time, without replacement. G1 = first card is green G2 = second card is green a. Draw a tree diagram of the situation. b. Find P(G1 AND G2). c. Find P(at least one green). d. Find P(G2\|G1). e. Are G2 and G1 independent events? Explain why or why not. Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) 242 Chapter 3 \| Probability Topics who are female is 48.60. Of the females, 5.03 percent are age 19 and under; 81.36 percent are age 20–64; 13.61 percent are age 65 or over. Of the licensed U.S. male drivers, 5.04 percent are age 19 and under; 81.43 percent are age 20–64; 13.53 percent are age 65 or over. 122. Complete the following: a. Construct a table or a tree diagram of the situation. b. Find P(driver is female). c. Find P(driver is age 65 or over\|driver is female). d. Find P(driver is age 65 or over AND female). e. f. Find P(driver is age 65 or over). g. Are being age 65 or over and being female mutually exclusive events? How do you know? In words, explain the difference between the probabilities in Part c and Part d. 123. Suppose that 10,000 U.S. licensed drivers are randomly selected. a. How many would you expect to be male? b. Using the table or tree diagram, construct a contingency table of gender versus age group. c. Using the contingency table, find the probability that out of the age 20–64 group, a randomly selected driver is female. 124. Approximately 86.5 percent of Americans commute to work by car, truck, or van. Out of that group, 84.6 percent drive alone and 15.4 percent drive in a carpool. Approximately 3.9 percent walk to work and approximately 5.3 percent take public transportation. a. Construct a table or a tree diagram of the situation. Include a branch for all other modes of transportation to work. b. Assuming that the walkers walk alone, what percent of all commuters travel alone to work? c. Suppose that 1,000 workers are randomly selected. How many would you expect to travel alone to work? d. Suppose that 1,000 workers are randomly selected. How many would you expect to drive in a carpool? 125. When the euro coin was introduced in 2002, two math professors had their statistics students test whether the Belgian one euro-coin was a fair coin. They spun the coin rather than tossing it and found that out of 250 spins, 140 showed a head (event H) while 110 showed a tail (event T). On that basis, they claimed that it is not a fair coin. a. Based on the given data, find P(H) and P(T). b. Use a tree to find the probabilities of each possible outcome for the experiment of spinning the coin twice. c. Use the tree to find the probability of obtaining exactly one head in two spins of the coin. d. Use the tree to find the probability of obtaining at least one head. 126. Use the following information to answer the next two exercises. The following are real data from Santa Clara County, California. As of a certain time, there had been a total of 3,059 documented cases of a disease in the county. They were grouped into the following categories, with risk factors of becoming ill with the disease labeled as Methods A, B, and C and Other: Method A Method B Method C Other Totals Female 0 Male 2,146 Totals ____ Table 3.26 70 463 ____ 136 60 ____ 49 135 ____ ____ ____ ____ Suppose a person with a disease in Santa Clara County is randomly selected. a. Find P(Person is female). b. Find P(Person has a risk factor of method C). c. Find P(Person is female OR has a risk factor of method B). d. Find P(Person is female AND has a risk factor of method A). e. Find P(Person is male AND has a risk factor of method B). f. Find P(Person is female GIVEN person got the disease from method C). g. Construct a Venn diagram. Make one group females and the other group method C. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 243 127. Answer these questions using probability rules. Do NOT use the contingency table. Three thousand fifty-nine cases of a disease had been reported in Santa Clara County, California, through a certain date. Those cases will be our population. Of those cases, 6.4 percent obtained the disease through method C and 7.4 percent are female. Out of the females with the disease, 53.3 percent got the disease from method C. a. Find P(Person is female). b. Find P(Person obtained the disease through method C). c. Find P(Person is female GIVEN person got the disease from method C) d. Construct a Venn diagram representing this situation. Make one group females and the other group method C. Fill in all values as probabilities. REFERENCES 3.1 Terminology Worldatlas. (2013). Countries list by continent. Retrieved from http://www.worldatlas.com/cntycont.htm 3.2 Independent and Mutually Exclusive Events Gallup. (n.d.). Retrieved from www.gallup.com/ Lopez, S., and Sidhu, P. (2013, March 28). U.S. teachers love their lives, but struggle in the workplace. Gallup Wellbeing. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx 3.3 Two Basic Rules of Probability Baseball Almanac. (2013). Retrieved from www.baseball-almanac.com DiCamillo, Mark, and Field, M. The file poll. Field Research Corporation. Retrieved from http://www.field.com/ fieldpollonline/subscribers/Rls2443.pdf Field Research Corporation. (n.d.). Retrieved from www.field.com/fieldpollonline Forum Research. (n.d.). Mayor’s approval down. Retrieved from http://www.forumresearch.com/forms/News Archives/ News Releases/74209_TO_Issues_-_Mayoral_Approval_%28Forum_Research%29%2820130320%29.pdf Rider, D. (2011, Sept. 14). Ford support plummeting, poll suggests. The Star. Retrieved from http://www.thestar.com/news/ gta/2011/09/14/ford_support_plummeting_poll_suggests.html Shin, H. B., and Kominski, R. A. (2010 A
pril). Language use in the United States: 2007 (American Community Survey Reports, ACS-12). Washington, DC: United States Census Bureau. Retrieved from http://www.census.gov/hhes/socdemo/ language/data/acs/ACS-12.pdf The Roper Center for Public Opinion Research. (n.d.). Archives. Retrieved from http://www.ropercenter.uconn.edu/ The Wall Street Journal. (n.d.). Retrieved from https://www.wsj.com/ U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/ Wikipedia. (n.d.). Roulette. Retrieved from http://en.wikipedia.org/wiki/Roulette 3.4 Contingency Tables American Red Cross. (2013). Blood Types. Retrieved from http://www.redcrossblood.org/learn-about-blood/bloodtypes Centers for Disease Control and Prevention/National Center for Health Statistics, United States Department of Health and Human Services. (n.d.). Retrieved from https://www.cdc.gov/nchs/ Haiman, C. A., et al. (2006, Jan. 26). Ethnic and racial differences in the smoking-related risk of lung cancer. The New England Journal of Medicine. Retrieved from http://www.nejm.org/doi/full/10.1056/NEJMoa033250 Samuels, T. M. (2013). Strange facts about RH negative blood. eHow Health. Retrieved from http://www.ehow.com/ facts_5552003_strange-rh-negative-blood.html The Disaster Center Crime Pages. (n.d.). United States: Uniform crime report – state statistics from 1960–2011. Retrieved 244 Chapter 3 \| Probability Topics from http://www.disastercenter.com/crime/ United Blood Services. (2011). Human blood types. Retrieved from http://www.unitedbloodservices.org/learnMore.aspx United States Senate. (n.d.). Retrieved from www.senate.gov 3.5 Tree and Venn Diagrams American Cancer Society. (n.d.). Retrieved from https://www.cancer.org/ Clara County Public Health Department. (n.d.). Retrieved from https://www.sccgov.org/sites/sccphd/en-us/pages/phd.aspx Federal Highway Administration, U.S. Department of Transportation. (n.d.). Retrieved from https://www.fhwa.dot.gov/ The Data and Story Library. (1996). Retrieved from http://lib.stat.cmu.edu/DASL/ The Roper Center http://www.ropercenter.uconn.edu/data_access/data/search_for_datasets.html Public Opinion Research. (2013). for Search for datasets. Retrieved from USA Today. (n.d.). Retrieved from https://www.usatoday.com/ U.S. Census Bureau. (n.d.). Retrieved from https://www.census.gov/ World Bank Group. (2013). Environment. Available online at http://data.worldbank.org/topic/environment SOLUTIONS 1 a. P(L′) = P(S) b. P(M OR S) c. P(F AND L) d. P(M\|L) e. P(L\|M) f. P(S\|F) g. P(F\|L) h. P(F OR L) i. P(M AND S) j. P(F) 3 P(N) = 15 42 = 5 14 = .36 5 P(C) = 5 42 = .12 7 P(G) = 20 150 = 2 15 = .13 9 P(R) = 22 150 = 11 75 = .15 11 P(O) = 150 − 22 − 38 − 20 − 28 − 26 150 = 16 150 = 8 75 = .11 13 P(E) = 47 194 = .24 15 P(N) = 23 194 = .12 17 P(S) = 12 194 = 6 97 = .06 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 245 19 13 52 = 1 4 = .25 21 3 6 = 1 2 = .5 23 P(R) = 4 8 = .5 25 P(O OR H) 27 P(H\|I) 29 P(N\|O) 31 P(I OR N) 33 P(I) 35 The likelihood that an event will occur given that another event has already occurred. 37 1 39 the probability of landing on an even number or a multiple of three 41 P(J) = .3 43 P(Q AND R) = P(Q)P(R) .1 = (.4)P(R) P(R) = .25 45 0.376 47 C\|L means, given the person chosen is a Latino Californian, the person is a registered voter who prefers life in prison without parole for a person convicted of first degree murder. 49 L AND C is the event that the person chosen is a voter of the ethnicity in question who prefers life without parole over the death penalty for a person convicted of first degree murder. 51 .6492 53 No, because P(L AND C) does not equal 0. 55 P(musician is a male AND had private instruction) = 15 130 = 3 26 = .12 57 P(being a female musician AND learning music in school) = 38 130 = 19 65 = .29 P(being a female musician)P(learning music in school) = ⎛ ⎝ 72 130 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 62 130 = 4, 464 16, 900 = 1, 116 4, 225 = .26 No, they are not independent because P(being a female musician AND learning music in school) is not equal to P(being a female musician)P(learning music in school). Chapter 3 \| Probability Topics 246 58 Figure 3.23 60 35,065 100,450 62 To pick one person from the study who is Japanese American AND uses the product 21 to 30 times a day means that the person has to meet both criteria: both Japanese American and uses the product 21 to 30 times a day. The sample space 4,715 100,450 should include everyone in the study. The probability is . 64 To pick one person from the study who is Japanese American given that person uses the product 21 to 30 times a day, means that the person must fulfill both criteria and the sample space is reduced to those who uses the product 21 to 30 times a day. The probability is 4715 15,273 . 67 a. You can't calculate the joint probability knowing the probability of both events occurring, which is not in the information given; the probabilities should be multiplied, not added; and probability is never greater than 100 percent b. A home run by definition is a successful hit, so he has to have at least as many successful hits as home runs. 69 0 71 .3571 73 .2142 75 Physician (83.7) 77 83.7 − 79.6 = 4.1 79 P(Occupation < 81.3) = .5 81 a. The Forum Research surveyed 1,046 Torontonians. b. 58 percent c. 42 percent of 1,046 = 439 (rounding to the nearest integer) d. e. .57 .60. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 247 82 a. yes; P(getting a pork chop) = P(not getting a chicken breast) b. getting a pork chop and getting a chicken breast c. no 83 a. 20/40 = 1/2 b. 5/40 = 1/8 c. 39/40 d. 4/40 = 1/10 e. 33/40 f. 15/40 = 3/8 g. 0/40 = 0 84 Compute the probabilities. a. 20/40 = 1/2 b. 8/40 = 1/5 c. 40/40 = 1 d. 16/40 = 2/5 e. 18/40 = 9/20 f. 40/40 = 1 85 a. {G1, G2, G3, G4, G5, Y1, Y2, Y3} b. c. d. e. No, because P(G AND E) does not equal 0. 87 NOTE The coin toss is independent of the card picked first. a. {(G,H) (G,T) (B,H) (B,T) (R,H) (R,T)} b. P(A) = P(blue)P(head) = ⎛ ⎝ ⎞ ⎠ 3 10 ⎞ ⎠ ⎛ ⎝ 1 2 = 3 20 c. Yes, A and B are mutually exclusive because they cannot happen at the same time; you cannot pick a card that is both blue and also (red or green). P(A AND B) = 0. d. No, A and C are not mutually exclusive because they can occur at the same time. In fact, C includes all of the outcomes of A; if the card chosen is blue it is also (red or blue). P(A AND C) = P(A) = 3 20. 248 89 Chapter 3 \| Probability Topics a. S = {(HHH), (HHT), (HTH), (HTT), (THH), (THT), (TTH), (TTT)} b. 4 8 c. Yes, because if A has occurred, it is impossible to obtain two tails. In other words, P(A AND B) = 0. 91 a. If Y and Z are independent, then P(Y AND Z) = P(Y)P(Z), so P(Y OR Z) = P(Y) + P(Z) – P(Y)P(Z). b. .5 93 iii; i; iv; ii 95 a. P(R) = .44 b. P(R\|E) = .56 c. P(R\|O) = .31 d. No, whether the money is returned is not independent of which class the money was placed in. There are several ways to justify this mathematically, but one is that the money placed in economics classes is not returned at the same overall rate; P(R\|E) ≠ P(R). e. No, this study definitely does not support that notion; in fact, it suggests the opposite. The money placed in the economics classrooms was returned at a higher rate than the money place in all classes collectively; P(R\|E) > P(R). 97 a. P(type O OR Rh–) = P(type O) + P(Rh–) – P(type O AND Rh–) 0.52 = 0.43 + 0.15 – P(type O AND Rh–); solve to find P(type O AND Rh–) = .06 6 percent of people have type O, Rh– blood b. P(NOT(type O AND Rh–)) = 1 – P(type O AND Rh–) = 1 – .06 = .94 94 percent of people do not have type O, Rh– blood 99 a. Let C = be the event that the cookie contains chocolate. Let N = the event that the cookie contains nuts. b. P(C OR N) = P(C) + P(N) – P(C AND N) = .36 + .12 – .08 = .40 c. P(NEITHER chocolate NOR nuts) = 1 – P(C OR N) = 1 – .40 = .60 101 0 103 10 67 105 10 34 107 d 110 b 112 a. b. c. 26 106 33 106 21 106 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 249 d. e. ⎛ ⎝ 26 106 ⎞ ⎠ + ⎛ ⎝ ⎞ ⎠ 33 106 – ⎛ ⎝ ⎞ ⎠ 21 106 = ⎛ ⎝ ⎞ ⎠ 38 106 21 33 114 a 117 a. P(C) = .4567 b. not enough information c. not enough information d. no, because over half (0.51) of men have at least one false-positive text 119 a. P(J OR K) = P(J) + P(K) − P(J AND K); .45 = .18 + .37 – P(J AND K); solve to find P(J AND K) = .10 b. P(NOT (J AND K)) = 1 – P(J AND K) = 1 – 010 = .90 c. P(NOT (J OR K)) = 1 – P(J OR K) = 1 – .45 = .55 120 Figure 3.24 a. b. P(GG = 25 64 c. P(at least one green) = P(GG) + P(GY) + P(YG) = 25 64 + 15 64 + 15 64 = 55 64 d. P(G\|G) = 5 8 e. Yes, they are independent because the first card is placed back in the bag before the second card is drawn. The composition of cards in the bag remains the same from draw one to draw two. 122a. 250 Chapter 3 \| Probability Topics <20 20–64 >64 Totals Female .0244 .3954 .0661 .486 Male .0259 .4186 .0695 .514 Totals .0503 .8140 .1356 1 Table 3.27 b. P(F) = .486 c. P(>64\|F) = .1361 d. P(>64 and F) = P(F) P(>64\|F) = (.486)(.1361) = .0661 e. P(>64\|F) is the percentage of female drivers who are 65 or older and P(>64 and F) is the percentage of drivers who are female and 65 or older. f. P(>64) = P(>64 and F) + P(>64 and M) = .1356 g. No, being female and 65 or older are not mutually exclusive because they can occur at the same time P(>64 and F) = .0661. 124 a. Car, Truck or Van Walk Public Transportation Other Totals Alone .7318 Not Alone .1332 Totals .8650 Table 3.28 .0390 .0530 .0430 1 b. If we assume that all walkers are alone and that none from the other two groups travel alone (which is a big assumption) we have: P(Alone) = .7318 + .0390 = .7708. c. Make the same assumptions as in (b) we have: (.7708)(1,000) = 771 d. (.1332)(1,000) = 133 126 The completed contingency table is as follows: Method A Method B Method C Other Totals Female 0 Male 2,146 Totals 2,146 Table 3.29 70 463 533 136 60 196 49 135 184 255 2,804 3,05
