INSTRUCTOR’S SOLUTIONS MANUAL A PATHWAY TO INTRODUCTORY STATISTICS SECOND EDITION. JAMES LAPP by digitaldownload87

Table of Contents Chapter 1: Performing Operations and Evaluating Expressions 1.1 Variables, Constants, Plotting Points, and Inequalities ................................................................ 1 1.2 Expressions................................................................................................................................... 4 1.3 Operations with Fractions and Proportions; Converting Units ..................................................... 7 1.4 Absolute Value and Adding Real Numbers................................................................................ 11 1.5 Change in a Quantity and Subtracting Real Numbers ................................................................ 14 1.6 Ratios, Percents, and Multiplying and Dividing Real Numbers ................................................. 16 1.7 Exponents, Square Roots, Order of Operations, and Scientific Notation ................................... 20 Review Exercises ....................................................................................................................... 24 Chapter Test ............................................................................................................................... 27 Chapter 2: Designing Observational Studies and Experiments 2.1 Simple Random Sampling .......................................................................................................... 31 2.2 Systematic, Stratified, and Cluster Sampling ............................................................................. 34 2.3 Observational Studies and Experiments ..................................................................................... 36 Review Exercises ....................................................................................................................... 40 Chapter Test ............................................................................................................................... 43 Chapter 3: Graphical and Tabular Displays of Data 3.1 Frequency Tables, Relative Frequency Tables, and Bar Graphs ................................................ 45 3.2 Pie Charts and Two-Way Tables ................................................................................................ 49 3.3 Dotplots, Stemplots, and Time-Series Plots ............................................................................... 52 3.4 Histograms.................................................................................................................................. 58 3.5 Misleading Graphical Displays of Data...................................................................................... 64 Review Exercises ....................................................................................................................... 66 Chapter Test ............................................................................................................................... 71 Chapter 4: Summarizing Data Numerically 4.1 Measures of Center ..................................................................................................................... 75 4.2 Measures of Spread .................................................................................................................... 80 4.3 Boxplots ..................................................................................................................................... 85 Review Exercises ....................................................................................................................... 90 Chapter Test ............................................................................................................................... 92 Chapter 5: Computing Probabilities 5.1 Meaning of Probability ............................................................................................................... 95 5.2 Complement and Addition Rules ............................................................................................... 97 5.3 Conditional Probability and the Multiplication Rule for Independent Events .......................... 100 5.4 Discrete Random Variables ...................................................................................................... 103 5.5 Finding Probabilities for a Normal Distribution ....................................................................... 106 5.6 Finding Values of Variables for Normal Distributions ............................................................ 110 Review Exercises ..................................................................................................................... 112 Chapter Test ............................................................................................................................. 114 Chapter 6: Describing Associations of Two Variables Graphically 6.1 Scatterplots ............................................................................................................................... 117 6.2 Determining the Four Characteristics of an Association .......................................................... 121 6.3 Modeling Linear Associations .................................................................................................. 125 Review Exercises ..................................................................................................................... 130 Chapter Test ............................................................................................................................. 133

Chapter 7: Graphing Equations of Lines and Linear Models; Rate of Change 7.1 Graphing Equations of Lines and Linear Models ..................................................................... 135 7.2 Rate of Change and Slope of a Line ......................................................................................... 138 7.3 Using Slope to Graph Equations of Lines and Linear Models ................................................. 142 7.4 Functions .................................................................................................................................. 148 Review Exercises ..................................................................................................................... 152 Chapter Test ............................................................................................................................. 156 Chapter 8: Solving Linear Equations and Inequalities to Make Prediction 8.1 Simplifying Expressions ........................................................................................................... 159 8.2 Solving Linear Equations in One Variable ............................................................................... 161 8.3 Solving Linear Equations to Make Predictions ........................................................................ 163 8.4 Solving Formulas ..................................................................................................................... 170 8.5 Solving Linear Inequalities to Make Predictions ...................................................................... 175 Review Exercises ..................................................................................................................... 181 Chapter Test ............................................................................................................................. 186 Chapter 9: Finding Equations of Linear Models 9.1 Using Two Points to Find an Equation of a Line ..................................................................... 191 9.2 Using Two Points to Find an Equation of a Linear Model ....................................................... 193 9.3 Linear Regression Model ......................................................................................................... 198 Review Exercises ..................................................................................................................... 204 Chapter Test ............................................................................................................................. 207 Chapter 10: Using Exponential Models to Make Predictions 10.1 Integer Exponents ..................................................................................................................... 209 10.2 Rational Exponents................................................................................................................... 211 10.3 Graphing Exponential Models .................................................................................................. 213 10.4 Using Two Points to Find an Equation of an Exponential Model ............................................ 216 10.5 Exponential Regression Model ................................................................................................. 221 Review Exercises ..................................................................................................................... 226 Chapter Test ............................................................................................................................. 230

Chapter 1: Performing Operations and Evaluating Expressions 1

Chapter 1: Performing Operations and Evaluating Expressions Homework 1.1 2. A constant is a symbol that represents a specific number. 4. Data are quantities or categories that describe people, animals, or things. 6. In 2017, about 37% of children aged 6–12 participated in a team sport (organized or unorganized) on a regular basis. 8. The temperature is 10F . That is, the temperature is 10 degrees below 0 (in Fahrenheit). 10. The statement t  3 represents the year 2012 (3 years before 2015). 12. Answers may vary. Example: Let s be the annual salary (in thousands of dollars) of a person. Then s can represent the numbers 25 and 32, but s cannot represent the numbers 15 and 9 . 14. Answers may vary. Example: Let n be the number of students enrolled in a prestatistics class. Then n can represent the numbers 15 and 28, but n cannot represent the numbers 20 or 0.5. 16. Answers may vary. Example: Let T be the temperature (in degrees Fahrenheit) in an oven. Then T can represent the numbers 300 and 450, but T cannot represent the numbers 300 or 450 . 18. a. Answers may vary. Some possible answers are shown below.

b. In the described situation, the symbols W and L are variables. Their values can change. c. In the described situation, the symbol A is a constant. Its value is fixed at 36 square feet. 20. a. Answers may vary. Some possible answers are shown below.

b. In the described situation, the symbols W, L, and A are all variables. All their values can change. c. In the described situation, none of the symbols are constants. All their values can change. 30. The integers between 6 and 3, inclusive, are 6,  5,  4,  3,  2,  1, 0, 1, 2, and 3.

