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Textbook by: Kathy Chu, Ph.D.. E-mail the author

# Exercises

Summary: Descriptive Statistics: Homework is part of the collection col10555 written by Barbara Illowsky and Susan Dean and provides homework questions related to lessons about descriptive statistics.

## Exercise 1

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

Table 1
# of movies Frequency Relative Frequency Cumulative Relative Frequency
0 5
1 9
2 6
3 4
4 1

• a. Find the sample mean x¯ x
• b. Find the sample standard deviation, ss
• c. Construct a histogram of the data.
• d. Complete the columns of the chart.
• e. Find the first quartile.
• f. Find the median.
• g. Find the third quartile.
• h. Construct a box plot of the data.
• i. What percent of the students saw fewer than three movies?
• j. Find the 40th percentile.
• k. Find the 90th percentile.
• l. Construct a line graph of the data.
• m. Construct a stem plot of the data.

• a. 1.48
• b. 1.12
• e. 1
• f. 1
• g. 2
• h.
• i. 80%
• j. 1
• k. 3

## Exercise 2

The median age for U.S. blacks currently is 30.9 years; for U.S. whites it is 42.3 years. ((Source: http://www.usatoday.com/news/nation/story/2012-05-17/minority-births-census/55029100/1))

• a. Based upon this information, give two reasons why the black median age could be lower than the white median age.
• b. Does the lower median age for blacks necessarily mean that blacks die younger than whites? Why or why not?
• c. How might it be possible for blacks and whites to die at approximately the same age, but for the median age for whites to be higher?

## Exercise 3

Forty randomly selected students were asked the number of pairs of sneakers they owned. Let X = the number of pairs of sneakers owned. The results are as follows:

Table 2
X Frequency Relative Frequency Cumulative Relative Frequency
1 2
2 5
3 8
4 12
5 12
7 1
• a. Find the sample mean x¯ x
• b. Find the sample standard deviation, ss
• c. Construct a histogram of the data.
• d. Complete the columns of the chart.
• e. Find the first quartile.
• f. Find the median.
• g. Find the third quartile.
• h. Construct a box plot of the data.
• i. What percent of the students owned at least five pairs?
• j. Find the 40th percentile.
• k. Find the 90th percentile.
• l. Construct a line graph of the data
• m. Construct a stem plot of the data

• a. 3.78
• b. 1.29
• e. 3
• f. 4
• g. 5
• h.
• i. 32.5%
• j. 4
• k. 5

## Exercise 4

600 adult Americans were asked by telephone poll, What do you think constitutes a middle-class income? The results are below. Also, include left endpoint, but not the right endpoint. (Source: Time magazine; survey by Yankelovich Partners, Inc.)

### Note:

"Not sure" answers were omitted from the results.
Table 3
Salary ($) Relative Frequency < 20,000 0.02 20,000 - 25,000 0.09 25,000 - 30,000 0.19 30,000 - 40,000 0.26 40,000 - 50,000 0.18 50,000 - 75,000 0.17 75,000 - 99,999 0.02 100,000+ 0.01 • a. What percent of the survey answered "not sure" ? • b. What percent think that middle-class is from$25,000 - $50,000 ? • c. Construct a histogram of the data 1. i: Should all bars have the same width, based on the data? Why or why not? 2. 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 ## Exercise 5 Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year (Source: San Jose Mercury News) 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% 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 were the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why? • i. Assume the population was the San Francisco 49ers. Find: • i. the population mean, μ μ . • ii. the population standard deviation, σ σ . • iii. the weight that is 2 standard deviations below the mean. • iv. When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard deviations above or below the mean was he? • j. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer? ### Solution • b. 241 • c. 205.5 • d. 272.5 • e. • f. 205.5, 272.5 • g. sample • h. population • i. • i. 236.34 • ii. 37.50 • iii. 161.34 • iv. 0.84 std. dev. below the mean • j. Young ## Exercise 6 An elementary school class ran 1 mile with a mean of 11 minutes and a standard deviation of 3 minutes. Rachel, a student in the class, ran 1 mile in 8 minutes. A junior high school class ran 1 mile with a mean of 9 minutes and a standard deviation of 2 minutes. Kenji, a student in the class, ran 1 mile in 8.5 minutes. A high school class ran 1 mile with a mean of 7 minutes and a standard deviation of 4 minutes. Nedda, a student in the class, ran 1 mile in 8 minutes. • a. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he? • b. Who is the fastest runner with respect to his or her class? Explain why. ## Exercise 7 In a survey of 20 year olds in China, Germany and America, people were asked the number of foreign countries they had visited in their lifetime. 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. • b. Explain how it is possible that more Americans than Germans surveyed have been to over eight foreign countries. • c. Compare the three box plots. What do they imply about the foreign travel of twenty year old residents of the three countries when compared to each other? ## Exercise 8 One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar. A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows: 3; 8; -1; 2; 0; 5; -3; 1; -1; 6; 5; -2 • a. What is the mean change score? • b. What is the standard deviation for this population? • c. What is the median change score? • d. Find the change score that is 2.2 standard deviations below the mean. ## Exercise 9 Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best G.P.A. when compared to his school? Explain how you determined your answer. Table 4 Student G.P.A. School Ave. G.P.A. School Standard Deviation Thuy 2.7 3.2 0.8 Vichet 87 75 20 Kamala 8.6 8 0.4 ### Solution Kamala ## Exercise 10 Given the following box plot: • a. Which quarter has the smallest spread of data? What is that spread? • b. Which quarter has the largest spread of data? What is that spread? • c. Find the Inter Quartile Range (IQR). • d. Are there more data in the interval 5 - 10 or in the interval 10 - 13? How do you know this? • e. Which interval has the fewest data in it? How do you know this? • I. 0-2 • II. 2-4 • III. 10-12 • IV. 12-13 ## Exercise 11 Given the following box plot: • a. Think of an example (in words) where the data might fit into the above box plot. In 2-5 sentences, write down the example. • b. What does it mean to have the first and second quartiles so close together, while the second to fourth quartiles are far apart? ## Exercise 12 Santa Clara County, CA, has approximately 27,873 Japanese-Americans. Their ages are as follows. (Source: West magazine) Table 5 Age Group Percent 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 • a. Construct a histogram of the Japanese-American community in Santa Clara County, CA. The bars will not be the same width for this example. Why not? • b. What percent of the community is under age 35? • c. Which box plot most resembles the information above? ## Exercise 13 Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, each asked adult consumers the number of fiction paperbacks they had purchased the previous month. The results are below. Table 6: Publisher A # of books Freq. Rel. Freq. 0 10 1 12 2 16 3 12 4 8 5 6 6 2 8 2 Table 7: Publisher B # of books Freq. Rel. Freq. 0 18 1 24 2 24 3 22 4 15 5 10 7 5 9 1 Table 8: Publisher C # of books Freq. Rel. Freq. 0-1 20 2-3 35 4-5 12 6-7 2 8-9 1 • a. Find the relative frequencies for each survey. Write them in the charts. • b. Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of 1. For Publisher C, make bar widths of 2. • c. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical. • d. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not? • e. Make new histograms for Publisher A and Publisher B. This time, make bar widths of 2. • f. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer. ## Exercise 14 Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all on-board transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Below is a summary of the bills for each group. Table 9: Singles Amount($) Frequency Rel. Frequency
51-100 5
101-150 10
151-200 15
201-250 15
251-300 10
301-350 5
Table 10: Couples
Amount($) Frequency Rel. 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 • 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.

