Section Exercises

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

84. The median age of the U.S. population in 1980 was [latex]30.0[/latex] years. In 1991, the median age was [latex]33.1[/latex] years.

1. What does it mean for the median age to rise?
2. Give two reasons why the median age could rise.
3. For the median age to rise, is the actual number of children less in 1991 than it was in 1980? Why or why not?

Measures of the Spread of the Data

Use the following information to answer the next two exercises:
The following data are the distances between [latex]20[/latex] retail stores and a large distribution center. The distances are in miles.  [latex]29[/latex]; [latex]37[/latex]; [latex]38[/latex]; [latex]40[/latex]; [latex]58[/latex]; [latex]67[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]76[/latex]; [latex]86[/latex]; [latex]87[/latex]; [latex]95[/latex]; [latex]96[/latex]; [latex]96[/latex]; [latex]99[/latex]; [latex]106[/latex]; [latex]112[/latex]; [latex]127[/latex]; [latex]145[/latex]; [latex]150[/latex]

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• [latex]μ[/latex] = [latex]1000[/latex] FTES
• median = [latex]1,014[/latex] FTES
• [latex]σ[/latex] = [latex]474[/latex] FTES
• first quartile = [latex]528.5[/latex] FTES
• third quartile = [latex]1,447.5[/latex] FTES
• [latex]n[/latex] = [latex]29[/latex] years

98. Compare the [latex]IQR[/latex] for the FTES for 1976–77 through 2004–2005 with the [latex]IQR[/latex] for the FTES for 2005-2006 through 2010–2011. Why do you suppose the [latex]IQR[/latex]s are so different?

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99. Three students were applying to the same graduate school. They came from schools with different grading systems. Which student had the best GPA when compared to other students at his school? Explain how you determined your answer.

Student	GPA	School Average GPA	School Standard Deviation
Thuy	[latex]2.7[/latex]	[latex]3.2[/latex]	[latex]0.8[/latex]
Vichet	[latex]87[/latex]	[latex]75[/latex]	[latex]20[/latex]
Kamala	[latex]8.6[/latex]	[latex]8[/latex]	[latex]0.4[/latex]

100. A music school has budgeted to purchase three musical instruments. They plan to purchase a piano costing $[latex]3,000[/latex], a guitar costing $[latex]550[/latex], and a drum set costing $[latex]600[/latex]. The mean cost for a piano is $[latex]4,000[/latex] with a standard deviation of $[latex]2,500[/latex]. The mean cost for a guitar is $[latex]500[/latex] with a standard deviation of $[latex]200[/latex]. The mean cost for drums is $[latex]700[/latex] with a standard deviation of $[latex]100[/latex]. 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.

1. Why is Kenji considered a better runner than Nedda, even though Nedda ran faster than he?
2. Who is the fastest runner with respect to his or her class? Explain why.

102. The most obese countries in the world have obesity rates that range from [latex]11.4[/latex]% to [latex]74.6[/latex]%. This data is summarized in Table 14.

Percent of Population Obese	Number of Countries
[latex]11.4–20.45[/latex]	[latex]29[/latex]
[latex]20.45–29.45[/latex]	[latex]13[/latex]
[latex]29.45–38.45[/latex]	[latex]4[/latex]
[latex]38.45–47.45[/latex]	[latex]0[/latex]
[latex]47.45–56.45[/latex]	[latex]2[/latex]
[latex]56.45–65.45[/latex]	[latex]1[/latex]
[latex]65.45–74.45[/latex]	[latex]0[/latex]
[latex]74.45–83.45[/latex]	[latex]1[/latex]

What is the best estimate of the average obesity percentage for these countries? What is the standard deviation for the listed obesity rates? The United States has an average obesity rate of 33.9%. Is this rate above average or below? How “unusual” is the United States’ obesity rate compared to the average rate? Explain.

