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Chapter 2: Descriptive Statistics

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 2: Descriptive Statistics

65 problems · 26 figures

2-1 §2.1 source ↗

For each of the following data sets, create a stemplot and identify any outliers.

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.

Answer

Stem Leaf
19 9 9
20 1 1 5 5 5 6 6 8 9
31 1 2 2 3 4 5 6 7 7 8 8 8 8
41 3 3
Table 2.87
Table 2.87 in the book ↗
Stem Leaf
19 9 9
20 1 1 5 5 5 6 6 8 9
31 1 2 2 3 4 5 6 7 7 8 8 8 8
41 3 3

2-3 §2.1 source ↗

For each of the following data sets, create a stemplot and identify any outliers.

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

Answer

Stem Leaf
25 5 6 7 7 8
30 0 1 2 3 3 5 5 5 7 7 9
41 6 9
56 7 7
61
Table 2.88
Table 2.88 in the book ↗
Stem Leaf
25 5 6 7 7 8
30 0 1 2 3 3 5 5 5 7 7 9
41 6 9
56 7 7
61

2-5 §2.1 source ↗


For the next three exercises, use the data to construct a line graph.

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.

Table 2.40 in the book ↗
Number of Times in StoreFrequency
14
210
316
46
54

Answer

Figure 2.53 in the book ↗

2-7 §2.1 source ↗


For the next three exercises, use the data to construct a line graph.

Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table 2.42.

Table 2.42 in the book ↗
Number of TV Shows Frequency
0 12
1 18
2 36
3 7
4 2

Answer

Figure 2.54 in the book ↗

2-9 §2.1 source ↗


For the next three exercises, use the data to construct a line graph.

Using the data from Mrs. Ramirez’s math class supplied in Exercise 2.8, construct a bar graph showing the percentages.

Answer

Figure 2.55 in the book ↗

2-11 §2.1 source ↗


For the next three exercises, use the data to construct a line graph.

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.

Answer

Figure 2.56 in the book ↗

2-13 §2.2 source ↗

What does the frequency column in Table 2.45 sum to? Why?
Table 2.45 in the book ↗
Data Value (Number of Cars) Frequency Relative Frequency Cumulative Relative Frequency

Answer

65

2-15 §2.2 source ↗

What is the difference between relative frequency and frequency for each data value in Table 2.45?
Table 2.45 in the book ↗
Data Value (Number of Cars) Frequency Relative Frequency Cumulative Relative Frequency

Answer

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.

2-17 §2.2 source ↗

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.
Table 2.45 in the book ↗
Data Value (Number of Cars) Frequency Relative Frequency Cumulative Relative Frequency
Figure 2.33 in the book ↗

Answer

Answers will vary. One possible histogram is shown below.
Figure 2.57 in the book ↗

2-19 §2.2 source ↗

Construct a frequency polygon from the frequency distribution for the 50 highest-ranked countries for depth of hunger.
Table 2.49 in the book ↗
Depth of Hunger Frequency
230–25921
260–28913
290–3195
320–3497
350–3791
380–4091
410–4391

Answer

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 in the book ↗

2-21 §2.2 source ↗

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.
Table 2.52 in the book ↗
Sex/Year 1855185618571858 185918601861
Female 45,54549,58250,25750,32451,91551,22052,403
Male 47,80452,23953,15853,694 54,62854,40954,606
Total 93,349101,821103,415 104,018106,543105,629 107,009
Table 2.53 in the book ↗
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
Male 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321
Total 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354
Table 2.54 in the book ↗
Sex/Year18711870187218711872182718741875
Female 56,099 56,431 57,472 56,099 57,472 58,233 60,109 60,146
Male 60,029 58,959 61,293 60,029 61,293 61,467 63,602 63,432
Total 116,128 115,390 118,765 116,128 118,765 119,700 123,711 123,578

Answer

Figure 2.59 in the book ↗

2-23 §2.3 source ↗

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.

Answer

a. The 40th percentile is 37 years.

b. The 78th percentile is 70 years.

2-25 §2.3 source ↗

Jesse was ranked 37th in his graduating class of 180 students. At what percentile is Jesse’s ranking?

Answer

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 n (100) = 143+.5(1) 180 (100) = 79.72. Jesse’s rank of 37 puts him at the 80th percentile.

2-27 §2.3 source ↗

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.

Answer

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).

2-29 §2.3 source ↗

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.

Answer

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.