9 a. b. c. 255 3059 196 3059 718 3059 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 3 \| Probability Topics 251 d. 0 e. f. 463 3059 136 196 g. Figure 3.25 252 Chapter 3 \| Probability Topics This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 253 4 \| DISCRETE RANDOM VARIABLES Figure 4.1 You can use probability and discrete random variables to calculate the likelihood of lightning striking the ground five times during a half-hour thunderstorm. (credit: Leszek Leszczynski) Introduction Chapter Objectives By the end of this chapter, the student should be able to do the following: • Recognize and understand discrete probability distribution functions, in general. • Calculate and interpret expected values. • Recognize the binomial probability distribution and apply it appropriately. • Recognize the poisson probability distribution and apply it appropriately. • Recognize the geometric probability distribution and apply it appropriately. • Recognize the hypergeometric probability distribution and apply it appropriately. • Classify discrete word problems by their distributions. A student takes a 10-question, true-false quiz. Because the student had such a busy schedule, he or she could not study and guesses randomly at each answer. What is the probability of the student passing the test with at least a 70 percent? Small companies might be interested in the number of long-distance phone calls their employees make during the peak time of the day. Suppose the average is 20 calls. What is the probability that the employees make more than 20 long-distance phone calls during the peak time? 254 Chapter 4 \| Discrete Random Variables These two examples illustrate two different types of probability problems involving discrete random variables. Recall that discrete data are data that you can count. A random variable is a variable whose values are numerical outcome of a probability experiment. We always describe a random variable in words and its values in numbers. The values of a random variable can vary with each repetition of an experiment. Random Variable Notation Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number. The following are examples of random variables: Example 1: Suppose a jar contains three marbles, one blue, one red, and one white. Randomly draw one marble from the jar. Let X = the possible number of red marbles to be drawn. The sample space for the drawing is red, white, and blue. Then, x = 0,1. If the marble we draw is red, then x = 1; otherwise, x = 0. Example 2: Let X = the number of female children in a randomly selected family with only two kids. Here we are only interested in families with two kids, not families with one kid or more than two kids. The sample space for the genders of two-kid families is MM, MF, FM, FF. Here the first letter represents the gender of the older child and the second letter represents the gender of the younger child. F represents a female child and M represents a male child. For example, FM represents that the older child is a girl and the younger child is a boy, while MF represents that the older child is a boy and the younger child is a girl. Then, x = 0,1,2. A family has 0 female children if it has two boys (MM), a family has one female child if it has one boy and one girl (MF or FM), and a family has two female children if both kids are girls (FF). Example 3: Let X = the number of heads you get when you toss three fair coins. The sample space for the toss of three fair coins is TTT, THH, HTH, HHT, HTT, THT, TTH, HHH. Here the first letter represents the result of the first toss, the second letter represents the result of the second toss, and the third letter represents the result of the third toss. T represents a tail and H represents a head. For example, THH means we get a tail in the first toss but a head in the second and third toss, while HHT means we get a head in the first and second toss but a tail in the third toss. Then, x = 0, 1, 2, 3. There are 0 heads if the result is TTT, one head if the result is THT, TTH, or HTT, two heads if the result is THH, HTH, or HHT, and three heads if the result is HHH. Toss a coin 10 times and record the number of heads. After all members of the class have completed the experiment (tossed a coin 10 times and counted the number of heads), fill in Table 4.1. Let X = the number of heads in 10 tosses of the coin. x Frequency of x Relative Frequency of x Table 4.1 a. Which value(s) of x occurred most frequently? b. If you tossed the coin 1,000 times, what values could x take on? Which value(s) of x do you think would occur most frequently? c. What does the relative frequency column sum to? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 255 4.1 \| Probability Distribution Function (PDF) for a Discrete Random Variable There are two types of random variables, discrete random variables and continuous random variables. The values of a discrete random variable are countable, which means the values are obtained by counting. All random variables we discussed in previous examples are discrete random variables. We counted the number of red balls, the number of heads, or the number of female children to get the corresponding random variable values. The values of a continuous random variable are uncountable, which means the values are not obtained by counting. Instead, they are obtained by measuring. For example, let X = temperature of a randomly selected day in June in a city. The value of X can be 68°, 71.5°, 80.6°, or 90.32°. These values are obtained by measuring by a thermometer. Another example of a continuous random variable is the height of a randomly selected high school student. The value of this random variable can be 5'2", 6'1", or 5'8". Those values are obtained by measuring by a ruler. A discrete probability distribution function has two characteristics: 1. Each probability is between zero and one, inclusive. 2. The sum of the probabilities is one. Example 4.1 A child psychologist is interested in the number of times a newborn baby's crying wakes its mother after midnight. For a random sample of 50 mothers, the following information was obtained. Let X = the number of times per week a newborn baby's crying wakes its mother after midnight. For this example, x = 0, 1, 2, 3, 4, 5. P(x) = probability that X takes on a value x. x P(x) 0 1 2 3 4 5 P(x = 0) = 2 50 P(x = 1) = 11 50 P(x = 2) = 23 50 P(x = 3) = 9 50 P(x = 4) = 4 50 P(x = 5) = 1 50 Table 4.2 X takes on the values 0, 1, 2, 3, 4, 5. This is a discrete PDF because we can count the number of values of x and also because of the following two reasons: a. Each P(x) is between zero and one, therefore inclusive b. The sum of the probabilities is one, that is, 2 50 + 11 50 + 23 50 + 9 50 + 4 50 + 1 50 = 1 256 Chapter 4 \| Discrete Random Variables 4.1 A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. Let X = the number of times a patient rings the nurse during a 12-hour shift. For this exercise, x = 0, 1, 2, 3, 4, 5. P(x) = the probability that X takes on value x. Why is this a discrete probability distribution function (two reasons)? X P(x) 0 P(x = 0) = 4 50 1 P(x = 1) = 8 50 2 P(x = 2) = 16 50 3 P(x = 3) = 14 50 4 P(x = 4) = 6 50 5 P(x = 5) = 2 50 Table 4.3 Example 4.2 Suppose Nancy has classes three days a week. She attends classes three days a week 80 percent of the time, two days 15 percent of the time, one day 4 percent of the time, and no days 1 percent of the time. Suppose one week is randomly selected. Describe the random variable in words. Let X = the number of days Nancy ________. Solution 4.2 a. Let X = the number of days Nancy attends class per week. b. In this example, what are possible values of X? Solution 4.2 b. 0, 1, 2, and 3 c. Suppose one week is randomly chosen. Construct a probability distribution table (called a PDF table) like the one in Example 4.1. The table should have two columns labeled x and P(x). Solution 4.2 c. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 257 x P(x) 0 1 2 3 .01 .04 .15 .80 Table 4.4 The sum of the P(x) column is 0.01+0.04+0.15+0.80 = 1.00. 4.2 Jeremiah has basketball practice two days a week. 90 percent of the time, he attends both practices. Eight percent of the time, he attends one practice. Two percent of the time, he does not attend either practice. What is X and what values does it take on? 4.2 \| Mean or Expected Value and Standard Deviation The expected value of a discrete random variable X, symbolized as E(X), is often referred to as the long-term average or mean (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average. NOTE To find the expected value, E(X), or mean μ of a discrete random variable X, simply multiply each value of the random variable by its probability and add the products. The formula is given as E(X) = μ = ∑ xP(x). Here x represents values of the random variable X, P(x) represents the corresponding probability, and symbol ∑ represents the sum of all products xP(x). Here we use symbol μ for the mean because it is a parameter. It represents the mean of a population. Example 4.3 A men's soccer team plays soccer zero, one, or two days a week. The proba
bility that they play zero days is .2, the probability that they play one day is .5, and the probability that they play two days is .3. Find the long-term average or expected value, μ, of the number of days per week the men's soccer team plays soccer. To do the problem, first let the random variable X = the number of days the men's soccer team plays soccer per week. X takes on the values 0, 1, 2. Construct a PDF table adding a column xP(x), the product of the value x with the corresponding probability P(x). In this column, you will multiply each x value by its probability. 258 Chapter 4 \| Discrete Random Variables x P(x) xP(x) 0 1 2 .2 .5 .3 (0)(.2) = 0 (1)(.5) = .5 (2)(.3) = .6 Table 4.5 Expected Value Table This table is called an expected value table. The table helps you calculate the expected value or longterm average. Add the last column x * P(x) to get the expected value/mean of the random variable X. E(X) = μ = ∑ xP(x) = 0 + .5 + .6 = 1.1 The expected value/mean is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long-term average or expected value if the men's soccer team plays soccer week after week after week. As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as σ 2 and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance σ 2 of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance σ 2 . The formulas are given as below. NOTE The formula of the variance σ 2 of a discrete random variable X is σ 2 = ∑ (x − μ)2 P(x). Here x represents values of the random variable X, μ is the mean of X, P(x) represents the corresponding probability, and symbol ∑ represents the sum of all products (x − μ)2 P(x). This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 259 To find the standard deviation, σ, of a discrete random variable X, simply take the square root of the variance x − μ)2 P(x) Example 4.4 A researcher conducted a study to investigate how a newborn baby’s crying after midnight affects the sleep of the baby's mother. The researcher randomly selected 50 new mothers and asked how many times they were awakened by their newborn baby's crying after midnight per week. Two mothers were awakened zero times, 11 mothers were awakened one time, 23 mothers were awakened two times, nine mothers were awakened three times, four mothers were awakened four times, and one mother was awakened five times. Find the expected value of the number of times a newborn baby's crying wakes its mother after midnight per week. Calculate the standard deviation of the variable as well. To do the problem, first let the random variable X = the number of times a mother is awakened by her newborn’s crying after midnight per week. X takes on the values 0, 1, 2, 3, 4, 5. Construct a PDF table as below. The column of P(x) gives the experimental probability of each x value. We will use the relative frequency to get the probability. For example, the probability that a mother wakes up zero times is 2 50 since there are two mothers out of 50 who were awakened zero times. The third column of the table is the product of a value and its probability, xP(x). x P(x) xP(x) 0 P(x = 0) = 2 50 ⎛ (0) ⎝ 2 50 ⎞ ⎠ = 0 1 P(x = 1) = 11 50 ⎛ (1) ⎝ 11 50 ⎞ ⎠ = 11 50 2 P(x = 2) = 23 50 ⎛ (2) ⎝ 23 50 ⎞ ⎠ = 46 50 3 P(x = 3) = 9 50 ⎛ (3) ⎝ 9 50 ⎞ ⎠ = 27 50 4 P(x = 4) = 4 50 ⎛ (4) ⎝ 4 50 ⎞ ⎠ = 16 50 5 P(x = 5) = 1 50 ⎛ (5) ⎝ 1 50 ⎞ ⎠ = 5 50 Table 4.6 We then add all the products in the third column to get the mean/expected value of X. E(X) = μ = ∑ xP(x) = 0 + 11 50 + 46 50 + 27 50 + 16 50 + 5 50 = 105 50 = 2.1 Therefore, we expect a newborn to wake its mother after midnight 2.1 times per week, on the average. To calculate the standard deviation σ, we add the fourth column (x-μ)2 and the fifth column (x - μ)2 ∙ P(x) to get the following table: 260 Chapter 4 \| Discrete Random Variables x P(x) xP(x) (x-µ)2 (x-µ)2•P(x) 0 P(x = 0) = 2 50 ⎛ (0) ⎝ 2 50 ⎞ ⎠ = 0 (0 − 2.1)2 = 4.41 4.41 • 2 50 = .1764 1 P(x = 1) = 11 50 ⎛ (1) ⎝ 11 50 ⎞ ⎠ = 11 50 (1 − 2.1)2 = 1.21 1.21 • 11 50 = .2662 2 P(x = 2) = 23 50 ⎛ (2) ⎝ 23 50 ⎞ ⎠ = 46 50 (2 − 2.1)2 = .01 3 P(x = 3) = 9 50 ⎛ (3) ⎝ 9 50 ⎞ ⎠ = 27 50 (3 − 2.1)2 = .81 .01 • 23 50 .81 • 9 50 = .0046 = .1458 4 P(x = 4) = 4 50 ⎛ (4) ⎝ 4 50 ⎞ ⎠ = 16 50 (4 − 2.1)2 = 3.61 3.61 • 4 50 = .2888 5 P(x = 5) = 1 50 ⎛ (5) ⎝ 1 50 ⎞ ⎠ = 5 50 (5 − 2.1)2 = 8.41 8.41 • 1 50 = .1682 Table 4.7 We then add all the products in the 5th column to get the variance of X. σ 2 = .1764 + .2662 + .0046 + .1458 + .2888 + .1682 = 1.05 To get the standard deviation σ, we simply take the square root of variance σ2. σ = σ 2 = 1.05 ≈ 1.0247 4.4 A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value? x P(x) 0 1 2 3 4 5 P(x = 0) = 4 50 P(x = 1) = 8 50 P(x = 2) = 16 50 P(x = 3) = 14 50 P(x = 4) = 6 50 P(x = 5) = 2 50 Table 4.8 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 261 Example 4.5 Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from zero to nine with replacement. You pay $2 to play and could profit $100,000 if you match all five numbers in order (you get your $2 back plus $100,000). Over the long term, what is your expected profit of playing the game? To do this problem, set up a PDF table for the amount of money you can profit. Let X = the amount of money you profit. If your five numbers match in order, you will win the game and will get your $2 back plus $100,000. That means your profit is $100,000. If your five numbers do not match in order, you will lose the game and lose your $2. That means your profit is -$2. Therefore, X takes on the values $100,000 and –$2. That is the second column x in the PDF table below. To win, you must get all five numbers correct, in order. The probability of choosing the correct first number is 1 10 because there are 10 numbers (from zero to nine) and only one of them is correct. The probability of choosing the correct second number is also 1 10 because the selection is done with replacement and there are still 10 numbers (from zero to nine) for you to choose. Due to the same reason, the probability of choosing the correct third number, the correct fourth number, and the correct fifth number are also 1 10 . The selection of one number does not affect the selection of another number. That means the five selections are independent. The probability of choosing all five correct numbers and in order is equal to the product of the probabilities of choosing each number correctly. ⎝choosing all five numbers correctly ⎞ P⎛ ⎝choosing 2nd number correctly⎞ P⎛ ⎠ • P⎛ ⎠ • P⎛ = ( 1 10 ) • ( 1 10 ) • ( 1 10 ⎝choosing 1st number correctly⎞ ⎝choosing 5th number correctly⎞ ) • ( 1 10 ) • ( 1 10 ) ⎠ ⎠ • = .00001 Therefore, the probability of winning is .00001 and the probability of losing is 1 − .00001 = .99999. That is how we get the third column P(x) in the PDF table below. To get the fourth column xP(x) in the table, we simply multiply the value x with the corresponding probability P(x). The PDF table is as follows: x P(x) x*P(x) Loss –2 .99999 (–2)(.99999) = –1.99998 Profit 100,000 .00001 (100000)(.00001) = 1 Table 4.9 We then add all the products in the last column to get the mean/expected value of X. E(X) = μ = ∑ xP(x) = − 1.99998 + 1 = − .9998. Since –.99998 is about –1, you would, on average, expect to lose approximately $1 for each game you play. However, each time you play, you either lose $2 or profit $100,000. The $1 is the average or expected loss per game after playing this game over and over. 262 Chapter 4 \| Discrete Random Variables 4.5 You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards. You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw. You pay $1 to play. If y
ou guess the right suit every time, you get your money back and $256. What is your expected profit of playing the game over the long term? Example 4.6 Suppose you play a game with a biased coin. You play each game by tossing the coin once. P(heads) = 2 3 and . If you toss a head, you pay $6. If you toss a tail, you win $10. If you play this game many times, P(tails) = 1 3 will you come out ahead? a. Define a random variable X. Solution 4.6 a. X = amount of profit b. Complete the following expected value table. Solution 4.6 b. x ____ ____ WIN 10 1 3 LOSE ____ ____ ____ –12 3 Table 4.10 x P(x) xP(x) 1 3 2 3 WIN 10 LOSE –6 Table 4.11 10 3 –12 3 c. What is the expected value, μ? Do you come out ahead? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 263 Solution 4.6 c. Add the last column of the table. The expected value E(X) = μ = 10 3 + ⎛ ⎝− 12 67 . You lose, on average, about 67 cents each time you play the game, so you do not come out ahead. 4.6 Suppose you play a game with a spinner. You play each game by spinning the spinner once. P(red) = 2 5 , P(blue) = 2 5 , and P(green) = 1 5 . If you land on red, you pay $10. If you land on blue, you don't pay or win anything. If you land on green, you win $10. Complete the following expected value table. –20 5 x P(x) Red Blue Green 10 Table 4.12 2 5 Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce rounding errors. For some probability distributions, there are shortcut formulas for calculating μ and σ. Example 4.7 Toss a fair, six-sided die twice. Let X = the number of faces that show an even number. Construct a table like Table 4.12 and calculate the mean μ and standard deviation σ of X. Solution 4.7 Tossing one fair six-sided die twice has the same sample space as tossing two fair six-sided dice. The sample space has 36 outcomes. (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) Table 4.13 Use the sample space to complete the following table. 264 Chapter 4 \| Discrete Random Variables x P(x) xP(x) (x – μ)2 ⋅ P(x) 0 1 2 9 36 18 36 9 36 0 18 36 18 36 (0 – 1)2 ⋅ 9 36 = 9 36 (1 – 1)2 ⋅ 18 36 = 0 (2 – 1)2 ⋅ 9 36 = 9 36 Table 4.14 Calculating μ and σ. Add the values in the third column to find the expected value: μ = 36 36 = 1. Use this value to complete the fourth column. Add the values in the fourth column and take the square root of the sum: σ = 18 36 ≈ .7071. Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson. Most elementary courses do not cover the geometric, hypergeometric, and Poisson. Your instructor will let you know if he or she wishes to cover these distributions. A probability distribution function is a pattern. You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations. These distributions are tools to make solving probability problems easier. Each distribution has its own special characteristics. Learning the characteristics enables you to distinguish among the different distributions. 4.3 \| Binomial Distribution (Optional) There are three characteristics of a binomial experiment: 1. There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials. 2. There are only two possible outcomes, called success and failure, for each trial. The outcome that we are measuring is defined as a success, while the other outcome is defined as a failure. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p + q = 1. 