22. 24. 26.

32.

28. The counting numbers between 1 and 5 are 2, 3, and 4.

34. The positive integers between 4 and 4 are 1, 2, and 3.

36. Answers may vary. Example: 2, 5 and 40 .

38. Answers may vary. Example: 2.1, 2.3, and 2.8 .

40. The temperature at the top of a skyscraper can be positive or negative, depending on the location of the skyscraper and the time of year. Temperature is not usually reported using fractions. So, among the choices, the integers are the smallest group of number that contains possible data. Copyright © 2021 Pearson Education, Inc.

2 ISM: A Pathway to Introductory Statistics

42. The commute time of an employee cannot be negative, but it can be measured in fractions. So, among the choices, the nonnegative real numbers are the smallest group of numbers that contains possible data. 44. McDonald’s sells hamburgers every day of every year and there is never just a portion of a hamburger sold. So, among the choices, the counting numbers is the smallest group of numbers that contains possible data. 50.

46.

48. 52. a. b. The number of hours of video uploaded to YouTube per minute increased between 2009 and 2014. The number of hours of video uploaded to YouTube per minute went up each year. c. The annual increases in the number of hours of video uploaded to YouTube per minute increased between 2009 and 2014. The annual increases are shown below. Years 2009 to 2010 2010 to 2011 2011 to 2012 2012 to 2013 2013 to 2014

Increase 25  14  11 48  25  23 73  48  25 100  73  27 300  100  200

54. a. b. The number of microbreweries increased from 2013 to 2017. c. The increases in the number of microbreweries stayed approximately constant from 2013 to 2017. The annual increases are shown below. Years

Increase

2013 to 2014 2014 to 2015

2.1  1.5  0.6 2.6  2.1  0.5

2015 to 2016

3.2  2.6  0.6

2016 to 2017

3.8  3.2  0.6

56. – 68.

70. The y-coordinate is 4 .

Chapter 1: Performing Operations and Evaluating Expressions 3 72. Point A is 2 units to the left of the origin and 4 units down. Thus, its coordinates are (2, 4) . Point B is 3 units to the left of the origin on the x-axis. Thus, its coordinates are (3, 0) . Point C is 5 units to the left of the origin and 4 units up. Thus, its coordinates are (5, 4) . Point D is 4 units to the right of the origin and 2 units up. Thus, its coordinates are (4, 2) . Point E is 3 units below the origin on the y-axis. Thus, its coordinates are (0, 3) . Point F is 3 units to the right of the origin and 2 units down. Thus, its coordinates are (3, 2) . 74. True. The number 2 lies to the right of 6 on a number line. 76. False. 5  5 , thus 5 is not strictly greater than 5 . 88. Inequality: x  3

78.

Interval notation:  ,3

80.

Graph:

82. 90. Inequality: x  1

84.

Interval notation:  1,  

86. Inequality: x  5

Graph:

Interval notation:  5,   Graph: 92. Inequality

numbers less than or equal to 6 numbers greater than 1

x  6

, 6

x 1

1,  

numbers greater than or equal to 4 numbers less than 5

x  4

 4,  

x5

(,5)

94.

Graph

Interval Notation

In Words

98.

96. 100. In Words numbers between –3 and 0 numbers between 1 and 4, as well as 1 numbers between –3 and 1, as well as 1 numbers between –4 and –1, inclusive

Inequality 3  x  0

Graph

Interval Notation (3, 0)

1 x  4

[1, 4)

3  x  1

(3,1]

4  x  1

[4, 1]

4 ISM: A Pathway to Introductory Statistics 102. The student completes the homework assignment in 30 or more minutes.

104. Inequality: h  44

110. The average gas mileage of a car on highways is between 35 and 40 miles per gallon.

Interval notation:  44,   Graph:

112. Inequality: 41  T  56 Interval notation:  41,56

106. Inequality: T  2 Interval notation:  , 2

Graph:

Graph: 114. Inequality: 140  w  145

108. Inequality: V  4.2

Interval notation: (140,145)

Interval notation:  4.2,  

Graph:

116. No. Answers may vary. Example: The numbers 2 and 5 are not “between 2 and 5.” The integers between 2 and 5 are simply 3 and 4. 118. The ordered pairs selected and plotted points may vary. The points will lie on the same horizontal line. Answers may vary. 120. Answers may vary. The inequality represents “4 is less than or equal to 4,” and 4 is equal to 4. 122. The types of numbers discussed in this section are real numbers, rational number, irrational numbers, integers, and counting numbers (or natural numbers). Answers may vary. Homework 1.2 2. We evaluate an expression by substituting a number for each variable in the expression and then calculating the result.