2-31 §2.3 source ↗

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.

Answer

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.

2-33 §2.3 source ↗

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?

Answer

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.

2-35 §2.3 source ↗

Use Exercise 2.25 to calculate the following values.

Second quartile = median = 50th percentile = ________

Answer

4

2-37 §2.3 source ↗

Use Exercise 2.25 to calculate the following values.

Interquartile range (IQR) = ________ – ________ = ________

Answer

6 – 4 = 2

2-39 §2.3 source ↗

Use Exercise 2.25 to calculate the following values.

70th percentile = ________

Answer

6

2-41 §2.4 source ↗

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.

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?

Answer

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.

2-43 §2.5 source ↗

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

Calculate the mean.

Answer

Mean: 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 = 738;

738 27 = 27.33

2-45 §2.5 source ↗

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

Identify the mode.

Answer

The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27

2-47 §2.5 source ↗


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.

median = ________

Answer

4

2-49 §2.6 source ↗

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.

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5

Answer

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.

2-51 §2.6 source ↗

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.

87, 87, 87, 87, 87, 88, 89, 89, 90, 91

Answer

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.

2-53 §2.6 source ↗

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.

When the data are symmetrical, what is the typical relationship between the mean and median?

Answer

When the data are symmetrical, the mean and median are close or the same.

2-55 §2.6 source ↗

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.

Describe the shape of this distribution.

Figure 2.34 in the book ↗

Answer

The distribution is skewed right because it looks pulled out to the right.

2-57 §2.6 source ↗

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.

Describe the relationship between the mean and the median of this distribution.

Figure 2.36 in the book ↗

Answer

The mean is 4.1 and is slightly greater than the median, which is 4.

2-59 §2.6 source ↗

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.

Describe the relationship between the mode and the median of this distribution.

Figure 2.38 in the book ↗

Answer

The mode and the median are the same. In this case, both 5.

2-61 §2.6 source ↗

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.

Describe the shape of this distribution.

Figure 2.40 in the book ↗

Answer

The distribution is skewed left because it looks pulled out to the left.

2-63 §2.6 source ↗

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.

Describe the relationship between the mean and the median of this distribution.

Figure 2.42 in the book ↗

Answer

Both the mean and the median are 6.

2-65 §2.6 source ↗

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.

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

Answer

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

2-67 §2.6 source ↗

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.

Of the three measures, which tends to reflect skewing the most, the mean, the mode, or the median? Why?

Answer

The mean tends to reflect skewing the most because it is affected the most by outliers.

2-69 §2.7 source ↗

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.

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?

Answer

sampling variability

2-70 §2.7 source ↗

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.

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?

Answer

induced variability

2-71 §2.7 source ↗

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.

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?

Answer

measurement variability

2-72 §2.7 source ↗

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.

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?

Answer

natural variability

2-73 §2.7 source ↗

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

Use a graphing calculator or computer to find the standard deviation and round to the nearest tenth.

Answer

s = 34.5

2-75 §2.7 source ↗

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

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?

Table 2.60 in the book ↗
Baseball Player Batting Average Team Batting Average Team Standard Deviation
Fredo .158 .166 .012
Karl .177 .189 .015

Answer

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.

2-77 §2.7 source ↗

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

Find the standard deviation for the following frequency tables using the formula. Check the calculations with the TI 83/84.

a.

b.

c.

Table 2.61 in the book ↗
GradeFrequency
49.5–59.52
59.5–69.53
69.5–79.58
79.5–89.512
89.5–99.55
Table 2.62 in the book ↗
Daily Low Temperature Frequency
49.5–59.5 53
59.5–69.5 32
69.5–79.5 15
79.5–89.5 1
89.5–99.5 0
Table 2.63 in the book ↗
Points per Game Frequency
49.5–59.5 14
59.5–69.5 32
69.5–79.5 15
79.5–89.5 23
89.5–99.5 2

Answer

a. sx= ∑ fm2 n− x¯ 2= 193,157.4530 −79.52 = 10.88

b. sx = ∑ fm2 n− x¯ 2= 380,945.3 101− 60.942= 7.62

c. sx= ∑ fm 2n − x¯2 = 440,051.5 86 − 70.662 = 11.14

2-79 §2.1 source ↗

Table 2.64 contains the 2010 rates for a specific disease in U.S. states and Washington, DC.

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.

Table 2.64 in the book ↗
State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1

Answer

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}.

Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

b.

c.