3. The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. Let us look at several examples of a binomial experiment. Example 1: Toss a fair coin once and record the result. This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, a head or a tail, of each trial. We can define a head as a success if we are measuring number of heads. For a fair coin, the probabilities of getting head or tail are both .5. So, p = q − .5. Both p and q remain the same from trial to trial. This experiment is also called a Bernoulli trial, named after Jacob Bernoulli who, in the late 1600s, studied such trials extensively. Any experiment that has characteristics two and three and where n = 1 is called a Bernoulli trial. A binomial experiment takes place when the number of successes is counted in one or more Bernoulli trials. Example 2: Randomly guess a multiple choice question has A, B, C and D four options. This is a binomial experiment since it meets all three characteristics. The number of trials n = 1. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. For a random This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 265 guess (you have no clue at all), the probability of guessing correct should be 1 4 because there are four options and only one option is correct. So, and p = 1 4 and . Both p and q remain the same from trial to trial. This experiment is also a Bernoulli trial. It meets the characteristics two and three and n = 1. Example 3: Toss a fair coin five times and record the result. This is a binomial experiment since it meets all three characteristics. The number of trials n = 5. There are only two outcomes, head or tail, of each trial. If we define head as a success, then p = q = 0.5. Both p and q remain the same for each trial. Since n = 5, this experiment is not a Bernoulli trial although it meets the characteristics two and three. Example 4: Randomly guess 10 multiple choice questions in an exam. Each question has A, B, C and D four options. This is a binomial experiment since it meets all three characteristics. The number of trials n = 10. There are only two outcomes, guess correctly or guess wrong, of each trial. We can define guess correctly as a success. As we explained in example 2, p = 1 4 . Both p and q remain the same for each guess. Since n = 10, this and experiment is not a Bernoulli trial. The next two experiments are not binomial experiments. Example 5: Randomly select two balls from a jar with five red balls and five blue balls without replacement. This means we select the first ball, and then without returning the selected ball into the jar, we will select the second ball. This is not a binomial experiment since the third characteristic is not met. The number of trials n = 2. There are only two outcomes, a red ball or a blue ball, of each trial. If we define selecting a red ball as a success, then selecting a blue ball is a failure. The probability of getting the first ball red is 5 10 since there are five red balls out of 10 balls. So, p = 5 10 and q = 1 − p = 1 − 5 10 = 5 10 . However, p and q do not remain the same for the second trial. If the first ball selected is red, then the probability of getting the second ball red is 4 9 since there are only four red balls out of nine balls. But if the first ball selected is blue, then the probability of getting the second ball red is 5 9 since there are still five red balls out of nine balls. Example 6: Toss a fair coin until a head appears. This is not a binomial experiment since the first characteristic is not met. The number of trials n is not fixed. n could be 1 if a head appears from the first toss. n could be 2 if the first toss is a tail and the second toss is a head. So on and so forth. More examples of binomial and non-binomial experiments will be discussed in this section later. The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a binomial probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book. Here n is the number of trials, p is the probability of a success, and q is the probability of a failure. μ = np, σ 2 = npq, σ = npq. Example 4.8 At ABC High School, the withdrawal rate from an elementary physics course is 30 percent for any given term. 266 Chapter 4 \| Discrete Random Variables This implies that, for any given term, 70 percent of the students stay in the class for the entire term. The random variable X = the number of students who withdraw from the randomly selected elementary physics class. Since we are measuring the number of students who withdrew, a success is defined as an individual who withdrew. 4.8 The state health board is concerned about the amount of fruit available in school lunches. Forty-eight percent of schools in the state offer fruit in their lunches every day. This implies that 52 percent do not. What would a success be in this case? Example 4.9 Suppose you play a game that you can only either win or lose. The probability that you win any game is 55 percent, and the probability that you lose is 45 percent. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, . . ., 20. The probability of a success is p = 0.55. The probability of a failure is q = .45. The number of trials is n = 20. The pro
bability question can be stated mathematically as P(x = 15). If you define X as the number of losses, then a success is defined as a loss and a failure is defined as a win. A success does not necessarily represent a good outcome. It is simply the outcome that you are measuring. X still takes on the values of 0, 1, 2, 3, . . ., 20. The probability of a success is p = .45 . The probability of a failure is q = .55 . 4.9 A trainer is teaching a dolphin to do tricks. The probability that the dolphin successfully performs the trick is 35 percent, and the probability that the dolphin does not successfully perform the trick is 65 percent. Out of 20 attempts, you want to find the probability that the dolphin succeeds 12 times. State the probability question mathematically. Example 4.10 A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than 10 heads? Let X = the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3, . . ., 15. Since the coin is fair, p = .5 and q = .5. The number of trials n = 15. State the probability question mathematically. Solution 4.10 P(x > 10) 4.10 A fair, six-sided die is rolled 10 times. Each roll is independent. You want to find the probability of rolling a one more than three times. State the probability question mathematically. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 267 Example 4.11 Approximately 70 percent of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly. a. This is a binomial problem because there is only a success or a ________, there are a fixed number of trials, and the probability of a success is .70 for each trial. Solution 4.11 a. failure b. If we are interested in the number of students who do their homework on time, then how do we define X? Solution 4.11 b. X = the number of statistics students who do their homework on time c. What values does x take on? Solution 4.11 c. 0, 1, 2, . . ., 50 d. What is a failure, in words? Solution 4.11 d. Failure is defined as a student who does not complete his or her homework on time. The probability of a success is p = .70. The number of trials is n = 50. e. If p + q = 1, then what is q? Solution 4.11 e. q = .30 f. The words at least translate as what kind of inequality for the probability question P(x ____ 40)? Solution 4.11 f. greater than or equal to (≥) The probability question is P(x ≥ 40). 4.11 Sixty-five percent of people pass the state driver’s exam on the first try. A group of 50 individuals who have taken the driver’s exam is randomly selected. Give two reasons why this is a binomial problem. Notation for the Binomial: B = Binomial Probability Distribution Function X ~ B(n, p) Read this as X is a random variable with a binomial distribution. The parameters are n and p: n = number of trials, p = probability of a success on each trial. 268 Chapter 4 \| Discrete Random Variables Example 4.12 It has been stated that about 41 percent of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education? Let X = the number of workers who have a high school diploma but do not pursue any further education. X takes on the values 0, 1, 2, . . ., 20 where n = 20, p = .41, and q = 1 – .41 = .59. X ~ B(20, .41) Find P(x ≤ 12). There is a formula to define the probability of a binomial distribution P(x). We can use the formula to find P(x ≤ 12) . But the calculation is tedious and time consuming, and people usually use a graphing calculator, software, or binomial table to get the answer. Use a graphing calculator, you can get P(x ≤ 12) = .9738 . The instruction of TI-83, 83+, 84, 84+ is given below. Go into 2nd DISTR. The syntax for the instructions are as follows: To calculate the probability of a value P(x = value) : use binompdf(n, p, number). Here binompdf represents binomial probability density function. It is used to find the probability that a binomial random variable is equal to an exact value. n is the number of trials, p is the probability of a success, and number is the value. If number is left out, which means use binompdf(n, p), then all the probabilities P(x = 0), P(x = 1), … , P(x = n) will be calculated. To calculate the cumulative probability P(x ≤ value) : use binomcdf(n, p, number). Here binomcdf represents binomial cumulative distribution function. It is used to determine the probability of at most type of problem, the probability that a binomial random variable is less than or equal to a value. n is the number of trials, p is the probability of a success, and number is the value. If number is left out, all the cumulative probabilities P(x ≤ 0), P(x ≤ 1), … , P(x ≤ n) will be calculated. To calculate the cumulative probability P(x ≥ value) : use 1 - binomcdf(n, p, number). n is the number of trials, p is the probability of a success, and number is the value. TI calculators do not have a built-in function to find the probability that a binomial random variable is greater than a value. However, we can use the fact that P(x > value) = 1 − P(x ≤ value) to find the answer. For this problem: After you are in 2nd DISTR, arrow down to binomcdf. Press ENTER. Enter 20,.41,12). The result is P(x ≤ 12) = .9738. NOTE If you want to find P(x = 12), use the pdf (binompdf). If you want to find P(x > 12), use 1 − binomcdf(20,.41,12). The probability that at most 12 workers have a high school diploma but do not pursue any further education is .9738. The graph of X ~ B(20, .41) is as follows. The previous graph is called a probability distribution histogram. It is made of a series of vertical bars. The x-axis of each bar is the value of X = the number of workers who have only a high school diploma, and the height of that bar is the probability of that value occurring. The number of adult workers that you expect to have a high school diploma but not pursue any further education This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 269 is the mean, μ = np = (20)(.41) = 8.2. The formula for the variance is σ2 = npq. The standard deviation is σ = npq . σ = (20)(.41)(.59) = 2.20. The following is the interpretation of the mean μ = 8.2 and standard deviation σ = 2.20 : If you randomly select 20 adult workers, and do that over and over, you expect around eight adult workers out of 20 to have a high school diploma but do not pursue any further education on average. And you expect that to vary by about two workers on average. 4.12 About 32 percent of students participate in a community volunteer program outside of school. If 30 students are selected at random, find the probability that at most 14 of them participate in a community volunteer program outside of school. Use the TI-83+ or TI-84 calculator to find the answer. Example 4.13 A store releases a 560-page art supply catalog. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X = the number of pages that feature signature artists. a. What values does x take on? b. What is the probability distribution? Find the following probabilities: i. ii. the probability that two pages feature signature artists the probability that at most six pages feature signature artists iii. the probability that more than three pages feature signature artists c. Using the formulas, calculate the (i) mean and (ii) standard deviation. Solution 4.13 a. x = 0, 1, 2, 3, 4, 5, 6, 7, 8 b. This is a binomial experiment since all three characteristics are met. Each page is a trial. Since we sample 100 pages, the number of trials is n = 100. For each page, there are two possible outcomes, features signature artists or does not feature signature artists. Since we are measuring the number of pages that feature signature artists, a page that features signature artists is defined as a success and a page that does not feature signature artists is defined as a failure. There are 8 out of 560 pages that feature signature artists. Therefore the probability of a success p = 8 560 and the probability of a failure q = 1 − p = 1 − 8 560 = 552 560 . Both p and q remain the same for each page. Therefore, X is a binomial random variable, and it can be ⎛ ⎝100, 8 written as X ~ B 560 ⎞ ⎠ . We can use a graphing calculator to answer Parts i to iii. i. P(x = 2) = binompdf ⎛ ⎝100, 8 560 , 2 ⎞ ⎠ = .2466 ii. P(x ≤ 6) = binomcdf ⎛ ⎝100, 8 560 , 6 ⎞ ⎠ = .9994 270 Chapter 4 \| Discrete Random Variables iii. P(x > 3) = 1 – P(x ≤ 3) = 1 – binomcdf ⎛ ⎝100, 8 560 , 3 ⎞ ⎠ = 1 – .9443 = .0557 c. i. mean = np = (100) ⎛ ⎝ ⎞ ⎠ 8 560 = 800 560 ≈ 1.4286 ii. ⎛ standard deviation = npq = (100) ⎝ 8 560 ⎞ ⎛ ⎝ ⎠ ⎞ ⎠ 552 560 ≈ 1.1867 4.13 According to a poll, 60 percent of American adults prefer saving over spending. Let X = the number of American adults out of a random sample of 50 who prefer saving to spending. a. What is the probability distribution for X? b. Use your calculator to find the following probabilities: i. The probability that 25 adults in the sample prefer saving over spending ii. The probability that at most 20 adults prefer saving iii. The probability that more than 30 adults prefer saving c. Using the formulas, calculate the (i) mean and (ii) standard deviation of X. Example 4.14 The lifetime risk of developing a specific disease is about 1 in 78 (1.28 percent). Suppose we randomly sample 200 people. Let X = the number of people who will develop the disease. a. What is the probability distribution for X? b. Using the formulas
, calculate the (i) mean and (ii) standard deviation of X. c. Use your calculator to find the probability that at most eight people develop the disease. d. Is it more likely that five or six people will develop the disease? Justify your answer numerically. Solution 4.14 a. This is a binomial experiment since all three characteristics are met. Each person is a trial. Since we sample 200 people, the number of trials is n = 200. For each person, there are two possible outcomes: will develop the disease or not. Since we are measuring the number of people who will develop the disease, a person who will develop the disease is defined as a success and a person who will not develop the disease is defined as a failure. The risk of developing the disease is 1.28 percent. Therefore the probability of a success, p = 1.28 percent, .0128 , and the probability of a failure0128 = .9872 . Both p and q remain the same for each person. Therefore, X is a binomial random variable and it can be written as X ~ B(200, .0128) . We can use a graphing calculator to answer Questions c and d. b. i. Mean = np = 200(.0128) = 2.56 ii. Standard Deviation = npq = (200)(0.128)(.9872) ≈ 1.5897 c. Using the TI-83, 83+, 84 calculator with instructions as provided in Example 4.12: P(x ≤ 8) = binomcdf(200, .0128, 8) = .9988 d. P(x = 5) = binompdf(200, .0128, 5) = .0707 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 271 P(x = 6) = binompdf(200, .0128, 6) = .0298 So P(x = 5) > P(x = 6); it is more likely that five people will develop the disease than six. 4.14 During the 2013 regular basketball season, a player had the highest field goal completion rate in the league. This player scored with 61.3 percent of his shots. Suppose you choose a random sample of 80 shots made by this player during the 2013 season. Let X = the number of shots that scored points. a. What is the probability distribution for X? b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X. c. Use your calculator to find the probability that this player scored with 60 of these shots. d. Find the probability that this player scored with more than 50 of these shots. Example 4.15 The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC High School has a student advisory committee made up of 10 staff members and six students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? The names of all committee members are put into a box, and two names are drawn without replacement. The first name drawn determines the chairperson and the second name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student on the first draw is 6 16 because there are six students out of 16 members (10 staff members + six students). If the first draw selects a student, then the probability of a student on the second draw is 5 16 because there are only five students out of 15 members. If the first draw selects a staff member, then the probability of a student on the second draw is 6 15 because there are still six students out of 15 members. The probability of drawing a student's name changes for each of the trials and, therefore, violates the condition of independence. 