### Solution

For pianos, the cost of the piano is 0.4 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 cost the most in comparison to the cost of other instruments of the same type.

## Exercise 33

Suppose that a publisher conducted a survey asking adult consumers the number of fiction paperback books they had purchased in the previous month. The results are summarized in the table below. (Note that this is the data presented for publisher B in homework exercise 13).

Table 15: Publisher B
# of books Freq. Rel. Freq.
0 18
1 24
2 24
3 22
4 15
5 10
7 5
9 1
1. Are there any outliers in the data? Use an appropriate numerical test involving the IQR to identify outliers, if any, and clearly state your conclusion.
2. If a data value is identified as an outlier, what should be done about it?
3. Are any data values further than 2 standard deviations away from the mean? In some situations, statisticians may use this criteria to identify data values that are unusual, compared to the other data values. (Note that this criteria is most appropriate to use for data that is mound-shaped and symmetric, rather than for skewed data.)
4. Do parts (a) and (c) of this problem give the same answer?
5. Examine the shape of the data. Which part, (a) or (c), of this question gives a more appropriate result for this data?
6. Based on the shape of the data which is the most appropriate measure of center for this data: mean, median or mode?

### Solution

• IQR = 4 – 1 = 3 ; Q1 – 1.5*IQR = 1 – 1.5(3) = -3.5 ; Q3 + 1.5*IQR = 4 + 1.5(3) = 8.5 ;The data value of 9 is larger than 8.5. The purchase of 9 books in one month is an outlier.
• The outlier should be investigated to see if there is an error or some other problem in the data; then a decision whether to include or exclude it should be made based on the particular situation. If it was a correct value then the data value should remain in the data set. If there is a problem with this data value, then it should be corrected or removed from the data. For example: If the data was recorded incorrectly (perhaps a 9 was miscoded and the correct value was 6) then the data should be corrected. If it was an error but the correct value is not known it should be removed from the data set.
• xbar – 2s = 2.45 – 2*1.88 = -1.31 ; xbar + 2s = 2.45 + 2*1.88 = 6.21 ; Using this method, the five data values of 7 books purchased and the one data value of 9 books purchased would be considered unusual.
• No: part (a) identifies only the value of 9 to be an outlier but part (c) identifies both 7 and 9.
• The data is skewed (to the right). It would be more appropriate to use the method involving the IQR in part (a), identifying only the one value of 9 books purchased as an outlier. Note that part (c) remarks that identifying unusual data values by using the criteria of being further than 2 standard deviations away from the mean is most appropriate when the data are mound-shaped and symmetric.
• The data are skewed to the right. For skewed data it is more appropriate to use the median as a measure of center.

**Exercises 32 and 33 contributed by Roberta Bloom

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