4. The quotient of a and b is a/b, where b is not zero. 6. Substitute 6 for x in 5  x : 5  6  11

12. Substitute 6 for x in 30  x : 30  (6)  5

8. Substitute 6 for x in x  4 : 6  4  2

14. Substitute 6 for x in x  x : 6  6  0

10. Substitute 6 for x in x(9) : 69  54

16. Substitute 6 for x in x  x : 6  6  1

18. Substitute 47 for r in r  29 : 47  29  76 . So, if 47% of Republicans favor gays to marry legally in 2017, then in that same year, about 76% of Democrats favor gays to marry legally. 20. Substitute 13.5 for U in U  6 : 13.5  6  7.5 . So, in 2016 if the average daily shipping volume for UPS was 13.5 million packages, in that same year, the average daily shipping volume for FedEx was about 7.5 million packages. 22. Substitute 17 for n in 599.99n : 599.99 17  10,199.83 . So, if 17 thousand Fender Standard Jazz Electric Bass Guitars with maple fingerboards are sold, the total revenue is about $10,200,000. 24. Substitute 328 for T in T  4 : 328  4  82 . So, if a student earns a total of 328 points on four tests, the student’s average test score is 82 points.

Chapter 1: Performing Operations and Evaluating Expressions 5 26. a.

Speed Limit (miles per hour)

Driving Speed (miles per hour)

35  5

40  5

45  5

50  5

s5

The expression s  5 represents the driving speed if the speed limit is s miles per hour. b. Substitute 65 for s in s  5 : 65  5  70 . So, if the speed limit is 65 miles per hour, the person will be driving 70 miles per hour. 28. a. Number of Shares

Total Value (dollars)

74.74 1

74.74  2

74.74  3

74.74  4

74.74n

The expression 74.74n represents the total value of the shares. b. Substitute 7 for n in 74.74n : 74.74 7   523.18 . So, the total value of 7 shares is $523.18. 30. a. Number of Siblings

Share of Cost (dollars)

3000  2

3000  3

3000  4

3000  5

3000  n

The expression 3000  n represents each sibling’s share of the cost in dollars. b. Substitute 6 for n in 3000  n : 3000  6  500 . So, the share of each sibling’s cost is $500. 32. a. We can write an expression 10  v to represent the total cost of parking and money spent on a vase. b. Substitute 25 for v in the expression 10  v : 10  25  35 . So, if $10 is spent on parking then the total cost of parking and money spent on a vase is $35. 34. a. We can write an expression r  2 to represent the net price of a shaver whose retail price is r dollars. b. Substitute 6 for r in the expression r  2 : 6  2  4 . So, if the retail price of a shaver is $6, then the net price is $4. 36. a. We can write an expression 105c to represent the total cost of tuition when enrolling in c credits of classes. b. Substitute 15 for c in the expression 105c : 105 15  1575 . So, if a student enrolls in 15 credits of classes, then the total cost of tuition is $1575. 38. a. We can write an expression 420  n to represent the equal share each of n siblings will receive of the inheritance. b. Substitute 3 for n in the expression 420  n : 420  3  140 . So, each of 3 siblings will receive an equal share of $140,000 of a $420,000 inheritance.

6 ISM: A Pathway to Introductory Statistics 40. 8  x ; substitute 8 for x in 8  x : 8  8  0 .

50. The quotient of 6 and the number

42. 6  x ; substitute 8 for x in 6  x : 6  8  14.

52. Two less than the number

44. x  15; substitute 8 for x in x  15: (8)  15  23.

54. The sum of 4 and the number 56. The product of the number and 5

46. x  7 ; substitute 8 for x in x  7 : 8  7  1 .

58. The sum of the number and 3

48. 5 x ; substitute 8 for x in 5 x : 5 8  40 .

60. The quotient of the number and 5

62. Substitute 6 for x and 3 for y in the expression y  x : 3  6  9 64. Substitute 6 for x and 3 for y in the expression xy : 63  18. 66. Substitute 6 for x and 3 for y in the expression x  y : 6  3  2. 68. x  y ; substitute 9 for x and 3 for y in the expression x  y : 9  3  12. 70. x  y ; substitute 9 for x and 3 for y in the expression x  y : 9  3  3. 72. Substitute 90.0 for c and 104.8 for r in the expression c  r : 90.0  104.8  194.8. So, in 2015 the average annual per-person consumption of chicken and red meat was 194.8 pounds. 74. Substitute 11.26 for w and 19.98 for a in the expression a  w : 19.98  11.26  8.72. So, in 2015 the college enrollments of all students who were not women was 8.72 million. 76. Substitute 2.5 for N and 1.8 for A in the expression NA : 2.5 1.8  4.5. So, in 2016 the average number of AP exams taken was 4.5 million. 78. Substitute 205,200 for s and 3.6 for n in the expression s  n : 205, 200  3.6  57, 000. So, in 2014 the average money earned by a teacher was about $57,000. 80. a. Substitute 4 for x in the expression x  2 : 4  2  6. Substitute 5 for x in the expression x  2 :

5  2  7 . Substitute 6 for x in the expression x  2 : 6  2  8 . b. Substitute 4 for x in the expression 2 x: 2  4  8 . Substitute 5 for x in the expression 2 x : 2 5  10 . Substitute 6 for x in the expression 2 x: 2 6  12 . c. Observe the values after substitution are different for the two expressions.

82. a. n 1 2 3 4

x2

42  6

2(4)  8

5 2  7

2(5)  10

62  8

2(6)  12

3n 3 1  3 3 2  6 33  9 3  4  12

The price of bread is $3, $6, $9, and $12 for 1, 2, 3, and 4 loaves, respectively. b. The cost per loaf of bread is $3. The cost per loaf is a constant while the number of loaves is a variable. In the expression 3n, the constant is 3 and the variable is n. c. Answers may vary. Example: For each additional loaf bought, the total price increases by $3.