4.15 A lacrosse team is selecting a captain. The names of all the seniors are put into a hat, and the first three that are drawn will be the captains. The names are not replaced once they are drawn (one person cannot be two captains). You want to see if the captains all play the same position. State whether this problem is binomial or not and state why. 4.4 \| Geometric Distribution (Optional) There are three main characteristics of a geometric experiment: 1. Repeating independent Bernoulli trials until a success is obtained. Recall that a Bernoulli trial is a binomial experiment with number of trials n = 1. In other words, you keep repeating what you are doing until the first success. Then you stop. For example, you throw a dart at a bull's-eye until you hit the bull's-eye. The first time you hit the bull's-eye is a success so you stop throwing the dart. It might take six tries until you hit the bull's-eye. You can think of the trials as failure, failure, failure, failure, failure, success, stop. 272 Chapter 4 \| Discrete Random Variables 2. In theory, the number of trials could go on forever. There must be at least one trial. 3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial. p + q = 1 and q = 1 − p. For example, the probability of rolling a three when you throw one fair die is 1 6 . This is true no matter how many times you roll the die. Suppose you want to know the probability of getting the first three on the fifth roll. On rolls one through four, you do not get a face with a three. The probability for each of the rolls is q = 5 6 , the probability of a failure. The probability of getting a three on the fifth roll is ⎛ ⎝ 0804. X = the number of independent trials until the first success. p = the probability of a success, q = 1 – p = the probability of a failure. There are shortcut formulas for calculating mean μ, variance σ2, and standard deviation σ of a geometric probability distribution. The formulas are given as below. The deriving of these formulas will not be discussed in this book. μ = 1 p, σ 2 = ( 1 p)( 1 p − 1), σ = ( 1 p)( 1 p − 1) Example 4.16 Suppose a game has two outcomes, win or lose. You repeatedly play that game until you lose. The probability of losing is p = 0.57. If we let X = the number of games you play until you lose (includes the losing game), then X is a geometric random variable. All three characteristics are met. Each game you play is a Bernoulli trial, either win or lose. You would need to play at least one game before you stop. X takes on the values 1, 2, 3, . . . (could go on indefinitely). Since we are measuring the number of games you play until you lose, we define a success as losing a game and a failure as winning a game. The probability of a success p = .57 and the probability of a failure q = 1 – p = 1 – 0.57 = 0.43. Both p and q remain the same from game to game. If we want to find the probability that it takes five games until you lose, then the probability could be written as P(x = 5). We will explain how to find a geometric probability later in this section. 4.16 You throw darts at a board until you hit the center area. Your probability of hitting the center area is p = 0.17. You want to find the probability that it takes eight throws until you hit the center. What values does X take on? Example 4.17 A safety engineer feels that 35 percent of all industrial accidents in her plant are caused by failure of employees to follow instructions. She decides to look at the accident reports (selected randomly and replaced in the pile after reading) until she finds one that shows an accident caused by failure of employees to follow instructions. If we let X = the number of accidents the safety engineer must examine until she finds a report showing an accident caused by employee failure to follow instructions, then X is a geometric random variable. All three characteristics are met. Each accident report she reads is a Bernoulli trial: the accident was either caused by failure of employees to follow instructions or not. She would need to read at least one accident report before she stops. X takes on the values 1, 2, 3, . . . (could go on indefinitely). Since we are measuring the number of reports she needs to read until one that shows an accident caused by failure of employees to follow instructions, we define a success as an accident caused by failure of employees to follow instructions. If an accident was caused by another reason, the report is defined as a failure. The probability of a success p = .35 and the probability of a failure q = 1 − p = 1 − .35 = .65 . Both p and q remain the same from report to report. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 273 If we want to find the probability that the safety engineer will have to examine at least three reports until she finds a report showing an accident caused by employee failure to follow instructions, then the probability could be written as p = .35 . If we want to find how many reports, on average, the safety engineer would expect to look at until she finds a report showing an accident caused by employee failure to follow instructions, we need to find the expected value E(x). We will explain how to solve these questions later in this section. 4.17 An instructor feels that 15 percent of students get below a C on their final exam. She decides to look at final exams (selected randomly and replaced in the pile after reading) until she finds one that shows a grade below a C. We want to know the probability that the instructor will have to examine at least 10 exams until she finds one with a grade below a C. What is the probability question stated mathematically? Example 4.18 Suppose that you are looking for a student at your college who lives within five miles of you. You know that 55 percent of the 25,000 students do live within five miles of you. You randomly contact students from the college until one says he or she lives within five miles of you. What is the probability that you need to contact four people? This is a geometric problem because you may have a number of failures before you have the one success you desire. Also, the probability of a success stays the same each time you ask a student if he or she lives within five miles of you. There is no definite number of trials (number of times you ask a student). a. Let X = the number of ________ you must ask ________ one says yes
. Solution 4.18 a. Let X = the number of students you must ask until one says yes. b. What values does X take on? Solution 4.18 b. 1, 2, 3, . . ., (total number of students) c. What are p and q? Solution 4.18 c. p = .55; q = .45 d. The probability question is P(_______). Solution 4.18 d. P(x = 4) 4.18 You need to find a store that carries a special printer ink. You know that of the stores that carry printer ink, 10 274 Chapter 4 \| Discrete Random Variables percent of them carry the special ink. You randomly call each store until one has the ink you need. What are p and q? Notation for the Geometric: G = Geometric Probability Distribution Function X ~ G(p) Read this as X is a random variable with a geometric distribution. The parameter is p; p = the probability of a success for each trial. Example 4.19 Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. How many components do you expect to test until one is found to be defective? Let X = the number of computer components tested until the first defect is found. X takes on the values 1, 2, 3, . . . where p = .02. X ~ G(.02) Find P(x = 7). There is a formula to define the probability of a geometric distribution P(x) . We can use the formula to find P(x = 7) . But since the calculation is tedious and time consuming, people usually use a graphing calculator or software to get the answer. Using a graphing calculator, you can get P(x = 7) = .0177 . The instruction of TI83, 83+, 84, 84+ is given below. Go into 2nd DISTR. The syntax for the instructions are as follows: To calculate the probability of a value P(x = value), use geometpdf(p, number). Here geometpdf represents geometric probability density function. It is used to find the probability that a geometric random variable is equal to an exact value. p is the probability of a success and number is the value. To calculate the cumulative probability P(x ≤ value), use geometcdf(p, number). Here geometcdf represents geometric cumulative distribution function. It is used to determine the probability of “at most” type of problem, the probability that a geometric random variable is less than or equal to a value. p is the probability of a success and number is the value. To find P(x = 7) , enter 2nd DISTR, arrow down to geometpdf(. Press ENTER. Enter .02,7). The result is P(x = 7) = .0177 . If we need to find P(x ≤ 7) enter 2nd DISTR, arrow down to geometcdf(. Press ENTER. Enter .02,7). The result is (x ≤ = 7) = .1319 . The graph of X ~ G(.02) is This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 275 Figure 4.2 The previous probability distribution histogram gives all the probabilities of X. The x-axis of each bar is the value of X = the number of computer components tested until the first defect is found, and the height of that bar is the probability of that value occurring. For example, the x value of the first bar is 1 and the height of the first bar is 0.02. That means the probability that the first computer components tested is defective is .02. p = 1 .02 The expected value or mean of X is E(X) = μ = 1 = 50 . The variance of X is σ 2 = ( 1 p)( 1 p − 1) = ( 1 .02 )( 1 .02 − 1) = (50)(49) = 2,450 The standard deviation of X is σ = σ 2 = 2,450 = 49.5 Here is how we interpret the mean and standard deviation. The number of components that you would expect to test until you find the first defective one is 50 (which is the mean). And you expect that to vary by about 50 computer components (which is the standard deviation) on average. 4.19 The probability of a defective steel rod is .01. Steel rods are selected at random. Find the probability that the first defect occurs on the ninth steel rod. Use the TI-83+ or TI-84 calculator to find the answer. Example 4.20 The lifetime risk of developing pancreatic cancer is about one in 78 (1.28 percent). Let X = the number of people you ask until one says he or she has pancreatic cancer. Then X is a discrete random variable with a geometric distribution: X ~ G ⎛ ⎝ or X ~ G(.0128). ⎞ ⎠ 1 78 a. What is the probability that you ask 10 people before one says he or she has pancreatic cancer? b. What is the probability that you must ask 20 people? c. Find the (i) mean and (ii) standard deviation of X. 276 Chapter 4 \| Discrete Random Variables Solution 4.20 a. P(x = 10) = geometpdf(.0128, 10) = .0114 b. P(x = 20) = geometpdf(.0128, 20) = .01 c. i. Mean = μ = 1 p = 1 .0128 = 78 ii0128 ⎞ ⎛ ⎝ ⎠ 1 .0128 − 1 ⎞ ⎠ = (78)(78 − 1) = 6,006 = 77.4984 ≈ 77 The number of people whom you would expect to ask until one says he or she has pancreatic cancer is 78. And you expect that to vary by about 77 people on average. 4.20 The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 12 percent. Let X = the number of Afghani women you ask until one says that she is literate. a. What is the probability distribution of X? b. What is the probability that you ask five women before one says she is literate? c. What is the probability that you must ask 10 women? d. Find the (i) mean and (ii) standard deviation of X. 4.5 \| Hypergeometric Distribution (Optional) There are five characteristics of a hypergeometric experiment: 1. You take samples from two groups. 2. You are concerned with a group of interest, called the first group. 3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players. 4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is 13 24 . The probability of picking a man second is 11 23 if a woman was picked first. It is 10 23 if a man was picked first. The probability of the second pick depends on what happened in the first pick. 5. You are not dealing with Bernoulli trials. The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. Example 4.21 A candy dish contains 100 jelly beans and 80 gumdrops. Fifty candies are picked at random. What is the probability that 35 of the 50 are gumdrops? The two groups are jelly beans and gumdrops. Since the probability question asks for the probability of picking gumdrops, the group of interest (first group) is gumdrops. The size of the group of interest (first group) is 80. The size of the second group is 100. The size of the sample is 50 (jelly beans or gumdrops). Let X = the number of gumdrops in the sample of 50. X takes on the values x = 0, 1, 2, . . . , 50. What is the probability statement written mathematically? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 277 Solution 4.21 P(x = 35) 4.21 A bag contains letter tiles. 44 of the tiles are vowels, and 56 are consonants. Seven tiles are picked at random. You want to know the probability that four of the seven tiles are vowels. What is the group of interest, the size of the group of interest, and the size of the sample? Example 4.22 Suppose a shipment of 100 DVD players is known to have 10 defective players. An inspector randomly chooses 12 for inspection. He is interested in determining the probability that, among the 12 players, at most two are defective. The two groups are the 90 non-defective DVD players and the 10 defective DVD players. The group of interest (first group) is the defective group because the probability question asks for the probability of at most two defective DVD players. The size of the sample is 12 DVD players. They may be non-defective or defective. Let X = the number of defective DVD players in the sample of 12. X takes on the values 0, 1, 2, . . . , 10. X may not take on the values 11 or 12. The sample size is 12, but there are only 10 defective DVD players. Write the probability statement mathematically. Solution 4.22 P(x ≤ 2) 4.22 A gross of eggs contains 144 eggs. A particular gross is known to have 12 cracked eggs. An inspector randomly chooses 15 for inspection. She wants to know the probability that, among the 15, at most three are cracked. What is X, and what values does it take on? Example 4.23 You are president of an on-campus special events organization. You need a committee of seven students to plan a special birthday party for the president of the college. Your organization consists of 18 women and 15 men. You are interested in the number of men on your committee. If the members of the committee are randomly selected, what is the probability that your committee has more than four men? This is a hypergeometric problem because you are choosing your committee from two groups (men and women). a. Are you choosing with or without replacement? Solution 4.23 a. without b. What is the group of interest? Solution 4.23 b. the men 278 Chapter 4 \| Discrete Random Variables c. How many are in the group of interest? Solution 4.23 c. 15 men d. How many are in the other group? Solution 4.23 d. 18 women e. Let X = ________ on the committee. What values does X take on? Solution 4.23 e. Let X = the number of men on the committee. x = 0, 1, 2, . . . , 7. f. The probability question is P(_______). Solution 4.23 f. P(x > 4) 4.23 A palette has 200 milk cartons. Of the 200 cartons, it is known that 10 of them have leaked and cannot be sold. A stock clerk randomly chooses 18 for inspection. He wants to know the probability that among the 18, no more than two are leaking. Give five reasons why this is a hypergeometric problem. Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function X ~ H(r, b, n) Read this as X is a random variable with a hypergeometric distribution. The parameters are r, b, and n
: r = the size of the group of interest (first group), b = the size of the second group, n = the size of the chosen sample. Example 4.24 A school site committee is to be chosen randomly from six men and five women. If the committee consists of four members chosen randomly, what is the probability that two of them are men? How many men do you expect to be on the committee? Let X = the number of men on the committee of four. The men are the group of interest (first group). X takes on the values 0, 1, 2, 3, 4, where r = 6, b = 5, and n = 4. X ~ H(6, 5, 4) Find P(x = 2). P(x = 2) = .4545 (calculator or computer) NOTE Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions. There are a number This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 279 of computer packages, including Microsoft Excel, that do. The probability that there are two men on the committee is about .45. The graph of X ~ H(6, 5, 4) is Figure 4.3 The y-axis contains the probability of X, where X = the number of men on the committee. You would expect m = 2.18 (about two) men on the committee. The formula for the mean is μ = nr r + b = (4)(6) 6 + 5 = 2.18. 4.24 An intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has 10 slots. You want to know the probability that eight of the players will be boys. What is the group of interest and the sample? 4.6 \| Poisson Distribution (Optional) There are two main characteristics of a Poisson experiment. 1. The Poisson probability distribution gives the probability of a number of events occurring in a fixed interval of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages. 2. The Poisson distribution may be used to approximate the binomial if the probability of success is small (such as .01) and the number of trials is large (such as 1,000). You will verify the relationship in the homework exercises. n is the number of trials, and p is the probability of a success. The random variable X = the number of occurrences in the interval of interest. Example 4.25 The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in five minutes. The time interval of interest is five minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in five minutes is three? Let X = the number of loaves of bread put on the shelf in five minutes. If the average number of loaves put on the 280 Chapter 4 \| Discrete Random Variables shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in five minutes is ⎛ ⎝ ⎞ ⎠ 5 30 (12) = 2 loaves of bread. The probability question asks you to find P(x = 3). 