Percent of Underweight Children	Number of Countries
[latex]16–21.45[/latex]	[latex]23[/latex]
[latex]21.45–26.9[/latex]	[latex]4[/latex]
[latex]26.9–32.35[/latex]	[latex]9[/latex]
[latex]32.35–37.8[/latex]	[latex]7[/latex]
[latex]37.8–43.25[/latex]	[latex]6[/latex]
[latex]43.25–48.7[/latex]	[latex]1[/latex]

What is the best estimate for the mean percentage of underweight children? What is the standard deviation? Which interval(s) could be considered unusual? Explain.

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

# of movies	Frequency
[latex]0[/latex]	[latex]5[/latex]
[latex]1[/latex]	[latex]9[/latex]
[latex]2[/latex]	[latex]6[/latex]
[latex]3[/latex]	[latex]4[/latex]
[latex]4[/latex]	[latex]1[/latex]

1. Find the sample mean [latex]\bar{x}[/latex].
2. Find the approximate sample standard deviation, [latex]s[/latex].

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

[latex]X[/latex]	Frequency
[latex]1[/latex]	[latex]2[/latex]
[latex]2[/latex]	[latex]5[/latex]
[latex]3[/latex]	[latex]8[/latex]
[latex]4[/latex]	[latex]12[/latex]
[latex]5[/latex]	[latex]12[/latex]
[latex]6[/latex]	[latex]0[/latex]
[latex]7[/latex]	[latex]1[/latex]

1. Find the sample mean [latex]\bar{x}[/latex]
2. Find the sample standard deviation, [latex]s[/latex]
3. Construct a histogram of the data.
4. Complete the columns of the chart.
5. Find the first quartile.
6. Find the median.
7. Find the third quartile.
8. Construct a box plot of the data.
9. What percent of the students owned at least five pairs?
10. Find the [latex]40[/latex]^th percentile.
11. Find the [latex]90[/latex]^th percentile.
12. Construct a line graph of the data
13. Construct a stemplot of the data

106. Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year. [latex]177[/latex]; [latex]205[/latex]; [latex]210[/latex]; [latex]210[/latex]; [latex]232[/latex]; [latex]205[/latex]; [latex]185[/latex]; [latex]185[/latex]; [latex]178[/latex]; [latex]210[/latex]; [latex]206[/latex]; [latex]212[/latex]; [latex]184[/latex]; [latex]174[/latex]; [latex]185[/latex]; [latex]242[/latex]; [latex]188[/latex]; [latex]212[/latex]; [latex]215[/latex]; [latex]247[/latex]; [latex]241[/latex]; [latex]223[/latex]; [latex]220[/latex]; [latex]260[/latex]; [latex]245[/latex]; [latex]259[/latex]; [latex]278[/latex]; [latex]270[/latex]; [latex]280[/latex]; [latex]295[/latex]; [latex]275[/latex]; [latex]285[/latex]; [latex]290[/latex]; [latex]272[/latex]; [latex]273[/latex]; [latex]280[/latex]; [latex]285[/latex]; [latex]286[/latex]; [latex]200[/latex]; [latex]215[/latex]; [latex]185[/latex]; [latex]230[/latex]; [latex]250[/latex]; [latex]241[/latex]; [latex]190[/latex]; [latex]260[/latex]; [latex]250[/latex]; [latex]302[/latex]; [latex]265[/latex]; [latex]290[/latex]; [latex]276[/latex]; [latex]228[/latex]; [latex]265[/latex]

1. Organize the data from smallest to largest value.
2. Find the median.
3. Find the first quartile.
4. Find the third quartile.
5. Construct a box plot of the data.
6. The middle [latex]50[/latex]% of the weights are from _______ to _______.
7. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why?
8. If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why?
9. Assume the population was the San Francisco 49ers. Find:
   1. the population mean, [latex]μ[/latex].
   2. the population standard deviation, [latex]σ[/latex].
   3. the weight that is two standard deviations below the mean.
   4. When Steve Young, quarterback, played football, he weighed [latex]205[/latex] pounds. How many standard deviations above or below the mean was he?
10. That same year, the mean weight for the Dallas Cowboys was [latex]240.08[/latex] pounds with a standard deviation of [latex]44.38[/latex] pounds. Emmit Smith weighed in at [latex]209[/latex] pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?