Chapter 3: Constructing Graphical and Tabular Displays of Data

46. a. The time-series plot is incorrect because the scaling on the vertical axis is not uniform.

b. No, the data points do not lie on a straight line. When a uniform scaling is used on the vertical axis, we see that the data points follow a curved shape. c. Yes, using non-uniform scaling distorts the shape of the distribution and therefore gives a false depiction of the changes over time. 48. A stemplot would not be the best type of diagram to describe the ages of the students because the students’ ages are all likely to be very similar, with some potential outliers. The stemplot would have a large number of repeat digits. 50. Using StatCrunch, the cutoff values are: A, 91; B, 75; C, 64; D, 46. The frequency distribution of the grades is Grade A B C D F

Frequency 5 11 7 6 3

52. a.

b. The private, not-for-profit colleges and universities tend to have larger tuitions and fees than the public colleges and universities. Most of the private, not-for-profit colleges and universities have higher tuitions than the most expensive public college. c. The smallest tuition and fees for a 4-year public college or university is Haskell Indian Nations University at $80. The largest tuition and fees for a 4-year public college or university is University of PittsburghMain Campus at $16,590. The difference is $16,510. Copyright © 2021 Pearson Education, Inc.

58 ISM: A Pathway to Introductory Statistics 52. (continued) d. The smallest tuition and fees for a 4-year private, not-for-profit college or university is Turtle Mountain Community College at $2250. The largest is Landmark College at $49,793. The difference is $47,543. e. The difference between the smallest and largest tuition and fees is nearly three times as large for private, not-for-profit colleges and universities as it is for public colleges and universities. Homework 3.4 2. For a density histogram, the area of a bar is equal to the relative frequency of the bar’s class. 4. If the right tail of a unimodal distribution is longer than the left tail, the distribution is skewed right. 6. The dotplot associated with given histogram is (a). The shape and spread of the distribution in the dotplot is most constant spread of the distribution in (a). 8. The dotplot associated with given histogram is (d). The shape and spread of the distribution in the dotplot is most consistent with the shape and spread of the distribution in (d). 10. The histogram associated with the given variable is (b). The age ranges and relative frequencies of those ages are consistent with that of students at the start of a school year at a certain high school. 12. The histogram associated with the given variable is (a). The age ranges and relative frequencies of those ages are consistent with that of homeowners in a certain city. 14. a. The variable is the time spent working per week (in hours). It is continuous because it can take on any values between two boundary values. b. The number who work between 9 and 25 hours per week is the sum of the frequencies of the classes 10–14, 15–19, and 20–24: 1  1  2  4. c. The number who work at least 15 hours per week is the sum of the frequencies of the classes 15–19, 20–24, and 25–29: 1  2  2  5. d. The proportion who work at most 5 hours per week is the frequency of the class 0–4 divided by the total number of observations: 4  10  0.4. e. The following are all in hours: 0, 1, 2, 3, 12, 16, 20, 24, 27, and 27; Answers may vary. 16. a. The histogram is skewed right. The 50th percentile belongs to the class 0–39 deaths. Therefore, the right tail is longer than the left tail. b. The number of fatal crashes in which no less than 120 people died is found by adding the frequencies of the classes 120–159, 160–199, and 240–279: 2  1  1  4. There were 4 fatal crashes in which no less than 120 people died. c. The class that represents the greatest number of deaths is 240–279. An estimate of the number of people who died is (240  280)  2  260. d. Since the class 0–39 contains more than half of the lowest observations (it contains 9 out of 18), the 50th percentile belongs to the class 0–39 deaths. Yes, observations in this class are typical values because over half of the observations are in the class 0–39. e. We cannot assume that the frequency of deaths resulting from crashes in any one year is representative of any other year. Because so many different factors impact flights, the conditions that result in a crash and fatalities one year are not likely to be repeated in another year. 18. a. The proportion whose ages were between 15 and 19 years, inclusive, is the relative frequency, 0.25. b. The proportion whose ages were under 29 is 0.03  0.25  0.20  0.16  0.64. c. Add the proportions of the classes with lower class limit greater than or equal to 0.20  0.16  0.11  0.08  0.06  0.04  0.04  0.02  0.01  0.72. d. We cannot assume the proportion of people stopped by police who were between the ages of 20 and 24 years, inclusive, would be the same amount on other days. In particular, the proportion on weekdays might be quite different than on weekends.

Chapter 3: Constructing Graphical and Tabular Displays of Data

20. a. Both of the distributions are skewed right. The salaries of players that are greater than the 50th percentile are much more spread out than the salaries of those that are less than the 50th percentile. b. Read the relative frequencies from each graph: the proportion of the Patriots who earn less than $1 million is about 0.51, and the proportion of the Rams who earn less than $1 million is about 0.75. c. The proportion of the Rams who earn at least $1 million is about 1  0.51  0.49. The proportion of the Rams who earn at least $1 million is about 1  0.75  0.25. d. The top-paid Rams player appears to have the larger salary. The largest Patriots salary is in the class 8 to 8,999,999 million dollars whereas the largest Rams salary is in the class 14 to 14,999,999 million dollars. e. The 70th percentile of the Patriots salaries is larger. The 70th percentile for the Patriots salaries is between 1 and $1,999,999 million whereas the 70th percentile for the Rams salaries is between 0 and $999,999. 22. a. From left to right, the areas of the bars are 0.05, 0.15, 0.3, 0.3, 0.15, and 0.05. b. The sum of the areas is 0.05  0.15  0.3  0.3  0.15  0.05  1. c. For any density histogram, the area of a bar is equal to the relative frequency of the bar’s class. And the sum of the relative frequencies of all the classes is equal to 1. d. Answers may vary.

