Problem 1

Do the following:

a. Group the data as indicated.

b. Prepare a frequency distribution with a column for intervals and frequencies.

c. Construct a histogram.

d. Construct a frequency polygon.

Use six intervals, starting with 0–24.

$\begin{array}{rrrrrr}{7} & {105} & {116} & {73} & {129} & {26} \\ {29} & {44} & {126} & {82} & {56} & {137} \\ {43} & {73} & {65} & {141} & {79} & {74} \\ {121} & {12} & {46} & {37} & {85} & {82} \\ {2} & {99} & {85} & {95} & {90} & {38} \\ {86} & {147} & {32} & {84} & {13} & {100}\end{array}$

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Problem 2

Do the following:

a. Group the data as indicated.

b. Prepare a frequency distribution with a column for intervals and frequencies.

c. Construct a histogram.

d. Construct a frequency polygon.

Use seven intervals, starting with 30–39.

$\begin{array}{llllllll}{79} & {71} & {78} & {87} & {69} & {50} & {63} & {51} \\ {60} & {46} & {65} & {65} & {56} & {88} & {94} & {56} \\ {74} & {63} & {87} & {62} & {84} & {76} & {82} & {67} \\ {59} & {66} & {57} & {81} & {93} & {93} & {54} & {88} \\ {55} & {69} & {78} & {63} & {63} & {48} & {89} & {81} \\ {98} & {42} & {91} & {66} & {60} & {70} & {64} & {70} \\ {61} & {75} & {82} & {65} & {68} & {39} & {77} & {81} \\ {67} & {62} & {73} & {49} & {51} & {76} & {94} & {54} \\ {83} & {71} & {94} & {45} & {73} & {95} & {72} & {66} \\ {71} & {77} & {48} & {51} & {54} & {57} & {69} & {87}\end{array}$

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Problem 3

Do the following:

a. Group the data as indicated.

b. Prepare a frequency distribution with a column for intervals and frequencies.

c. Construct a histogram.

d. Construct a frequency polygon.

Repeat Exercise 1 using eight intervals, starting with 0–19.

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Problem 4

Do the following:

a. Group the data as indicated.

b. Prepare a frequency distribution with a column for intervals and frequencies.

c. Construct a histogram.

d. Construct a frequency polygon.

Repeat Exercise 2 using six intervals, starting with 39–48.

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Problem 6

Discuss the advantages and disadvantages of the mean as a measure of central tendency.

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Problem 9

Find the mean for each list of numbers.

$30,200 ; 23,700 ; 33,320 ; 29,410 ; 24,600 ; 27,750 ; 27,300 ; 32,680$

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Problem 10

Find the mean for each list of numbers.

$38,500 ; 39,720 ; 42,183 ; 21,982 ; 43,250$

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Problem 11

Find the mean for each list of numbers.

$9.4,11.3,10.5,7.4,9.1,8.4,9.7,5.2,1.1,4.7$

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Problem 12

Find the mean for each list of numbers.

$15.3,27.2,14.8,16.5,31.8,40.1,18.9,28.4,26.3,35.3$

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Problem 13

Find the mean for the following.

$\begin{array}{ll}{\text { Value }} & {\text { Frequency }} \\ \hline 4 & {6} \\ \hline 6 & {1} \\ \hline 9 & {3} \\ \hline 15 & {2}\end{array}$

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Problem 14

Find the mean for the following.

$\begin{array}{cc}{\text { Value }} & {\text { Frequency }} \\ \hline 9 & {3} \\ \hline 12 & {5} \\ \hline 15 & {1} \\ {18} & {1}\end{array}$

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Problem 15

Find the mean for the data in Exercise 1 from the grouped frequency distribution found in each of the following exercises.

a. Exercise 1

b. Exercise 3

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Problem 16

Find the mean for the data in Exercise 2 from the grouped frequency distribution found in each of the following exercises.

a. Exercise 2

b. Exercise 4

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Problem 21

Find the median for each list of numbers.

$28.4,9.1,3.4,27.6,59.8,32.1,47.6,29.8$

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Problem 22

Find the median for each list of numbers.

$0.2,1.4,0.6,0.2,2.5,1.9,0.8,1.5$

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Problem 23

Use a graphing calculator or spreadsheet to calculate the mean and median for the data in the indicated exercises.

Exercise 1

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Problem 24

Use a graphing calculator or spreadsheet to calculate the mean and median for the data in the indicated exercises.

Exercise 2

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Problem 26

Find the mode or modes for each list of numbers.

$16,15,13,15,14,13,11,15,14$

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Problem 27

Find the mode or modes for each list of numbers.

$55,62,62,71,62,55,73,55,71$

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Problem 28

Find the mode or modes for each list of numbers.

$158,162,165,162,165,157,163$

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Problem 29

Find the mode or modes for each list of numbers.

$6.8,6.3,6.3,6.9,6.7,6.4,6.1,6.0$

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Problem 30

Find the mode or modes for each list of numbers.

$22.35,14.90,17.85,15.46,14.91,17.85,21.35$

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Problem 32

Under what circumstances would the mode be an appropriate measure of central tendency?