Figure 2.60 in the book ↗
Figure 2.61 in the book ↗
Figure 2.62 in the book ↗

2-81 §2.2 source ↗

Table 2.68 Singles in the book ↗
Amount ($) Frequency Relative Frequency
51–100 5
101–150 10
151–200 15
201–250 15
251–300 10
301–350 5
Table 2.69 Couples in the book ↗
Amount ($) Frequency Relative Frequency
100–150 5
201–250 5
251–300 5
301–350 5
351–400 10
401–450 10
451–500 10
501–550 10
551–600 5
601–650 5

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:

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 singles as they do person by person as a couple? Explain why in one or two complete sentences.

Answer

Amount($) Frequency Relative Frequency
51–100 5 .08
101–150 10 .17
151–200 15 .25
201–250 15 .25
251–300 10 .17
301–350 5 .08
Table 2.89 Singles
Amount ($) Frequency Relative Frequency
100–150 5 .07
201–250 5 .07
251–300 5 .07
301–350 5 .07
351–400 10 .14
401–450 10 .14
451–500 10 .14
501–550 10 .14
551–600 5 .07
601–650 5 .07
Table 2.90 Couples

a. See Table 2.89 and Table 2.90.

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.

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.

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 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.

Table 2.89 Singles in the book ↗
Amount($) Frequency Relative Frequency
51–100 5 .08
101–150 10 .17
151–200 15 .25
201–250 15 .25
251–300 10 .17
301–350 5 .08
Table 2.90 Couples in the book ↗
Amount ($) Frequency Relative Frequency
100–150 5 .07
201–250 5 .07
251–300 5 .07
301–350 5 .07
351–400 10 .14
401–450 10 .14
451–500 10 .14
501–550 10 .14
551–600 5 .07
601–650 5 .07
Figure 2.63 in the book ↗
Figure 2.64 in the book ↗

2-83 §2.2 source ↗

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.

A histogram showing the results of a survey. Of 111 respondents, 5 own 1 t-shirt costing more than $19, 17 own 2, 23 own 3, 39 own 4, 25 own 5, 2 own 6, and no respondents own 7.

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

Answer

c

2-85 §2.2 source ↗

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.

A histogram showing the results of a survey. Of 111 respondents, 5 own 1 t-shirt costing more than $19, 17 own 2, 23 own 3, 39 own 4, 25 own 5, 2 own 6, and no respondents own 7.

Following are the 2010 obesity rates by U.S. states and Washington, DC.

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.

Table 2.71 in the book ↗
State Percent (%) State Percent (%) State Percent (%)
Alabama 32.2 Kentucky 31.3 North Dakota 27.2
Alaska 24.5 Louisiana 31.0 Ohio 29.2
Arizona 24.3 Maine 26.8 Oklahoma 30.4
Arkansas 30.1 Maryland 27.1 Oregon 26.8
California 24.0 Massachusetts 23.0 Pennsylvania 28.6
Colorado 21.0 Michigan 30.9 Rhode Island 25.5
Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
Delaware 28.0 Mississippi 34.0 South Dakota 27.3
Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
Florida 26.6 Montana 23.0 Texas 31.0
Georgia 29.6 Nebraska 26.9 Utah 22.5
Hawaii 22.7 Nevada 22.4 Vermont 23.2
Idaho 26.5 New Hampshire 25.0 Virginia 26.0
Illinois 28.2 New Jersey 23.8 Washington 25.5
Indiana 29.6 New Mexico 25.1 West Virginia 32.5
Iowa 28.4 New York 23.9 Wisconsin 26.3
Kansas 29.4 North Carolina 27.8 Wyoming 25.1

Answer

Answers will vary.

2-87 §2.3 source ↗

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.

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.

Table 2.72 in the book ↗
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

Answer

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.

2-89 §2.3 source ↗

The following box plot shows the ages of the U.S. population for 1990, the latest available year:

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)?

Figure 2.44 in the book ↗

Answer

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

2-91 §2.4 source ↗

Given the following box plot, answer the questions.

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?

Figure 2.46 in the book ↗

Answer

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.

2-93 §2.4 source ↗

Figure 2.48 in the book ↗

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:

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

Answer

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.

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.

2-96 §2.5 source ↗

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?
Table 2.75 in the book ↗
Percentage of Children with the Condition Number of Countries
16–21.45 23
21.45–26.9 4
26.9–32.35 9
32.35–37.8 7
37.8–43.25 6
43.25–48.7 1

Answer

the mean percentage, x ¯ = 1,328.65 50 = 26.75

2-98 §2.7 source ↗


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.