24. a. The 50th percentile is in the class 2–3 days. Observations in that class are typical values since the class contains almost a quarter (23%) of the observations. b. The response time was at least 2 days for the following proportion of requests: 0.23  0.15  0.11  0.08  0.06  0.04  0.03  0.70. c. The proportion of requests where the response time was less than 4 days is 0.05  0.25  0.23  0.15  0.68. d. The proportion of requests where the response time was less than 3 days is 0.05  0.25  0.23  0.53. e. We cannot assume the distribution in 2019 will be the same as it was for the period of time represented by the distribution. 26. a. The variable is tuitions of 2-year, public colleges. It is discrete because there are gaps between successive possible values. b. The proportion between $3000 and $3999 is 0.23. c. The proportion with tuitions less than $6,000 is 0.13  0.12  0.23  0.31  0.15  0.94. d. The 50th percentile is in the class $4000–$4999. Yes, observations in this class are typical values; there are more observations in the class $4000–$4999 than any other class. e. The proportion with tuitions at least $6000 is the proportion with tuitions less than $6000 (found in part (c)) subtracted from 1: 1  0.94  0.06. 28. a. From the histogram, we see that 25% (or 0.13  0.12  0.25 ) of the observations are at most $2999 so we estimate $2996 is in the 25th percentile. b. From the histogram, we see that 0.13  0.12  0.23  0.31  0.79 or about 79% of the observations are $4999, so we estimate $4980 is in the 79th percentile.

60 ISM: A Pathway to Introductory Statistics 28. (continued) c. The 48th percentile 0.13  0.12  0.23  0.48 is in the class $3,000–$3999. We estimate the tuition at Lake Erie College to be $3999. d. The 94th percentile 0.13  0.12  0.23  0.31  0.15  0.94 is in the class $5000–$5999. We estimate the tuition at Southeast Technical Institute to be $5999. 30. a. Class (Number of Reports) Frequency 0  0.49 2 0.5  0.99 5 1.0  1.49 3 1.5  1.99 3 2.0  2.49 0 2.5  3.99 1 Total 14

Class

Relative

(Number of Reports)

Frequency

0  0.49 0.5  0.99 1.0  1.49 1.5  1.99 2.0  2.49 2.5  2.99 Total

2  0.143 14 5  0.357 14 3  0.214 14 3  0.214 14 0 0 14 1  0.071 14 14 1 14

c. The number of observations that are no less than 1 report of cheating per 100 domestic students is the sum of the frequencies for the classes 1.0–1.49, 1.5–1.99, 2.0–2.49, and 2.5–2.99: 3  3  0  1  7. d. The proportion of observations that are at least 1.5 reports of cheating per 100 domestic students is the 3 0 1 4 sum of the proportions for the classes 1.5–1.99, 2.0–2.49, and 2.5–2.99:     0.286. 14 14 14 14 e. Since University of California Davis had the greatest number of reports of cheating per 100 students, we assume that the observation 2.6 represents University of California Davis since 2.6 is the greatest number among the data given. Since there were 2.6 reports of cheating per 100 students and during the 2018 fall domestic enrollment there were 31,333 students, we estimate the number of reports of cheating at University of California Davis during the 2018–2019 academic year to be 31,133  100  2.6  809.46 or about 809 reports. b.

32. a.

c. The distribution is skewed right.

Chapter 3: Constructing Graphical and Tabular Displays of Data

32. (continued) d. The proportion of shark attacks that are no more than 24 shark attacks per year is the sum of the relative frequencies for the classes 10–14.99, 15–19.99, and 20–24.99: 0.353  0  0.235  0.588. e. The proportion of the observations that are at least 30 shark attacks per year is the sum of the relative frequencies for the classes: 30–34.99 and 35–39.99: 0.118  0.059  0.177. 34. a. From the histogram of Chile, we see that 59% (0.59) of the observations are in the class 0–2.99. So the 50th percentile is in the class 0–2.99. From the histogram of Japan, we see that about 80% (0.80) of the observations are in the class 0–9.99. So the 50th percentile is in the class 0–9.99. Since in each histogram the 50th percentile is in the first class and the relative frequency for the first bar in that class exceeds 50%, we are unable to pinpoint what the observation is for the 50th percentile. So, we cannot define which is larger. b. When we compare the two histograms, the spreads of the distributions are very similar. c. Both distributions are skewed right since the right tails are longer than the left tails. We interpret this to mean that most deaths caused by individual volcanic eruptions in Japan and Chile since 1900 tended to be low; however, there were some volcanic eruptions that occurred that resulted in significant fatalities, especially in Japan. d. Identify the class containing the highest number of fatalities. That is 140–149 deaths. To estimate the number of people who died, we find the average of the lowest and highest numbers of fatalities in the class: (140  150)  2  145. We estimate the number of people who died at Mount Tokachi in 1926 to be about 145. e. On the basis of the relative frequency histograms alone, we cannot conclude that more people died from volcanic eruptions in Japan than in Chile since 1900. Without knowing how many volcanic eruptions actually occurred in each country, we cannot determine the actual numbers of fatalities. 36. a. Obesity Rate (Percent)

Frequency

20–21.9

22–23.9

24–25.9

26–27.9

28–29.9

30–31.9

32–33.9

34–35.9

Relative Frequency 1  0.05 20 1  0.05 20 4  0.20 20 52  0.25 20 3  0.15 20 2  0.10 20 2  0.10 20 2  0.10 20 20 1 20

Total

62 ISM: A Pathway to Introductory Statistics 36. (continued) b.

d. The distribution is unimodal, skewed right. e. The student is incorrect. In this case, the differences between the groups are not large enough for the combined distribution to be bimodal. 38. a. The best approximation is the average of 100 and 200, or 150. b.

d. The proportion with between 400 and 799 Facebook friends is approximately 0.17  0.17  0.34. e. The proportion who have at most 799 Facebook friends is approximately 0.17  0.27  0.17  0.17  0.78. 40. a.

c. The distribution is skewed right. d. The proportion with prices between $300 and $399, inclusive, is 0.1. e. The class $100–$199.99 has the largest relative frequency, 0.30. 30% of the selected smartphones are priced in this range, more than in any of the other ranges. 42. The sum of the heights of the bars in a relative frequency histogram equals 1 because the whole is the sum of the parts. 44. If a frequency histogram has equal class widths of 1, then the frequency of each class is the number of observations of the same value. This is the same information that is given in a dotplot.