4.25 The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes? Example 4.26 A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let X = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day. Write a mathematical statement for the probability question. Solution 4.26 P(x < 5) 4.26 An electronics store expects to have 10 returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day. State the probability question mathematically. Example 4.27 You notice that a news reporter says "uh," on average, two times per broadcast. What is the probability that the news reporter says "uh" more than two times per broadcast? This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast. a. What is the interval of interest? Solution 4.27 a. one broadcast b. What is the average number of times the news reporter says "uh" during one broadcast? Solution 4.27 b. 2 c. Let X = ________. What values does X take on? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 281 Solution 4.27 c. Let X = the number of times the news reporter says "uh" during one broadcast. x = 0, 1, 2, 3, . . . d. The probability question is P(______). Solution 4.27 d. P(x > 2) 4.27 An emergency room at a particular hospital gets an average of five patients per hour. A doctor wants to know the probability that the ER gets more than five patients per hour. Give the reason why this would be a Poisson distribution. Notation for the Poisson: P = Poisson Probability Distribution Function X ~ P(μ) Read this as X is a random variable with a Poisson distribution. The parameter is μ (or λ); μ (or λ) = the mean for the interval of interest. Example 4.28 Leah's answering machine receives about six telephone calls between 8 a.m. and 10 a.m. What is the probability that Leah receives more than one call in the next 15 minutes? Let X = the number of calls Leah receives in 15 minutes. The interval of interest is 15 minutes or 1 4 hour. x = 0, 1, 2, 3, . . . If Leah receives, on the average, six telephone calls in two hours, and there are eight 15-minute intervals in two hours, then Leah receives (6) = .75 calls in 15 minutes, on average. So, μ = .75 for this problem(.75) Find P(x > 1). P(x > 1) = .1734 (calculator or computer) NOTE The TI calculators use λ (lambda) for the mean. • Press 1 – and then press 2nd DISTR. • Arrow down to poissoncdf. Press ENTER. • Enter (.75,1). • The result is P(x > 1) = .1734. 282 Chapter 4 \| Discrete Random Variables The probability that Leah receives more than one telephone call in the next 15 minutes is about .1734 or P(x > 1) = 1 − poissoncdf(.75, 1). The graph of X ~ P(.75) is Figure 4.4 The y-axis contains the probability of x where X = the number of calls in 15 minutes. 4.28 A customer service center receives about 10 emails every half-hour. What is the probability that the customer service center receives more than four emails in the next six minutes? Use the TI-83+ or TI-84 calculator to find the answer. Example 4.29 According to Baydin, an email management company, an email user gets, on average, 147 emails per day. Let X = the number of emails an email user receives per day. The discrete random variable X takes on the values x = 0, 1, 2 . . . . The random variable X has a Poisson distribution: X ~ P(147). The mean is 147 emails. a. What is the probability that an email user receives exactly 160 emails per day? b. What is the probability that an email user receives at most 160 emails per day? c. What is the standard deviation? Solution 4.29 a. P(x = 160) = poissonpdf(147, 160) ≈ .0180 b. P(x ≤ 160) = poissoncdf(147, 160) ≈ .8666 c. Standard Deviation = σ = μ = 147 ≈ 12.1244 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 283 4.29 According to a recent poll girls between the ages of 14 and 17 send an average of 187 text messages each day. Let X = the number of texts that a girl aged 14 to 17 sends per day. The discrete random variable X takes on the values x = 0, 1, 2 …. The random variable X has a Poisson distribution: X ~ P(187). The mean is 187 text messages. a. What is the probability that a teen girl sends exactly 175 texts per day? b. What is the probability that a teen girl sends at most 150 texts per day? c. What is the standard deviation? Example 4.30 Text message users receive or send an average of 41.5 text messages per day. a. How many text messages does a text message user receive or send per hour? b. What is the probability that a text message user receives or sends two messages per hour? c. What is the probability that a text message user receives or sends more than two messages per hour? Solution 4.30 a. Let X = the number of texts that a user sends or receives in one hour. The average number of texts received per hour is 41.5 24 ≈ 1.7292. b. X ~ P(1.7292), so P(x = 2) = poissonpdf(1.7292, 2) ≈ .2653 c. P(x > 2) = 1 – P(x ≤ 2) = 1 – poissoncdf(1.7292, 2) ≈ 1 – .7495 = .2505 4.30 Scientists recently researched the busiest airport in the world. On average, there are 2,500 arrivals and departures each day. a. How many airplanes arrive and depart the airport per hour? b. What is the probability that there are exactly 100 arrivals and departures in one hour? c. What is the probability that there are at most 100 arrivals and departures in one hour? Example 4.31 On May 13, 2013, starting at 4:30 p.m., the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02 percent. Use this information for the next 200 days to find the probability that there will be low seismic activity in 10 of the next 200 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close? Solution 4.31 Let X = the number of days with low seismic activity. Using the binomial distribution • P(x = 10) = binompdf(200, .0102, 10) ≈ .000039 Using the Poisson distribution 284 Chapter 4 \| Discrete Random Variables • Calculate μ = np = 200(.0102) ≈ 2.04 • P(x = 10) = poissonpdf(2.04, 10) ≈ .000045 We expect the approximation to be good because n is large (greater than 20) and p is small (less than .05). The results are close—both probabilities reported are almost 0. 4.31 On May 13, 2013, starting at 4:30 p.m., the probability of moderat
e seismic activity for the next 48 hours in the Kuril Islands off the coast of Japan was reported at about 1.43 percent. Use this information for the next 100 days to find the probability that there will be low seismic activity in 5 of the next 100 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close? 4.7 \| Discrete Distribution (Playing Card Experiment) This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 285 4.1 Discrete Distribution (Playing Card Experiment) Student Learning Outcomes • The student will compare empirical data and a theoretical distribution to determine if an everyday experiment fits a discrete distribution. • The student will compare technology-generated simulation and a theoretical distribution. • The student will demonstrate an understanding of long-term probabilities. Supplies • One full deck of playing cards • Programmable calculator Procedure for Empirical Data The experimental procedure for empirical data is to pick one card from a deck of shuffled cards. 1. The theoretical probability of picking a diamond from a deck is ________. 2. Shuffle a deck of cards. 3. Pick one card from it. 4. Record whether it was a diamond or not a diamond. 5. Put the card back and reshuffle. 6. Do this a total of 10 times. 7. Record the number of diamonds picked. 8. Let X = number of diamonds. Theoretically, X ~ B(_____,_____) Procedure for Simulation Repeat the experimental procedure using a programmable calculator. 1. Use the randInt function to generate data. Consider 1 to be spades, 2 to be hearts, 3 to be diamonds, and 4 to be clubs. Generate 10 draws of cards with four suits with randInt(1,4,10). 2. Let X = number of diamonds . Theoretically, X ~ B(_____,_____). Organize the Empirical Data 1. Record the number of diamonds picked for your class with playing cards in Table 4.15. Then calculate the relative frequency. x Frequency Relative Frequency 0 1 2 3 4 5 __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ 286 Chapter 4 \| Discrete Random Variables x 6 7 8 9 Frequency Relative Frequency __________ __________ __________ __________ __________ __________ __________ __________ 10 __________ __________ Table 4.15 2. Calculate the following: a. b. x¯ = ________ s = ________ 3. Construct a histogram of the empirical data. Figure 4.5 Organize the Simulation Data 1. Use Table 4.16 to record the number of diamonds picked for your class using the calculator simulation. Calculate the relative frequency. X Frequency Relative Frequency 0 1 2 3 4 5 6 __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ Table 4.16 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 287 X Frequency Relative Frequency 7 8 9 __________ __________ __________ __________ __________ __________ 10 __________ __________ Table 4.16 2. Calculate the following: a. b. x¯ = ________ s = ________ 3. Construct a histogram of the simulation data. Figure 4.6 Theoretical Distribution a. Build the theoretical PDF chart based on the distribution in the Procedure section. x P(x 288 Chapter 4 \| Discrete Random Variables x P(x) 9 10 Table 4.17 b. Calculate the following: a. μ = ____________ b. σ = ____________ c. Construct a histogram of the theoretical distribution. Figure 4.7 Using the Data NOTE RF = relative frequency Use the table from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places. • P(x = 3) = ________ • P(1 < x < 4) = ________ • P(x ≥ 8) = ________ Use the data from the Organize the Empirical Data section to calculate the following answers. Round your answers to four decimal places. • RF(x = 3) = ________ • RF(1 < x < 4) = ________ • RF(x ≥ 8) = ________ Use the data from the Organize the Simulation Data section to calculate the following answers. Round your answers to four decimal places. • RF(x = 3) = ________ This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 289 • RF(1 < x < 4) = ________ • RF(x ≥ 8) = ________ Discussion Questions For Questions 1 and 2, think about the shapes of the two graphs, the probabilities, the relative frequencies, the means, and the standard deviations. 1. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical, empirical, and simulation distributions. Use complete sentences. 2. Describe the three most significant differences between the graphs or distributions of the theoretical, empirical, and simulation distributions. 3. Using your answers from Questions 1 and 2, does it appear that the two sets of data fit the theoretical distribution? In complete sentences, explain why or why not. 4. Suppose that the experiment had been repeated 500 times. Would you expect Table 4.15, Table 4.16, or Table 4.17 to change, and how would it change? Why? Why wouldn’t the other table(s) change? 4.8 \| Discrete Distribution (Lucky Dice Experiment) 290 Chapter 4 \| Discrete Random Variables 4.2 Discrete Distribution (Lucky Dice Experiment) Student Learning Outcomes • The student will compare empirical data and a theoretical distribution to determine if a Tet gambling game fits a discrete distribution. • The student will demonstrate an understanding of long-term probabilities. Supplies • One “Lucky Dice” game or three regular dice • One programming calculator Procedure Round answers to relative frequency and probability problems to four decimal places. 1. The experimental procedure is to bet on one object. Then, roll three Lucky Dice and count the number of matches. The number of matches will decide your profit. 2. What is the theoretical probability of one die matching the object? 3. Choose one object to place a bet on. Roll the three Lucky Dice. Count the number of matches. 4. Let X = number of matches. Theoretically, X ~ B(______,______) 5. Let Y = profit per game. Organize the Data In Table 4.18, fill in the y-value that corresponds to each x-value. Next, record the number of matches picked for your class. Then, calculate the relative frequency. 1. Complete the table. y Frequency Relative Frequency x 0 1 2 3 Table 4.18 2. Calculate the following: a. b. c. d. x¯ = _______ sx = ________ y¯ = _______ sy = _______ 3. Explain what x¯ represents. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 291 4. Explain what y¯ represents. 5. Based upon the experiment, answer the following questions: a. What was the average profit per game? b. Did this represent an average win or loss per game? c. How do you know? Answer in complete sentences. 6. Construct a histogram of the empirical data. Figure 4.8 Theoretical Distribution Build the theoretical PDF chart for x and y based on the distribution from the Procedure section. 1. y P(x) = P(y) x 0 1 2 3 Table 4.19 2. Calculate the following: a. μx = ________ b. σx = ________ c. μx = ________ 3. Explain what μx represents. 4. Explain what μy represents. 5. Based upon theory, answer the following questions: a. What was the expected profit per game? b. Did the expected profit represent an average win or loss per game? c. How do you know? Answer in complete sentences. 292 Chapter 4 \| Discrete Random Variables 6. Construct a histogram of the theoretical distribution. Figure 4.9 Use the Data NOTE RF = relative frequency Use the data from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places. 1. P(x = 3) = ________ 2. P(0 < x < 3) = ________ 3. P(x ≥ 2) = ________ Use the data from the Organize the Data section to calculate the following answers. Round your answers to four decimal places. 1. RF(x = 3) = ________ 2. RF(0 < x < 3) = ________ 3. RF(x ≥ 2) = ________ Discussion Question For Questions 1 and 2, consider the graphs, the probabilities, the relative frequencies, the means, and the standard deviations. 1. Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences. 2. Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions. 3. Thinking about your answers to Questions 1 and 2, does it appear that the data fit the theoretical distribution? In complete sentences, explain why or why not. 4. Suppose that the experiment had been repeated 500 times. Would you expect Table 4.18 or Table 4.19 to change, and how would it change? Why? Why wouldn’t the other table change? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 293 KEY TERMS Bernoulli trials an experiment with the following characteristics: 1. There are only two possible outcomes called success and failure for each trial 2. The probability p of a success is the same for any trial (so the probability q = 1 − p of a failure is the same for any trial) binomial experiment a statistical experiment that satisfies the following three conditions: 1. There are a fixed number of trials, n 2. There are only two possible outcomes, called success and, failure, for each trial; the letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial 3. The n trials are independent and are repeated using identical conditions a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed binomial probability distribution number, n, of independent trials Independent means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted u
nder the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X ~ B(n, p). The mean is μ = np and the standard deviation is σ = npq . The probability of the following exactly x successes in n trials is n P(X = x) = ⎛ ⎝ x ⎞ ⎠ pxqn − x expected value expected arithmetic average when an experiment is repeated many times; also called the mean; notations μ; for a discrete random variable (RV) with probability distribution function P(x),the definition can also be written in the form μ = ∑ xP(x) geometric distribution until the first success. The geometric variable X is defined as the number of trials until the first success. Notation X ~ G(p). The mean is μ a discrete random variable (RV) that arises from the Bernoulli trials; the trials are repeated = 1 p and the standard deviation is ⎞ ⎠ . The probability of exactly x failures before the first success is given by the formula . P(X = x) = p(1 – p) x – 1 geometric experiment a statistical experiment with the following properties: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success 2. In theory, the number of trials could go on foreve; there must be at least one trial 3. The probability, p, of a success and the probability, q, of a failure do not change from trial to trial hypergeometric experiment a statistical experiment with the following properties: 1. You take samples from two groups 2. You are concerned with a group of interest, called the first group 3. You sample without replacement from the combined groups 4. Each pick is not independent, since sampling is without replacement 5. You are not dealing with Bernoulli trials hypergeometric probability a discrete random variable (RV) that is characterized by the following: 1. The experiment uses a fixed number of trials. 