107. One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of [latex]12[/latex] 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 [latex]12[/latex] change scores are as follows:
[latex]3[/latex]; [latex]8[/latex]; [latex]–1[/latex]; [latex]2[/latex]; [latex]0[/latex]; [latex]5[/latex]; [latex]–3[/latex]; [latex]1[/latex]; [latex]–1[/latex]; [latex]6[/latex]; [latex]5[/latex]; [latex]–2[/latex]

1. What is the mean change score?
2. What is the standard deviation for this population?
3. What is the median change score?
4. Find the change score that is [latex]2.2[/latex] standard deviations below the mean.

108. Refer to the figure to determine which of the following are true and which are false. Explain your solution to each part in complete sentences.

1. The medians for all three graphs are the same.
2. We cannot determine if any of the means for the three graphs is different.
3. The standard deviation for graph b is larger than the standard deviation for graph a.
4. We cannot determine if any of the third quartiles for the three graphs is different.

1. Organize the data in a chart.
2. Find the median, the first quartile, and the third quartile.
3. Find the [latex]65[/latex]^th percentile.
4. Find the [latex]10[/latex]^th percentile.
5. Construct a box plot of the data.
6. The middle [latex]50[/latex]% of the conferences last from _______ days to _______ days.
7. Calculate the sample mean of days of engineering conferences.
8. Calculate the sample standard deviation of days of engineering conferences.
9. Find the mode.
10. If you were planning an engineering conference, which would you choose as the length of the conference: mean; median; or mode? Explain why you made that choice.
11. Give two reasons why you think that three to five days seem to be popular lengths of engineering conferences.

110. A survey of enrollment at [latex]35[/latex] community colleges across the United States yielded the following figures:
[latex]6414[/latex]; [latex]1550[/latex]; [latex]2109[/latex]; [latex]9350[/latex]; [latex]21828[/latex]; [latex]4300[/latex]; [latex]5944[/latex]; [latex]5722[/latex]; [latex]2825[/latex]; [latex]2044[/latex]; [latex]5481[/latex]; [latex]5200[/latex]; [latex]5853[/latex]; [latex]2750[/latex]; [latex]10012[/latex]; [latex]6357[/latex]; [latex]27000[/latex]; [latex]9414[/latex]; [latex]7681[/latex]; [latex]3200[/latex]; [latex]17500[/latex]; [latex]9200[/latex]; [latex]7380[/latex]; [latex]18314[/latex]; [latex]6557[/latex]; [latex]13713[/latex]; [latex]17768[/latex]; [latex]7493[/latex]; [latex]2771[/latex]; [latex]2861[/latex]; [latex]1263[/latex]; [latex]7285[/latex]; [latex]28165[/latex]; [latex]5080[/latex]; [latex]11622[/latex]

1. Organize the data into a chart with five intervals of equal width. Label the two columns "Enrollment" and "Frequency."
2. Construct a histogram of the data.
3. If you were to build a new community college, which piece of information would be more valuable: the mode or the mean?
4. Calculate the sample mean.
5. Calculate the sample standard deviation.
6. A school with an enrollment of [latex]8000[/latex] would be how many standard deviations away from the mean?

111. The [latex]80[/latex]^th percentile is _____

1. [latex]5[/latex]
2. [latex]80[/latex]
3. [latex]3[/latex]
4. [latex]4[/latex]

112. The number that is [latex]1.5[/latex] standard deviations BELOW the mean is approximately _____

1. [latex]0.7[/latex]
2. [latex]4.8[/latex]
3. [latex]–2.8[/latex]
4. Cannot be determined

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113. 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.

# of books	Freq.	Rel. Freq.
[latex]0[/latex]	[latex]18[/latex]
[latex]1[/latex]	[latex]24[/latex]
[latex]2[/latex]	[latex]24[/latex]
[latex]3[/latex]	[latex]22[/latex]
[latex]4[/latex]	[latex]15[/latex]
[latex]5[/latex]	[latex]10[/latex]
[latex]7[/latex]	[latex]5[/latex]
[latex]9[/latex]	[latex]1[/latex]

1. Are there any outliers in the data? Use an appropriate numerical test involving the [latex]IQR[/latex] 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 two 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?