Chapter 3: Constructing Graphical and Tabular Displays of Data 63 46. A histogram with large class widths will have fewer bars than one with smaller widths. If the distribution is bimodal, the two separate peaks will disappear if the observations they represent are merged into the same class. 50.

48.

52. Stemplots and histograms are similar in that they both present observations within classes. If the leaf of each observation in a stemplot is the ones digit, the width of each class of a similar histogram would be the difference in the stems (tens digit). Since the width defines each class, by subtracting one stem from another, which is the succession of one class to another, it is possible to determine the width. If the leaf of each observation in a split stem is the ones digit, the width of each class of a similar histogram is the difference between successive, unique stems. 54. The distribution of the age of a car (in use) would be skewed right. The reason is most people drive cars that are fairly new or moderately new, but only a few drive cars that are very old. 56. The distribution of a year on a penny in circulation is skewed left. The reason is most pennies in circulation are older. Newer pennies take time to circulate, so there are fewer. 58. The distribution of the price (in dollars) of a gallon of gasoline at the gas station is symmetric. The reason is the price of different grades of fuel is usually priced accordingly. So the lowest grade is the cheapest, the highest grade the most expensive, with the middle grade priced in between. 60. A relative frequency bar graph or a pie chart would be appropriate for a single categorical variable. 62. Because the data vary over the years from 2000 to 2019, a time series chart is appropriate. 64. There are two categorical variables involved–gender and favorite type of music–so a two-way table or a multiple bar graph would be appropriate. 66. a. Answers may vary. b. Answers may vary. c. Answers may vary.

d. Answers may vary. e. Answers may vary.

68. a.

Royal Caribbean, Carnival, and Princess own 0.1456  0.1392  0.1076  0.3924 of the cruise ships. They 3  0.15 of the companies. Since the proportion of cruise ships owned is more than twice the are 20 proportion of all the companies, these three companies are likely three of the largest cruise ship companies. Copyright © 2021 Pearson Education, Inc.

64 ISM: A Pathway to Introductory Statistics 68. (continued) b.

The distribution of the lengths is unimodal and skewed left. c.

The two outliers are Royal Caribbean’s Oasis, with a crew size of 2100, and Carnival’s Conquest, with a crew size of 1910. d. The distribution of the crew sizes is unimodal and skewed left, which is the same as the distribution of the lengths. This makes sense because larger ships will tend to have larger crew sizes. e. The 50th percentile of crew size is between 800 and 1000 crew members. The 50th percentile of ship length is between 800 and 900 feet. A cruise ship will need about 1 crew member per foot. Homework 3.5 2. True. If the vertical axis of a time-series plot does not start at 0, the changes in the variable described by that axis are being emphasized.

4. Graphs in three dimensions can be misleading. 6. a. The histogram with class widths of 3 percent is bimodal and skewed left. b. The histogram with class widths of 4 percent is unimodal. c. The histogram with class widths of 3 percent suggests the difference between the Western states and the rest of the country, with one mound centered around the class 64–67 and another centered around the class 70–73. 49  52  50.5%. 2 e. The histogram with class widths of 3 percent makes it easy to estimate the number of states where the percentage of adults who exercise is between 70% and 73%; the frequency of that class is 11 states.

d. For the smallest class, an estimate of the single value is

8. a. Boston’s mayor’s office would use the bar graph with the vertical axis starting at 11%. Because the scaling on the vertical axis increases by a smaller amount than in the other graph, this bar graph emphasizes the differences in percentages and seems to show that the percentage of commuters who walk or bike to work in Boston is greater than that of the other cities (especially San Francisco, Seattle, and Portland). b. San Francisco’s mayor’s office would use the bar graph with the vertical axis starting at 0%. Because the scaling on the vertical axis increases by a larger amount than in the other graph, this bar graph deemphasizes the differences in percentages and seems to show San Francisco’s percentage of walking or biking commuters to be similar to Boston’s.

Chapter 3: Constructing Graphical and Tabular Displays of Data 65 8. (continued) c. The bar graph with the vertical axis starting at 11% gives a better estimate of the percentage of San Francisco’s walking or biking commuters. The graph is more “zoomed in” on the vertical axis. The estimate percentage is 13.9%. d. We estimate the number of commuters in Seattle who walk or bike to work to be approximately 0.129  425.8  54.928 thousand. e. Boston has about 16.7% of commuters walking or biking for a total of 0.167  368,800  61,589.60 or 61,590 (61.590 thousand) walking or biking commuters. San Francisco has 13.9% of commuters walking or biking for a total of 0.139  511, 400  71, 084.6 or about 71,085 (71.085 thousand) walking or biking commuters. San Francisco’s mayor’s office would want to show the results that it has 71,085 (71.085 thousand) walking or biking commuters while Boston has 61,590 (61.590 thousand). 10. a. If the college wants to deemphasize how much its tuition has increased, it would use the time-series plot with vertical axis starting at $13.2 thousand. The differences between the tuitions are deemphasized because the scaling increases by a larger amount than in the other time-series plot. b. A better estimate of the tuition in 2016 comes from the time-series plot with the vertical axis starting at $13.2 thousand. It is easier to estimate the tuition because the scaling on the vertical axis increases by a smaller amount than in the other time-series plot. The 2016 tuition is approximately $13.850 thousand. c. The tuition increased the most from 2016 to 2017, about $0.35 thousand. d. The change in tuition from 2013 to 2017 was about $14, 200  $13,350  $850 or about $0.85 thousand. If the same change occurred from 2017 to 2021, the 2021 tuition would be $15.05 thousand. We are not at all sure that this would be the case. Many factors could influence the cost of tuition so we cannot assume the changes in tuition over time will be similar. 12. a. To de-emphasize how much McDonald’s revenue has decreased, it should display the time-series plot with vertical axis starting at $0 billion. The differences in revenue are de-emphasized because the scaling on the vertical axis increases by a larger amount than in the other time-series plot. b. To emphasize how McDonald’s revenue has decreased, Burger King should display the time-series plot with vertical axis starting at $21 billion. The differences in revenue are emphasized because the scaling on the vertical axis increases by a smaller amount than in the other time-series plot. c. The bar graph with the vertical axis starting at $21 billion makes it easier to estimate the revenue in 2014. The graph is more “zoomed in” on the vertical axis. The estimated revenue in 2014 is $28.2 billion. d. The change in revenue from 2015 to 2016 was about $25.1  $26.0  $0.9 billion. e. The change in revenue from 2014 to 2018 was about $21.6  $28.2  $6.6 billion. If the revenue were to change by the same amount from 2018 to 2022, the revenue in 2022 would be $21.6  $6.6  $15 billion. We are not at all sure that this would be the case. Many factors could influence the revenue over time. 14. a. The bar graph is misleading because the scaling on the horizontal axis is not uniform. b.