2. The probability of success is not the same from trial to trial We sample from two groups of items when we are interested in only one group. X is defined as the number of 294 Chapter 4 \| Discrete Random Variables successes out of the total number of items chosen. Notation X ~ H(r, b, n), where r = the number of items in the group of interest, b = the number of items in the group not of interest, and n = the number of items chosen. mean a number that measures the central tendency; a common name for mean is average The term mean is a shortened form of arithmetic mean. By definition, the mean for a sample (denoted by x¯ ) is x¯ = is μ = and the mean for a population (denoted by μ) Sum of all values in the sample Number of values in the sample Sum of all values in the population Number of values in the population . mean of a probability distribution the long-term average of many trials of a statistical experiment Poisson probability distribution a discrete random variable (RV) that counts the number of times a certain event will occur in a specific interval; characteristics of the variable: • The probability that the event occurs in a given interval is the same for all intervals • The events occur with a known mean and independently of the time since the last event The distribution is defined by the mean μ of the event in the interval. Notation X ~ P(μ). The mean is μ = np. The μ x x ! standard deviation is σ = μ . The probability of having exactly x successes in r trials is P(X = x) = (e −μ ) . The Poisson distribution is often used to approximate the binomial distribution, when n is large and p is small (a general rule is that n should be greater than or equal to 20 and p should be less than or equal to .05). probability distribution function (PDF) a mathematical description of a discrete random variable (RV), given either in the form of an equation (formula) or in the form of a table listing all the possible outcomes of an experiment and the probability associated with each outcome random variable (RV) a characteristic of interest in a population being studied; common notation for variables are uppercase Latin letters X, Y, Z, . . . ; common notation for a specific value from the domain (set of all possible values of a variable) are lowercase Latin letters x, y, and z For example, if X is the number of children in a family, then x represents a specific integer 0, 1, 2, 3, . . . ; variables in statistics differ from variables in intermediate algebra in the two following ways: • The domain of the random variable (RV) is not necessarily a numerical set; the domain may be expressed in words; for example, if X = hair color then the domain is {black, blond, gray, green, orange} • We can tell what specific value x the random variable X takes only after performing the experiment standard deviation of a probability distribution experiment are from the mean of the distribution a number that measures how far the outcomes of a statistical the law of large numbers as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency probability approaches zero CHAPTER REVIEW 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable The characteristics of a probability distribution function (PDF) for a discrete random variable are as follows: 1. Each probability is between zero and one, inclusive (inclusive means to include zero and one) 2. The sum of the probabilities is one 4.2 Mean or Expected Value and Standard Deviation The expected value, or mean, of a discrete random variable predicts the long-term results of a statistical experiment that has been repeated many times. The standard deviation of a probability distribution is used to measure the variability of possible outcomes. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 295 4.3 Binomial Distribution (Optional) A statistical experiment can be classified as a binomial experiment if the following conditions are met: 1. There are a fixed number of trials, n 2. There are only two possible outcomes, called success and failure, for each trial; the letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial 3. The n trials are independent and are repeated using identical conditions The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X = the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ = np, and the standard deviation is given by the formula σ = npq . 4.4 Geometric Distribution (Optional) There are three characteristics of a geometric experiment: 1. There are one or more Bernoulli trials with all failures except the last one, which is a success 2. In theory, the number of trials could go on forever; there must be at least one trial 3. The probability, p, of a success and the probability, q, of a failure are the same for each trial In a geometric experiment, define the discrete random variable X as the number of independent trials until the first success. We say that X has a geometric distribution and write X ~ G(p) where p is the probability of success in a single trial. The mean of the geometric distribution X ~ G(p) is μ = 1 − p p2 = .5 Hypergeometric Distribution (Optional) A hypergeometric experiment is a statistical experiment with the following properties: 1. You take samples from two groups 2. You are concerned with a group of interest, called the first group 3. You sample without replacement from the combined groups 4. Each pick is not independent, since sampling is without replacement 5. You are not dealing with Bernoulli trials The outcomes of a hypergeometric experiment fit a hypergeometric probability distribution. The random variable X = the number of items from the group of interest. The distribution of X is denoted X ~ H(r, b, n), where r = the size of the group of interest (first group), b = the size of the second group, and n = the size of the chosen sample. It follows that n ≤ r + b. The mean of X is μ = nr r + b and the standard deviation is σ = rbn(r + b − n) (r + b)2 (r + b − 1) . 4.6 Poisson Distribution (Optional) A Poisson probability distribution of a discrete random variable gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen at a known average rate and independently of the time since the last event. The Poisson distribution may be used to approximate the binomial, if the probability of success is small (less than or equal to .05) and the number of trials is large (greater than or equal to 20). FORMULA REVIEW 296 Chapter 4 \| Discrete Random Variables 4.2 Mean or Expected Value and Standard Deviation Mean or Expected Value: μ = ∑ x ∈ X xP(x) Standard Deviation: σ = ∑ x ∈ X (x − μ)2 P(x) 4.3 Binomial Distribution (Optional) X ~ B(n, p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p. X = the number of successes in n independent trials n = the number of independent trials X takes on the values x = 0, 1, 2, 3, . . . , n p = the probability of a success for any trial q = the probability of a failure for any trial The mean of X is μ = np. The standard deviation of X is σ = npq . 4.4 Geometric Distribution (Optional) X ~ G(p) means that the discrete random variable X has a geometric probability distribution with probability of success in a single trial p. X = the number of independent trials until the first success X takes on the values x = 1, 2, 3, . . . p = the probability of a success for any trial q = the probability of a failure for any trial PRACTICE The mean is μ = 1 p . The standard deviation is σ = 1 – p p2 = .5 Hypergeometric Distribution (Optional) X ~ H(r, b, n) means that the discrete random variable X has a hypergeometric probability distribution with r = the size of the group of interest (first group), b = the size of the second group,
and n = the size of the chosen sample. X = the number of items from the group of interest that are in the chosen sample, and X may take on the values x = 0, 1, . . . , up to the size of the group of interest. The minimum value for X may be larger than zero in some instances. n ≤ r + b The mean of X is given by the formula μ = nr r + b and the standard deviation is = rbn(r + b − n) (r + b)2(r + b − 1) . 4.6 Poisson Distribution (Optional) X ~ P(μ) means that X has a Poisson probability distribution where X = the number of occurrences in the interval of interest. X takes on the values x = 0, 1, 2, 3, . . . The mean μ is typically given. The variance is σ2 = μ, and the standard deviation is σ = μ . When P(μ) is used to approximate a binomial distribution, μ = np where n represents the number of independent trials and p represents the probability of success in a single trial. 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution: Let X = the number of years a new hire will stay with the company. Let P(x) = the probability that a new hire will stay with the company x years. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 297 1. Complete Table 4.20 using the data provided. x P(x) 0 1 2 3 4 5 6 .12 .18 .30 .15 .10 .05 Table 4.20 2. P(x = 4) = ________ 3. P(x ≥ 5) = ________ 4. On average, how long would you expect a new hire to stay with the company? 5. What does the column “P(x)” sum to? Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution. x P(x) 1 2 3 4 .15 .35 .40 .10 Table 4.21 6. Define the random variable X. 7. What is the probability the baker will sell more than one batch? P(x > 1) = ________ 8. What is the probability the baker will sell exactly one batch? P(x = 1) = ________ 9. On average, how many batches should the baker make? Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random. 10. Define the random variable X. 11. Construct a probability distribution table for the data. 12. We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic? Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent 298 Chapter 4 \| Discrete Random Variables of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time. 13. Define the random variable X. 14. What values does x take on? 15. Construct a PDF table. 16. Find the probability that Javier volunteers for fewer than three events each month. P(x < 3) = ________ 17. Find the probability that Javier volunteers for at least one event each month. P(x > 0) = ________ 4.2 Mean or Expected Value and Standard Deviation 18. Complete the expected value table. x P(x) xP(x) 0 1 2 3 .2 .2 .4 .2 Table 4.22 19. Find the expected value from the expected value table. 20. Find the standard deviation. x P(x) xP(x) 2 4 6 8 .1 .3 .4 .2 2(.1) = .2 4(.3) = 1.2 6(.4) = 2.4 8(.2) = 1.6 Table 4.23 x P(x) xP(x) (x – μ)2P(x) 2 4 6 8 0.1 0.3 0.4 0.2 2(.1) = .2 (2–5.4)2(.1) = 1.156 4(.3) = 1.2 (4–5.4)2(.3) = .588 6(.4) = 2.4 (6–5.4)2(.4) = .144 8(.2) = 1.6 (8–5.4)2(.2) = 1.352 Table 4.24 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 299 21. Identify the mistake in the probability distribution table. x P(x) xP(x) 1 2 3 4 5 .15 .25 .30 .20 .15 .15 .50 .90 .80 .75 Table 4.25 22. Identify the mistake in the probability distribution table. x P(x) xP(x) 1 2 3 4 5 .15 .25 .25 .20 .15 .15 .40 .65 .85 1 Table 4.26 Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution: x P(x) xP(x) 1 2 3 4 5 6 .35 .20 .15 .10 .05 Table 4.27 23. Define the random variable X. 24. Define P(x), or the probability of x. 25. Find the probability that a physics major will do postgraduate research for four years. P(x = 4) = ________ 26. Find the probability that a physics major will do postgraduate research for at most three years. P(x ≤ 3) = ________ 27. On average, how many years would you expect a physics major to spend doing postgraduate research? Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution: • Let X = the number of years a student will study ballet with the teacher. 300 Chapter 4 \| Discrete Random Variables • Let P(x) = the probability that a student will study ballet x years. 28. Complete Table 4.28 using the data provided. x P(x) xP(x) 1 2 3 4 5 6 7 .10 .05 .10 .30 .20 .10 Table 4.28 29. In words, define the random variable X. 30. P(x = 4) = ________ 31. P(x < 4) = ________ 32. On average, how many years would you expect a child to study ballet with this teacher? 33. What does the column P(x) sum to and why? 34. What does the column xP(x) sum to and why? 35. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. What is the expected value of playing the game? 36. You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. Should you play the game? 4.3 Binomial Distribution (Optional) Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming firsttime, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law. 37. In words, define the random variable X. 38. X ~ _____(_____,_____) 39. What values does the random variable X take on? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 301 40. Construct the probability distribution function (PDF). x P(x) Table 4.29 41. On average (μ), how many would you expect to answer yes? 42. What is the standard deviation (σ)? 43. What is the probability that at most five of the freshmen reply yes? 44. What is the probability that at least two of the freshmen reply yes? 4.4 Geometric Distribution (Optional) Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming firsttime, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask. 45. In words, define the random variable X. 46. X ~ _____(_____,_____) 47. What values does the random variable X take on? 48. Construct the probability distribution function (PDF). Stop at x = 6. x P(x) 1 2 3 4 5 6 Table 4.30 49. On average (μ), how many freshmen would you expect to have to ask until you found one who replies yes? 50. What is the probability that you will need to ask fewer than three freshmen? 4.5 Hypergeometric Distribution (Optional) Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into 302 Chapter 4 \| Discrete Random Variables two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample. 51. In words, define the random variable X. 52. X ~ _____(_____,_____) 53. What values does X take on? 54. Find the standard deviation. 55. On average (μ), how many would you expect to be business majors? 4.6 Poisson Distribution (Optional) Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day. 56. Assume the event occurs independently in any given day. Define the random variable X. 57. What values does X take on? 58. What is the probability of getting 150 customers in one day? 59. What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day. 60. What is the probability that the store will have more than 12 customers in the first hour? 61. What is the probability that the store will have fewer than 12 customers in the first two hours? 62. Which type of distribution can the Poisson model
be used to approximate? When would you do this? Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age. 63. Assume the event occurs independently in any given day. In words, define the random variable X. 64. X ~ _____(_____,_____) 65. What values does X take on? 66. For the given values of the random variable X, fill in the corresponding probabilities. 67. Is it likely that there will be no teens killed from motor vehicle injuries on any given day in the United States? Justify your answer numerically. 68. Is it likely that there will be more than 20 teens killed from motor vehicle injuries on any given day in the United States? Justify your answer numerically. HOMEWORK This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 303 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable 69. Suppose that the PDF for the number of years it takes to earn a bachelor of science (B.S.) degree is given in Table 4.31. x P(x) 3 4 5 6 7 .05 .40 .30 .15 .10 Table 4.31 In words, define the random variable X. a. b. What does it mean that the values 0, 1, and 2 are not included for x in the PDF? 4.2 Mean or Expected Value and Standard Deviation 70. A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150. In words, define the random variable X. a. What are you interested in here? b. c. List the values that X may take on. d. Construct a PDF. e. If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle? 71. A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails. • • • If the card is a face card, and the coin lands on heads, you win $6. If the card is a face card, and the coin lands on tails, you win $2. If the card is not a face card, you lose $2, no matter what the coin shows. a. Find the expected value for this game (expected net gain or loss). b. Explain what your calculations indicate about your long-term average profits and losses on this game. c. Should you play this game to win money? 72. You buy a ticket to a raffle that costs $10 per ticket. There are only 100 tickets available to be sold in this raffle. In this raffle there are one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss. 73. Complete the PDF and answer the questions. x P(x) xP(x) 0 1 2 3 .3 .2 .4 Table 4.32 a. Find the probability that x = 2. b. Find the expected value. 304 Chapter 4 \| Discrete Random Variables 74. Suppose that you are offered the following deal: You roll a die. If you roll a six, you win $10. If you roll a four or five, you win $5. If you roll a one, two, or three, you pay $6. In words, define the random variable X. a. What are you ultimately interested in here (the value of the roll or the money you win)? b. c. List the values that X may take on. d. Construct a PDF. e. Over the long run of playing this game, what are your expected average winnings per game? f. Based on numerical values, should you take the deal? Explain your decision in complete sentences. 75. A venture capitalist, willing to invest $1,000,000, has three investments to choose from: The first investment, a software company, has a 10 percent chance of returning $5,000,000 profit, a 30 percent chance of returning $1,000,000 profit, and a 60 percent chance of losing the million dollars. The second company, a hardware company, has a 20 percent chance of returning $3,000,000 profit, a 40 percent chance of returning $1,000,000 profit, and a 40 percent chance of losing the million dollars. The third company, a biotech firm, has a 10 percent chance of returning $6,000,000 profit, a 70 percent of no profit or loss, and a 20 percent chance of losing the million dollars. a. Construct a PDF for each investment. b. Find the expected value for each investment. c. Which is the safest investment? Why do you think so? d. Which is the riskiest investment? Why do you think so? e. Which investment has the highest expected return, on average? 76. Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children married people have. x 0 1 2 3 4 5 P(x) xP(x) .10 .20 .30 .10 .05 6 (or more) .05 Table 4.33 In words, what does the expected value in this example represent? a. Find the probability that a married adult has three children. b. c. Find the expected value. d. Is it more likely that a married adult will have two to three children or four to six children? How do you know? This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 305 77. Suppose that the PDF for the number of years it takes to earn a bachelor of science (B.S.) degree is given as in Table 4.34. x P(x) 3 4 5 6 7 .05 .40 .30 .15 .10 Table 4.34 On average, how many years do you expect it to take for an individual to earn a B.S.? 306 Chapter 4 \| Discrete Random Variables 78. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video to Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs. x P(x) 0 1 2 3 4 5 .03 .50 .24 .70 .04 Table 4.35 a. Describe the random variable X in words. b. Find the probability that a customer rents three DVDs. c. Find the probability that a customer rents at least four DVDs. d. Find the probability that a customer rents at most two DVDs. Another shop, Entertainment Headquarters, rents DVDs and video games. The probability distribution for DVD rentals per customer at this shop is given as follows. They also have a five-DVD limit per customer. x P(x) 0 1 2 3 4 5 .35 .25 .20 .10 .05 .05 Table 4.36 e. At which store is the expected number of DVDs rented per customer higher? f. If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form. If Video to Go expects 300 customers next week, and Entertainment Headquarters projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain. g. h. Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that? 79. A “friend” offers you the following deal: For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift. • Ten of the coupons are for a free gift worth $6. • Eighty of the coupons are for a free gift worth $8. • Six of the coupons are for a free gift worth $12. • Four of the coupons are for a free gift worth $40. Based upon the financial gain or loss over the long run, should you play the game? a. Yes, I expect to come out ahead in money. b. No, I expect to come out behind in money. It doesn’t matter. I expect to break even. c. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 307 80. A university has 14 statistics classes scheduled for its Summer 2013 term. One class has space available for 30 students, eight classes have space for 60 students, one class has space for 70 students, and four classes have space for 100 students. a. What is the average class size assuming each class is filled to capacity? b. Space is available for 980 students. Suppose that each class is filled to capacity and select a statistics student at random. Let the random variable X equal the size of the student’s class. Define the PDF for X. c. Find the mean of X. d. Find the standard deviation of X. 81. In a raffle, there are 250 prizes of $5, 50 prizes of $25, and 10 prizes of $100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to charge to break even? 4.3 Binomial Distribution (Optional) 82. According to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per 1,000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1,000 babies in an intensive care nursery. Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf. Use the following information to answer the next four exercises: Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. 83. Define the random variable and list its possible values. 84. State the distribution of X. 85. Find the probability that at least four of the 25 patients actually have the flu. 86. On average, for every 25 patients calling in, how many do you expect to have the flu? 87. People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video to Go is given Table 4.37. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs. x P(x) 0 1 2 3 4 5 .03 .50 .24 .07 .04 Table 4.37 a. Describe the random variable X in words. b. Find the probability that a customer rents three DVDs. c. Find the probability that a customer rents at least four DVDs. d. Find the probability that a customer rents at most two DVDs. 3
08 Chapter 4 \| Discrete Random Variables 88. A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18 percent of students attend Tet festivities. We are interested in the number of students who will attend the festivities. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many of the 12 students do we expect to attend the festivities? e. Find the probability that at most four students will attend. f. Find the probability that more than two students will attend. Use the following information to answer the next three exercises: The probability that a local hockey team will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. 89. What is the expected number of wins for that upcoming month? a. 1.67 b. 12 c. 382 1043 d. 4.43 Let X = the number of games won in that upcoming month. 90. What is the probability that the team wins six games in that upcoming month? a. b. c. d. .1476 .2336 .7664 .8903 91. What is the probability that the team wins at least five games in that upcoming month a. b. c. d. .3694 .5266 .4734 .2305 92. A student takes a 10-question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70 percent of the questions correct. 93. A student takes a 32-question multiple choice exam, but did not study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the student guesses more than 75 percent of the questions correctly. 94. Six different colored dice are rolled. Of interest is the number of dice that show a one. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. On average, how many dice would you expect to show a one? e. Find the probability that all six dice show a one. f. Is it more likely that three or that four dice will show a one? Use numbers to justify your answer numerically. In words, define the random variable X. 95. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. On average, how many schools would you expect to offer such courses? e. Find the probability that at most 10 offer such courses. f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 309 96. Suppose that about 85 percent of graduating students attend their graduation. A group of 22 graduating students is randomly chosen. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to attend their graduation? e. Find the probability that 17 or 18 attend. f. Based on numerical values, would you be surprised if all 22 attended graduation? Justify your answer numerically. 97. At the Fencing Center, 60 percent of the fencers use the foil as their main weapon. We randomly survey 25 fencers at the Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to not to use the foil as their main weapon? e. Find the probability that six do not use the foil as their main weapon. f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically. 98. Approximately 8 percent of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-school sports all four years of high school. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many seniors are expected to have participated in after-school sports all four years of high school? e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all In words, define the random variable X. four years of high school? Justify your answer numerically. f. Based upon numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. In words, define the random variable X. 99. The chance of an IRS audit for a tax return reporting more than $25,000 in income is about 2 percent per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many audits are expected in a 20-year period? e. Find the probability that a person is not audited at all. f. Find the probability that a person is audited more than twice. 100. It has been estimated that only about 30 percent of California residents have adequate earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. What is the probability that at least eight have adequate earthquake supplies? e. f. How many residents do you expect will have adequate earthquake supplies? Is it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why? 310 Chapter 4 \| Discrete Random Variables 101. There are two similar games played for Chinese New Year and Vietnamese New Year. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The house rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his or her $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his or her $1 bet, plus $3 profit. Let X = number of matches and Y = profit per game. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities. e. Calculate the average expected matches over the long run of playing this game for the player. f. Calculate the average expected earnings over the long run of playing this game for the player. g. Determine who has the advantage, the player or the house. 102. According to the World Bank, only 9 percent of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 150 people in Uganda. Let X = the number of people who have access to electricity. a. What is the probability distribution for X? b. Using the formulas, calculate the mean and standard deviation of X. c. Use your calculator to find the probability that 15 people in the sample have access to electricity. d. Find the probability that at most 10 people in the sample have access to electricity. e. Find the probability that more than 25 people in the sample have access to electricity. 103. The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate in Afghanistan is 28.1 percent. Suppose you choose 15 people in Afghanistan at random. Let X = the number of people who are literate. a. Sketch a graph of the probability distribution of X. b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X. c. Find the probability that more than five people in the sample are literate. Is it more likely that three people or four people are literate? 4.4 Geometric Distribution (Optional) 104. A consumer looking to buy a used red sports car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28 percent. We are interested in the number of dealerships she must call. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. On average, how many dealerships would we expect her to have to call until she finds one that has the car? e. Find the probability that she must call at most four dealerships. f. Find the probability that she must call three or four dealerships. 105. Suppose that the probability that an adult in America will watch the Super Bowl is 40 percent. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bo
wl. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl? e. Find the probability that you must ask seven people. f. Find the probability that you must ask three or four people. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 311 106. It has been estimated that only about 30 percent of California residents have adequate earthquake supplies. Suppose we are interested in the number of California residents we must survey until we find a resident who does not have adequate earthquake supplies. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. What is the probability that we must survey just one or two residents until we find a California resident who does not have adequate earthquake supplies? e. What is the probability that we must survey at least three California residents until we find a California resident who does not have adequate earthquake supplies? f. How many California residents do you expect to need to survey until you find a California resident who does not have adequate earthquake supplies? g. How many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies? 107. In one of its spring catalogs, a retailer advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked more than once. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many pages do you expect to advertise footwear on them? e. Is it probable that all 20 will advertise footwear on them? Why or why not? f. What is the probability that fewer than 10 will advertise footwear on them? g. Reminder: A page may be picked more than once. We are interested in the number of pages that we must randomly survey until we find one that has footwear advertised on it. Define the random variable X and give its distribution. h. What is the probability that you only need to survey at most three pages in order to find one that advertises footwear on it? i. How many pages do you expect to need to survey in order to find one that advertises footwear? 108. Suppose that you are performing the probability experiment of rolling one fair six-sided die. Let F be the event of rolling a four or a five. You are interested in how many times you need to roll the die to obtain the first four or five as the outcome. • p = probability of success (event F occurs) • q = probability of failure (event F does not occur) a. Write the description of the random variable X. b. What are the values that X can take on? c. Find the values of p and q. d. Find the probability that the first occurrence of event F (rolling a four or five) is on the second trial. 109. Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random. What values does X take on? 110. Researchers investigate the prevalence of a particular infectious disease in countries around the world. According to their data, “Prevalence of this disease refers to the percentage of people ages 15 to 49 who are infected with it.” In South Africa, the prevalence of this disease is 17.3 percent. Let X = the number of people you test until you find a person infected with this disease. a. Sketch a graph of the distribution of the discrete random variable X. b. What is the probability that you must test 30 people to find one with this disease? c. What is the probability that you must ask 10 people? d. Find the (i) mean and (ii) standard deviation of the distribution of X. 312 Chapter 4 \| Discrete Random Variables 111. According to a recent poll, 75 percent of millennials (people born between 1981 and 1995) have a profile on a social networking site. Let X = the number of millennials you ask until you find a person without a profile on a social networking site. a. Describe the distribution of X. b. Find the (i) mean and (ii) standard deviation of X. c. What is the probability that you must ask 10 people to find one person without a social networking site? d. What is the probability that you must ask 20 people to find one person without a social networking site? e. What is the probability that you must ask at most five people? 4.5 Hypergeometric Distribution (Optional) 112. A group of martial arts students is planning on participating in an upcoming demonstration. Six are students of tae kwon do, and seven are students of shotokan karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of shotokan karate students in that first demonstration. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many shotokan karate students do we expect to be in that first demonstration? 113. In one of its spring catalogs, a retailer advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many pages do you expect to advertise footwear on them? e. Calculate the standard deviation. 114. Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that 10 people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many instructors do you expect on the committee who are not technically proficient? e. Find the probability that at least five on the committee are not technically proficient. f. Find the probability that at most three on the committee are not technically proficient. 115. Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the local basketball team and four volunteers from the local football team. We are interested in the number of football players picked. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. Are you choosing the nine athletes with or without replacement? 116. A bridge hand is defined as 13 cards selected at random and without replacement from a deck of 52 cards. In a standard deck of cards, there are 13 cards from each suit: hearts, spades, clubs, and diamonds. What is the probability of being dealt a hand that does not contain a heart? a. What is the group of interest? b. How many are in the group of interest? c. How many are in the other group? d. Let X = _________. What values does X take on? e. The probability question is P(_______). f. Find the probability in question. g. Find the (i) mean and (ii) standard deviation of X. This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 313 4.6 Poisson Distribution (Optional) 117. The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let X = the number of calls received at noon. a. Find the mean and standard deviation of X. b. What is the probability that the office receives at most six calls at noon on Monday? c. Find the probability that the law office receives six calls at noon. What does this mean to the law office staff who get, on average, 5.5 incoming phone calls at noon? d. What is the probability that the office receives more than eight calls at noon? 118. The maternity ward at a hospital in the Philippines is one of the busiest in the world with an average of 60 births per day. Let X = the number of births in an hour. a. Find the mean and standard deviation of X. b. Sketch a graph of the probability distribution of X. c. What is the probability that the maternity ward will deliver three babies in one hour? d. What is the probability that the maternity ward will deliver at most three babies in one hour? e. What is the probability that the maternity ward will deliver more than five babies in one hour? 119. A manufacturer of decorative string lights knows that 3 percent of its bulbs are defective. Using both the binomial and Poisson distributions, find the probability that a string of 100 lights contains at most four defective bulbs. 120. The average number of children a Japanese woman has in her lifetime is 1.37. Suppose that one Japanese woman is randomly chosen. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. Find the probability that she has no children. e. Find the probability that she has fewer children than the Japanese average. f. Find the probability that she has more children than the Japanese average. 121. The average number of children a Spanish woman has in her lifetime is 1.47. Suppose that one Spanish woman is randomly chosen. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribut