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.

Answer

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.

2-100 §2.7 source ↗


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.

The population standard deviation = ______.

Answer

474 FTES

2-102 §2.7 source ↗


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.

What is the IQR? What does the IQR represent?

Answer

919

2-104 §2.7 source ↗


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.

Calculate the mean, median, standard deviation, the first quartile, the third quartile, and the IQR. Round to one decimal place.

Answer

• mean = 1,809.3

• median = 1,812.5

• standard deviation = 151.2

• first quartile = 1,690

• third quartile = 1,935

• IQR = 245

2-106 §2.7 source ↗


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.

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?

Answer

Hint: think about the number of years covered by each time period and what happened to higher education during those periods.

2-108 §2.7 source ↗


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.

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.

Answer

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.

2-110 §2.7 source ↗


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.

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.

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.

Table 2.78 in the book ↗
Percentage of Population with Disease Number of Countries
11.4–20.4529
20.45–29.4513
29.45–38.454
38.45–47.450
47.45–56.452
56.45–65.451
65.45–74.450
74.45–83.451

Answer

• 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.

2-112 §2.BIT source ↗

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.

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?

Table 2.80 in the book ↗
Age-Group Percentage of Community
0–17 18.9
18–24 8.0
25–34 22.8
35–44 15.0
45–54 13.1
55–64 11.9
65+ 10.3
Figure 2.49 in the book ↗

Answer

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.

2-114 §2.BIT source ↗

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.

What is the IQR?

a. 8

b. 11

c. 15

d. 35

Table 2.82 in the book ↗
Number of Years Frequency Number of Years Frequency
7 1 22 1
14 3 23 1
15 1 26 1
18 1 40 2
19 4 42 2
20 3
Total = 20

Answer

a

2-116 §2.BIT source ↗

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.

Is this a sample or the entire population?

a. sample

b. entire population

c. neither

Table 2.82 in the book ↗
Number of Years Frequency Number of Years Frequency
7 1 22 1
14 3 23 1
15 1 26 1
18 1 40 2
19 4 42 2
20 3
Total = 20

Answer

b

2-117 §2.BIT source ↗

Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows:

a. Find the sample mean x¯ .

b. Find the approximate sample standard deviation, s.

Table 2.83 in the book ↗
Number of Movies Frequency
0 5
1 9
2 6
3 4
4 1

Answer

a. 1.48

b. 1.12

2-119 §2.BIT source ↗

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?

h. 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?

i. Assume the population was a California-based football team. Find

i. the population mean, μ,

ii. the population standard deviation, σ, and

iii. the weight that is two standard deviations below the mean.

iv. 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?

Answer

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. A box plot with a whisker between 174 and 205.5, a solid line at 205.5, a dashed line at 241, a solid line at 272.5, and a whisker between 272.5 and 302.

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

2-121 §2.BIT source ↗

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.

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.

Figure 2.52 in the book ↗

Answer

a. true

b. true

c. true

d. false

2-123 §2.BIT source ↗

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?

Answer

a.

EnrollmentFrequency
1,000–5,00010
5,000–10,00016
10,000–15,0003
15,000–20,0003
20,000–25,0001
25,000–30,0002
Table 2.91

b. Check student’s solution.

c. mode

d. 8,628.74

e. 6,943.88

f. –0.09

Table 2.91 in the book ↗
EnrollmentFrequency
1,000–5,00010
5,000–10,00016
10,000–15,0003
15,000–20,0003
20,000–25,0001
25,000–30,0002

2-125 §2.BIT source ↗


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.

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

Table 2.85 in the book ↗
X Frequency
0 3
1 12
2 33
3 28
4 11
5 9
6 4

Answer

a

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Problems and answers are adapted from Statistics by Barbara Illowsky and Susan Dean, attributed to the Texas Education Agency (TEA) and published by OpenStax, and are used under the CC BY 4.0 licence. Access for free at https://openstax.org/books/statistics/pages/1-introduction. The original material is available at https://www.texasgateway.org/book/tea-statistics. Changes were made to the original material, including updates to art, structure, and other content updates. Art is credited in its own caption wherever the book credits it. This adaptation is neither affiliated with nor endorsed by OpenStax, Rice University or the Texas Education Agency (TEA).