This graph is not misleading because the scaling on the horizontal axis is uniform. c. If someone does not look carefully, the bar graph makes it seem like the annual revenue is increasing by greater and greater amounts. The fact that the horizontal axis is not uniform gives the impression that the revenue is increasing by greater and greater amounts. d. The revenue in 2013 is about $1.5 billion.

66 ISM: A Pathway to Introductory Statistics 14. (continued) e. The annual revenue from 2007 to 2017 increased about $2.48  $0.4  $2.08 billion. If it increased by the same amount from 2017 to 2027, the annual revenue in 2027 would be about $2.48  $2.08  $4.56 billion. No, we should not have much faith; we cannot assume the increase in revenue from 2017 to 2027 will be the same as from 2007 to 2017. Many factors could influence the revenue over time. 16. a. The confusing thing about the bar graph is that it is difficult to tell how the tops of the bars line up with the scaling on the vertical axis. b. A (two-dimensional) bar graph would be more straightforward.

c. The wine consumption per person for the United States is 861  328  2.63 gallons per person. The wine consumption per person for France is 713  65  10.97 gallons per person. France has the larger consumption per person. d. Italy consumes 597  59  10.12 gallons per person; Germany, 534  82  6.51 gallons per person; China, 473  1410  0.34 gallons per person. Of these five countries, France has the largest wine consumption per person; China has the least. e. It is possible for some other country to have the largest wine consumption per person because the ratio is not dependent on population. For example, a country consisting of ten people, each of whom consumes 12 gallons of wine in a year, will have a larger consumption per person than France. Since it would still have to consume fewer than 473 million gallons (China’s consumption), its population would have to be less than 473  12  39 million people. 18. If the numbers on the vertical axis do not increase by the same amount and are not equally spaced, the visual information given by the bar graph or time-series plot will not reflect the true differences in values of the variable measured on that axis. 20. If the categories for a bar graph are various years and the years do not increase by the same amount, the graph will not give an accurate picture of how much the observations change from year to year. A time-series plot is a better choice. 22. Answers may vary. Chapter 3 Review Exercises 1. The distance a student drives to school is numerical. There is a possible value between two different possible values.

2. A person’s favorite flavor of ice cream is a categorical variable. The variable assigns labels to different flavors. 3. a. The proportion who say they are late every day is 0.05. b. The proportion who say they are NOT late every day is 1  0.05  0.95. c. The proportion who say they are late less than once a month OR never is 0.21  0.47  0.68. d. The proportion who say they are late at least once a week is 0.05  0.11  0.04  0.20. e. Response bias is probably present because some surveyed adults might underestimate how often they are late to work because they are in denial. Also, if the survey is not anonymous, some survey adults might try to impress the data collector. Without response bias, the “Never” category would probably be smaller.

Chapter 3: Constructing Graphical and Tabular Displays of Data 67 4. a. Category

Frequency

Alternative

Country

Dance

Hip-hop

Pop

R&B/Soul

Total

Relative Frequency 1 10

1 20

1 20 1 2 1

4 1 20

 0.10  0.05  0.05  0.50  0.25  0.05

20 20

1

d. The proportion that are NOT Pop is 1  0.25  0.75. e. The proportion that are hip-hop OR alternative is 0.5  0.1  0.60. 5. a. The proportion who chose Wendy’s is 0.07. b. The proportion who said the restaurant with best fries was NOT Wendy’s is 1  0.07  0.93. c. The proportion who chose Arby’s OR McDonald’s is 0.07  0.34  0.41. d. Since the “Other” category has 24% of the votes, there must be at least 24%  2%  12 other restaurants, making at least 12  8  20 restaurants. 6. a. The proportion who think the government should take no action is 9489  28, 468  0.333. b. The proportion who are Independents is 10, 240  28, 468  0.360. c. The proportion of the Democrats who think the government should reduce the income difference is 5895  10,313  0.572. d. The proportion of the Republicans who think the government should reduce the income difference is 2344  7513  0.312. The proportion of Republicans who think the government should reduce the income difference is less than the proportion of Democrats who feel that way. e. The proportion of those who think the government should reduce the income difference who are Republicans is 2344  13, 257  0.177. f. The student is incorrect because in part (d) we are treating all Republicans as a whole and finding a certain fraction of that whole, but in part (e) we are treating the surveyed adults who think the government should reduce the income difference as the whole, and finding a certain fraction of that whole. Copyright © 2021 Pearson Education, Inc.