ion of X. X ~ _____(_____,_____) d. Find the probability that she has no children. e. Find the probability that she has fewer children than the Spanish average. f. Find the probability that she has more children than the Spanish average. 122. Fertile, female cats produce an average of three litters per year. Suppose that one fertile, female cat is randomly chosen. Answer the questions about the cat's probability of litters in one year. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _______ d. Find the probability that she has no litters in one year. e. Find the probability that she has at least two litters in one year. f. Find the probability that she has exactly three litters in one year. 123. The chance of having an extra fortune in a fortune cookie is about 3 percent. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem, but only use one distribution to solve the problem. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many cookies do we expect to have an extra fortune? e. Find the probability that none of the cookies have an extra fortune. f. Find the probability that more than three have an extra fortune. g. As n increases, what happens involving the probabilities using the two distributions? Explain in complete sentences. 314 Chapter 4 \| Discrete Random Variables 124. According to the South Carolina Department of Mental Health website, for every 200 U.S. women, the average number who suffer from a particular disease is one. Out of a randomly chosen group of 600 U.S. women. Determine the following: In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to suffer from this disease? e. Find the probability that no one suffers from this disease. f. Find the probability that more than four suffer from this disease. 125. The chance of an IRS audit for a tax return reporting more than $25,000 in income is about 2 percent per year. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in one year. Use a Poisson distribution to anwer the following questions. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many are expected to be audited? e. Find the probability that no one was audited. f. Find the probability that at least three were audited. 126. Approximately 8 percent of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-school sports all four years of high school. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many seniors are expected to have participated in after-school sports all four years of high school? e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. f. Based on numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically. 127. On average, Pierre, an amateur chef, drops three pieces of eggshell into every two cake batters he makes. Suppose that you buy one of his cakes. In words, define the random variable X. a. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. On average, how many pieces of eggshell do you expect to be in the cake? e. What is the probability that there will not be any pieces of eggshell in the cake? f. Let’s say that you buy one of Pierre’s cakes each week for six weeks. What is the probability that there will not be any eggshell in any of the cakes? g. Based upon the average given for Pierre, is it possible for there to be seven pieces of shell in the cake? Why? Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week. 128. In words, what is the random variable X? a. b. c. d. the number of times Mrs. Plum’s cats wake her up each week the number of times Mrs. Plum’s cats wake her up each hour the number of times Mrs. Plum’s cats wake her up each night the number of times Mrs. Plum’s cats wake her up 129. Find the probability that her cats will wake her up no more than five times next week. a. b. c. d. .5000 .9329 .0378 .0671 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 315 4.7 Discrete Distribution (Playing Card Experiment) 130. Use a programmable calculator to simulate a binomial distribution. a. How would you use the randInt function to simulate the number of successes in five trials of an experiment with two outcomes, each of which has a .5 probability of occurring? b. Use the randInt function to simulate 10 observations of the random variable in Part A. c. Find the sample mean and sample standard deviation. d. Compare the sample mean and sample standard deviation to the theoretical mean and the theoretical standard deviation. REFERENCES 4.2 Mean or Expected Value and Standard Deviation Florida State University. (n.d.). Class catalogue at the Florida State University. Retrieved from https://apps.oti.fsu.edu/ RegistrarCourseLookup/SearchFormLegacy World Earthquakes. http://www.worldearthquakes.com/index.php?option=ethq_prediction earthquakes: Live (2012). World earthquake news and highlights. Retrieved from 4.3 Binomial Distribution (Optional) American Cancer Society. http://www.cancer.org/cancer/pancreaticcancer/detailedguide/pancreatic-cancer-key-statistics (2013). What are the key statistics about pancreatic cancer? Retrieved from Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworldfactbook/geos/af.html ESPN NBA. (2013). NBA statistics – 2013. Retrieved from http://espn.go.com/nba/statistics/_/seasontype/2 Newport, F. (2013). Americans still enjoy saving rather than spending: Few demographic differences seen in these views other than by income. GALLUP Economy. Retrieved from http://www.gallup.com/poll/162368/americansenjoy-savingrather-spending.aspx Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles, CA: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/ Monographs/TheAmericanFreshman2011.pdf Wikipedia. (n.d.). Distance education. Retrieved from http://en.wikipedia.org/wiki/Distance_education World Bank Group. (2013). Access to electricity (% of population). Retrieved from http://data.worldbank.org/indicator/ EG.ELC.ACCS.ZS?order=wbapi_data_value_2009%20wbapi_data_value%20wbapi_data_value-first&sort=asc 4.4 Geometric Distribution (Optional) Central Intelligence Agency. (n.d.). The world factbook. Retrieved from https://www.cia.gov/library/publications/theworldfactbook/geos/af.html Pew Research Center. (n.d.). Millennials: A portrait of generation next. Retrieved from http://www.pewsocialtrends.org/ files/2010/10/millennials-confident-connected-open-to-change.pdf Pew Research. (2013). Millennials: confident. Executive Summary: Pew Research Social & Demographic Trends. Retrieved from http://www.pewsocialtrends.org/2010/02/24/millennials-confident-connected-open-tochange/ Pryor, J. H., et al. (2011). The American freshman: National norms fall 2011. Los Angeles: Cooperative Institutional Research Program, Higher Education Research Institute. Retrieved from http://heri.ucla.edu/PDFs/pubs/TFS/Norms/ Monographs/ TheAmericanFreshman2011.pdf The European Union and ICON-Institute. (2007/8). Summary of the national risk and vulnerability assessment 2007/ http://ec.europa.eu/europeaid/where/asia/documents/ profile 8: afgh_brochure_summary_en.pdf Afghanistan. Retrieved from of A The World Bank. (2013). Prevalence of HIV, total (% of populations ages 15-49). Retrieved from http://data.worldbank.org/ 316 Chapter 4 \| Discrete Random Variables indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc UNICEF Television. (n.d.). UNICEF reports on female literacy centers in Afghanistan established to teach women and girls basic reading and writing skills. (Video). Retrieved from http://www.unicefusa.org/assets/video/afghan-femaleliteracycenters.html 4.6 Poisson Distribution (Optional) Centers for Disease Control and Prevention. (2012, Oct. 2). Teen drivers: Get the facts. Retrieved from http://www.cdc.gov/ Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html Daily Mail. (2011, June 9). One born every minute: the maternity unit where mothers are THREE to a bed. Retrieved http://www.dailymail.co.uk/news/article-2001422/Busiest-maternity-ward-planet-averages-60-babies-dayfrom mothersbed.html Department of Aviation at the Hartsfield-Jackson Atlanta International Airport. (2013). ATL fact sheet. Retrieved from http://www.atlanta-airport.com/Airport/ATL/ATL_FactSheet.aspx Lenhart, A. (2012). Teens, smartphones & testing: Texting volume is up while the frequency of voice calling is down. About one in four teens say they own smartphones. Pew Internet. Retrieved from http://www.pewinternet.org/~/media/Files/ Reports/2012/PIP_Teens_Smartphones_and_Texting.pdf Ministry of Health, Labour, and Welfare. (n.d.). Children and childrearing. Retrieved from http://www.mhlw.go.j
p/english/ policy/children/children-childrearing/index.html Pew Internet. (2013). How Americans use text messaging. Retrieved from http://pewinternet.org/Reports/2011/Cell-PhoneTexting-2011/Main-Report.aspx South Carolina Department of Mental Health. (2006). Eating disorder statistics. Retrieved from http://www.state.sc.us/dmh/ anorexia/statistics.htm The Guardian. (2011, June 8). Giving birth in Manila: The maternity ward at the Dr Jose Fabella Memorial Hospital in Manila, in the Philippines, where there is an average of 60 births a day. Retrieved from http://www.theguardian.com/world/gallery/2011/jun/08/philippines-health#/?picture=375471900&index=2 the busiest Vanderkam, 8). http://management.fortune.cnn.com/2012/10/08/stop-checking-your-email-now/ (2012, Oct. checking email, Stop your L. now. CNNMoney. Retrieved from World Earthquakes. http://www.worldearthquakes.com/index.php?option=ethq_prediction earthquakes: Live (2012). World earthquake news and highlights. Retrieved from SOLUTIONS 1 x P(x) 0 1 2 3 4 5 6 .12 .18 .30 .15 .10 .10 .05 Table 4.38 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 317 3 .10 + .05 = .15 5 1 7 .35 + .40 + .10 = .85 9 1(.15) + 2(.35) + 3(.40) + 4(.10) = .15 + .70 + 1.20 + .40 = 2.45 11 x P(x) 0 1 2 3 .03 .04 .08 .85 Table 4.39 13 Let X = the number of events Javier volunteers for each month. 15 x P(x) 0 1 2 3 4 5 .05 .05 .10 .20 .25 .35 Table 4.40 17 1 – .05 = .95 19 .2 + 1.2 + 2.4 + 1.6 = 5.4 21 The values of P(x) do not sum to one. 23 Let X = the number of years a physics major will spend doing postgraduate research. 25 1 – .35 – .20 – .15 – .10 – .05 = .15 27 1(.35) + 2(.20) + 3(.15) + 4(.15) + 5(.10) + 6(.05) = .35 + .40 + .45 + .60 + .50 + .30 = 2.6 years 29 X is the number of years a student studies ballet with the teacher. 31 .10 + .05 + .10 = .25 33 The sum of the probabilities sum to one because it is a probability distribution. 35 −2 ⎛ ⎝ 40 52 ⎞ ⎠ + 30 ⎛ ⎝ 12 52 ⎞ ⎠ = − 1.54 + 6.92 = 5.38 37 X = the number that reply yes 39 0, 1, 2, 3, 4, 5, 6, 7, 8 41 5.7 318 43 .4151 Chapter 4 \| Discrete Random Variables 45 X = the number of freshmen selected from the study until one replied yes to the law that was passed. 47 1,2,… 49 1.4 51 X = the number of business majors in the sample. 53 2, 3, 4, 5, 6, 7, 8, 9 55 6.26 57 0, 1, 2, 3, 4, … 59 .0485 61 .0214 63 X = the number of United States teens who die from motor vehicle injuries per day. 65 0, 1, 2, 3, 4, ... 67 no 71 The variable of interest is X, or the gain or loss, in dollars. The face cards jack, queen, and king. There are (3)(4) = 12 face cards and 52 – 12 = 40 cards that are not face cards. We first need to construct the probability distribution for X. We use the card and coin events to determine the probability for each outcome, but we use the monetary value of X to determine the expected value. Card Event X net gain/loss P(X) Face Card and Heads Face Card and Tails 6 2 (Not Face Card) and (H or T) –2 Table 4.41 ⎛ ⎝ 12 52 ⎞ ⎛ ⎝ ⎠ 1 2 ⎞ ⎠ = ⎛ ⎝ 6 52 ⎞ ⎠ ⎛ ⎝ 12 52 ⎞ ⎛ ⎝ ⎠ 1 2 ⎞ ⎠ = ⎛ ⎝ 6 52 ⎞ ⎠ ⎛ ⎝ 40 52 ⎞ ⎠(1) = ⎛ ⎝ 40 52 ⎞ ⎠ ⎛ • Expected value = (6) ⎝ 6 52 ⎛ ⎞ ⎠ + (2) ⎝ 6 52 ⎛ ⎞ ⎠ + ( − 2) ⎝ 40 52 ⎞ ⎠ = – 32 52 • Expected value = –$0.62, rounded to the nearest cent • If you play this game repeatedly, over a long string of games, you would expect to lose 62 cents per game, on average. • You should not play this game to win money because the expected value indicates an expected average loss. 73 a. .1 b. 1.6 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 319 75 a. Software Company x 5,000,000 1,000,000 –1,000,000 Table 4.42 P(x) .10 .30 .60 Hardware Company x 3,000,000 1,000,000 –1,000,00 Table 4.43 P(x) .20 .40 .40 Biotech Firm x 6,000,000 0 P(x) .10 .70 –1,000,000 .20 Table 4.44 b. $200,000; $600,000; $400,000 c. d. e. third investment because it has the lowest probability of loss first investment because it has the highest probability of loss second investment 77 4.85 years 79 b 81 Let X = the amount of money to be won on a ticket. The following table shows the PDF for X: x 0 5 P(x) .969 250 10,000 = .025 Table 4.45 320 Chapter 4 \| Discrete Random Variables x P(x) 25 100 50 10,000 10 10,000 = .005 = .001 Table 4.45 Calculate the expected value of X. 0(.969) + 5(.025) + 25(.005) + 100(.001) = .35 A fair price for a ticket is $0.35. Any price over $0.35 will enable the lottery to raise money. 83 X = the number of patients calling in claiming to have the flu, who actually have the flu. X = 0, 1, 2, ...25 85 .0165 87 a. X = the number of DVDs a Video to Go customer rents b. c. d. .12 .11 .77 89 d. 4.43 91 c 93 • X = number of questions answered correctly • X ~ B ⎛ ⎝32, 1 3 ⎞ ⎠ • We are interested in MORE THAN 75 percent of 32 questions correct. 75 percent of 32 is 24. We want to find P(x > 24). The event more than 24 is the complement of less than or equal to 24. • Using your calculator's distribution menu: 1 – binomcdf ⎛ ⎝32, 1 3 , 24 ⎞ ⎠ • P(x > 24) = 0 • The probability of getting more than 75 percent of the 32 questions correct when randomly guessing is very small and practically zero. 95 a. X = the number of college and universities that offer online offerings. b. 0, 1, 2, …, 13 c. X ~ B(13, 0.96) d. 12.48 e. .0135 f. P(x = 12) = .3186 P(x = 13) = 0.5882 More likely to get 13. 97 a. X = the number of fencers who do not use the foil as their main weapon b. 0, 1, 2, 3,... 25 c. X ~ B(25,.40) d. 10 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 321 e. .0442 f. The probability that all 25 not use the foil is almost zero. Therefore, it would be very surprising. 99 a. X = the number of audits in a 20-year period b. 0, 1, 2, …, 20 c. X ~ B(20, .02) d. e. f. .4 .6676 .0071 101 1. X = the number of matches 2. 0, 1, 2, 3 3. X ~ B ⎛ ⎝3, 1 6 ⎞ ⎠ 4. 5. In dollars: −1, 1, 2, 3 1 2 6. Multiply each Y value by the corresponding X probability from the PDF table. The answer is −.0787. You lose about eight cents, on average, per game. 7. The house has the advantage. 103 a. X ~ B(15, .281) Figure 4.10 b. i. Mean = μ = np = 15(.281) = 4.215 ii. Standard Deviation = σ = npq = 15(.281)(.719) = 1.7409 c. P(x > 5) = 1 – P(x ≤ 5) = 1 – binomcdf(15, .281, 5) = 1 – 0.7754 = .2246 P(x = 3) = binompdf(15, .281, 3) = .1927 P(x = 4) = binompdf(15, .281, 4) = .2259 It is more likely that four people are literate than three people are. 322 105 Chapter 4 \| Discrete Random Variables a. X = the number of adults in America who are surveyed until one says he or she will watch the Super Bowl. b. X ~ G(.40) c. 2.5 d. e. .0187 .2304 107 a. X = the number of pages that advertise footwear b. X takes on the values 0, 1, 2, ..., 20 c. X ~ B(20, 29 192 ) d. 3.02 e. no f. .9997 g. X = the number of pages we must survey until we find one that advertises footwear. X ~ G( 29 192 ) h. .3881 i. 6.6207 pages 109 0, 1, 2, and 3 111 a. X ~ G(.25) b. i. mean = μ = 1 p = 1 0.25 = 4 ii. standard deviation = σ = 1 − p p2 = 1 − .25 .252 ≈ 3.4641 c. P(x = 10) = geometpdf(.25, 10) = .0188 d. P(x = 20) = geometpdf(.25, 20) = .0011 e. P(x ≤ 5) = geometcdf(.25, 5) = .7627 113 a. X = the number of pages that advertise footwear b. 0, 1, 2, 3, ..., 20 c. X ~ H(29, 163, 20), r = 29, b = 163, n = 20 d. 3.03 e. 1.5197 115 a. X = the number of Patriots picked b. 0, 1, 2, 3, 4 c. X ~ H(4, 8, 9) d. without replacement 117 a. X ~ P(5.5); μ = 5.5; σ = 5.5 ≈ 2.3452 This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 4 \| Discrete Random Variables 323 b. P(x ≤ 6) = poissoncdf(5.5, 6) ≈ .6860 c. There is a 15.7 percent probability that the law staff will receive more calls than they can handle. d. P(x > 8) = 1 – P(x ≤ 8) = 1 – poissoncdf(5.5, 8) ≈ 1 – .8944 = .1056 119 Let X = the number of defective bulbs in a string. Using the Poisson distribution: • μ = np = 100(.03) = 3 • X ~ P(3) • P(x ≤ 4) = poissoncdf(3, 4) ≈ .8153 Using the binomial distribution • X ~ B(100, .03) • P(x ≤ 4) = binomcdf(100, .03, 4) ≈ .8179 The Poisson approximation is very good—the difference between the probabilities is only .0026. 121 a. X = the number of children for a Spanish woman b. 0, 1, 2, 3,... c. X ~ P(1.47) d. e. f. .2299 .5679 .4321 123 a. X = the number of fortune cookies that have an extra fortune b. 0, 1, 2, 3,... 144 c. X ~ B(144, .03) or P(4.32) d. 4.32 e. f. .0124 or .0133 .6300 or .6264 g. As n gets larger, the probabilities get closer together. 125 a. X = the number of people audited in one year b. 0, 1, 2, ..., 100 c. X ~ P(2) d. 2 e. f. .1353 .3233 127 a. X = the number of shell pieces in one cake b. 0, 1, 2, 3,... c. X ~ P(1.5) d. 1.5 e. f. .2231 .0001 g. yes 324 129 d 130 Chapter 4 \| Discrete Random Variables a. You can use randInt (0,1,5) to generate five trials of the experiment. Count the number of 1’s generated to determine the number of successes. b. Student answers may vary. c. Student answers may vary. d. The theoretical mean is (5)(.5) = 2.5 . The theoretical standard deviation is (5)(.5)(0.5) = 1.25 . This OpenStax book is available for free at http://cnx.org/content/col30309/1.8 Chapter 5 \| Continuous Random Variables 325 5 \| CONTINUOUS RANDOM VARIABLES Figure 5.1 The heights of these radish plants are continuous random variables. (credit: Rev Stan) Introduction Chapter Objectives By the end of this chapter, the student should be able to do the following: • Recognize and understand continuous probability density functions in general • Recognize the uniform probability distribution and apply it appropriately • Recognize the exponential probability distribution and apply it appropriately Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a longdistance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The fiel

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