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Problem 33

For grouped data, the modal class is the interval containing the most data values. Find the modal class for each collection of grouped data.

Use the distribution in Exercise 1.

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Problem 34

For grouped data, the modal class is the interval containing the most data values. Find the modal class for each collection of grouped data.

Use the distribution in Exercise 2.

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Problem 35

To predict the outcome of the next congressional election, you take a survey of your friends. Is this a random sample of the voters in your congressional district? Explain why or why not.

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Problem 38

The total compensation (in millions of dollars) for the 10 highest paid CEOs in 2009 is given in the following table. Source: The Huffington Post.

a. Find the mean total compensation for this group of people.

b. Find the median total compensation for this group of people.

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Problem 39

The total income for African American households making under $\$ 250,000$ in 2008 is given in the following table. Source: U.S. Census Bureau.

Use the table to estimate the mean income for African American households in 2008.

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Problem 40

The total income for white households making under $\$ 250,000$ in 2008 is given in the following table. Source: U.S. Census Bureau.

a. Use this table to estimate the mean income for white households in 2008.

b. Compare this estimate with the estimate found in Exercise 39. Discuss whether this provides evidence that white American households have higher earnings than African American households.

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Problem 41

The number of consumer complaints against the top U.S. airlines during the first six months of 2010 is given in the following table. Source: U.S. Department of Transportation.

a. By considering the numbers in the column labeled “Complaints,” calculate the mean and median number of complaints per airline.

b. Explain why the averages found in part a are not meaningful.

c. Find the mean and median of the numbers in the column labeled “Complaints per 100,000 Passengers Boarding.” Discuss whether these averages are meaningful.

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Problem 42

The size of the home ranges (in square kilometers) of several pandas were surveyed over a year’s time, with the following results.

a. Sketch a histogram and frequency polygon for the data.

b. Find the mean for the data.

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Problem 43

The number of recognized blood types varies by species, as indicated by the table below. Find the mean, median, and mode of this data. Source: The Handy Science Answer Book.

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Problem 44

The following table gives the number of days in June and July of recent years in which the temperature reached 90 degrees or higher in New York’s Central Park. Source: The New York Times and Accuweather.com.

a. Prepare a frequency distribution with a column for intervals and frequencies. Use six intervals, starting with 0–4.

b. Sketch a histogram and a frequency polygon, using the intervals in part a.

c. Find the mean for the original data.

d. Find the mean using the grouped data from part a.

e. Explain why your answers to parts c and d are different.

f. Find the median and the mode for the original data.

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Problem 45

The table below gives the average monthly temperatures in degrees Fahrenheit for a certain area.

Find the mean and median for the following.

a. The maximum temperature

b. The minimum temperature

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Problem 46

The number of nations participating in the winter Olympic games, from 1968 to 2010, is given below. Find the following measures for the data. Source: New York Times 2010 Almanac and International Olympic Committee.

a. Mean

b. Median

c. Mode

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Problem 47

When Russian billionaire Roman Abramovich became governor of the Russian province Chukotka (in the Bering Straits, opposite Alaska), it instantly became the fourth most prosperous region in Russia, even though its 80,000 other residents are poor. Mr. Abramovich was then worth $\$ 5.7$ billion. Suppose each of the $80,000$ other residents of Chukotka was worth $\$ 100 .$ Source: National Public Radio.

a. Calculate the average worth of a citizen of Chukotka.

b. What does this example tell you about the use of the mean to describe an average?

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Problem 48

Washington Post writer John Schwartz pointed out that if Microsoft Corp. cofounder Bill Gates, who, at the time, was reportedly worth $\$ 10$ billion, lived in a town with 10,000 totally penniless people, the average personal wealth in the town would make it seem as if everyone were a millionaire. Source: The Washington Post.

a. Verify Schwartz’s statement.

b. What would be the median personal wealth in this town?

c. What would be the mode for the personal wealth in this town?

d. In this example, which average is most representative: the mean, the median, or the mode?

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Problem 49

The Major League Baseball team with the highest payroll in 2010 (and most other years) was the New York Yankees. The following table gives the salary of each Yankee in 2010. Source: About.com.

a. Find the mean, median, and mode of the salaries.

b. Which average best describes this data?

c. Why is there such a difference between the mean and the median?

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Problem 50

Given the following sequence of numbers*

$$1, a, a^{2}, a^{3}, \ldots, a^{n}$$

*Permission to reprint SAT materials does not constitute review or endorsement by Educational Testing Service or the College Board of this publication as a whole or of any other questions or testing information it may contain. This problem appeared, minus the additional assumption, on an SAT in 1996. Colin Rizzio, a high school student at the time, became an instant celebrity when he noticed that the additional assumption was needed to complete the problem. Source: The New York Times. where $n$ is a positive even integer, with the additional assumption that $a$ is a positive number, the median is best described as

a. greater than $a^{n / 2}$

b. smaller than $a^{n / 2}$

c. equal to $a^{n / 2}$

d. The relationship cannot be determined from the information given.

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