68 ISM: A Pathway to Introductory Statistics 7. a. The proportion who are NOT Republicans is 10,313  10, 240  402  28, 468  0.736. b. The proportion who are Independents AND think the government should not take action is 3212  28, 468  0.113. c. The proportion who are Independents OR think the government should not take action is 10, 240  9489  3212  28, 468  0.580. d. The proportion of all Americans who think the government should reduce the income difference may not equal 47% due to sampling bias and sampling error. e. It would be stratified sampling. 8. The length is a continuous variable. 9. The number of musicians is a discrete variable. 10. a. The distribution is symmetric. b. The 50th percentile is in the class 4–4.99 surface-wave magnitude. The size of a typical earthquake is between 4 and 4.99 surface-wave magnitudes. c. The number with surface-wave magnitudes that were at least 3 and less than 5 is approximately 800  1190  1990 earthquakes. d. The number with surface-wave magnitudes greater than or equal to 4 is approximately 1200  390  50  1640 earthquakes; round down to 1600 earthquakes. e. The proportion of U.S. earthquakes occurring in California in 1985 is approximately 862  2550  0.338 or 34%. 11. a. Reading from the density histogram, 0.15 or 15% of the islands have between 50 and 59 species, inclusive. b. The percentage of the islands that have between 30 and 49 species, inclusive, is 0.08  0.13  100%  21%. c. The labeled relative frequencies add up to 0.98, so the unlabeled bar has relative frequency 0.02. The proportion with at least 60 species is thus 0.17  0.06  0.02  0.25. d. 0.12  0.19  0.08  0.08  0.13  0.60 of the islands have no more than 49 species. 12. a. Since there are two peaks, the distribution is bimodal. b. The 50th percentile is in the class 40–49 species. c. Adding the relative frequencies for classes containing at most 69 species gives 0.92, so Malaita Island is at the 92nd percentile. d. Since the proportion of islands having at most 19 species is 0.12  0.19  0.31, Samarai Island has 19 species. 13. a.

b. If there were one outlier, it would be 60 miles. This means that a government employee commutes 60 miles to work. c. The proportion of the commute times between 10 and 20 miles inclusive is d. The proportion of commute times that are at most 10 miles is

15  0.6. 25

5  0.2. 25

Chapter 3: Constructing Graphical and Tabular Displays of Data 69 14. a.

4  0.4. 10 c. The 30th percentile is $15 million; the earnings $15 million is greater than or equal to approximately 30% of the top-10 annual earnings of DJs in 2018. d. Since half of the values are less than or equal to $22 million, and the other half is greater than $22 million, $22 million is the 50th percentile. e. $46 million is at the 90th percentile. The earnings $46 million is greater than or equal to approximately 90% of the top-10 annual earnings of DJs in 2018.

b. The proportion of the observations that are no more than $20 million is

15. a.

b. Annual per-person wine consumption has generally increased. From the graph we see increases from 1995-2015 at which point the consumption stays the same. c. The change in annual per-person wine consumption from 1995 to 2000 is 2.0  1.8  0.2 gallons per person. This means that annual per-person wine consumption increased by 0.2 gallons from 1995 to 2000. d. The change in annual per-person wine consumption from 2010 to 2017 is 2.9  2.6  0.3. If annual perperson wine consumption changes by the same amount from 2017 to 2024, the per-person wine consumption in 2024 will be 2.9  0.3  3.2 gallons. No, we cannot assume the change in annual perperson wine consumption from 2017 to 2024 will be the same as from 2010 to 2017. 16. a. Class (numbers of species) Frequency Relative Frequency 13  0.481 0  49 13 27 7  0.259 50  99 7 27 5  0.185 100  149 5 27 1  0.037 150  199 1 27 0 0 200  250 0 27 0 0 250  300 0 27 1  0.037 300  350 1 27 27  0.999 Total 27 27

70 ISM: A Pathway to Introductory Statistics 16. (continued) b.

d. The distribution is skewed right. e. The 50th percentile is in the class 50–99 threatened species. f. The outlier would be 310 threatened species. 17. a. The salesperson would refer to the bar graph with the vertical axis starting at 25 thousand cars; the differences between the numbers of stolen cars are emphasized because the scaling on the vertical axis increases by a smaller amount than in the other bar graph. b. The salesperson would refer to the bar graph with the vertical axis starting at 0 thousand cars; the differences between the numbers of stolen cars are de-emphasized because the scaling on the vertical axis increases by a larger amount than in the other bar graph. c. The better estimate comes from the bar graph with the vertical axis starting at 25 thousand cars; it is easier to make the estimation because the scaling on the vertical axis increases by a smaller amount than in the other bar graph. We estimate the number of Honda Accords stolen to be 44 thousand. d. We estimate that 44  35  9 thousand more Honda Accords were stolen than full-size Ford Pickups. 18. a. The graph is misleading because the scaling on the horizontal axis is not uniform. b. This graph is not misleading because the scaling on the horizontal axis is uniform.

c. The bar graph makes it seem like the enrollment is increasing by greater and greater amounts because the scaling on the horizontal axis is not uniform. d. We estimate the charter-school enrollment in 2013 to be about 2.3 million students. e. The increase from 2007 to 2017 is approximately 3.2  1.3  1.9 million students. If it increases by the same amount from 2017 to 2027, there will be approximately 3.2  1.9  5.1 million students enrolled. No, we should not have much faith; we cannot assume the change in the charter-school enrollment from 2017 to 2027 will be the same as from 2007 to 2017.

Chapter 3: Constructing Graphical and Tabular Displays of Data 71 19. a. The graph is confusing because it is difficult to line up the tops of the boxes with the scaling on the vertical axis. b. A relative frequency bar graph would give a straightforward description.

c. The proportion who were not married couples is 1  0.55  0.45. d. The proportion who were single women or single men is 0.19  0.11  0.3. Chapter 3 Test 1. The cost of a car is a numerical variable because it describes a measurable quantity.

2. a.