Ch02

1. Briefly explain the difference between ungrouped and grouped data. Give one example of each type of such data.

2. You have asked 20 persons whether they think they are financially better off than their parents.

The following are the responses of these 20 persons. (B indicates that the person is better off, W represents that the person is worse off, and S stands for the same.)

B W

W S

B B

S B

B B

B W

B W

S B

W S

S B

a. b. c. d.

Construct a frequency distribution table.

Calculate the relative frequencies and percentages for all categories.

What percentage of these persons said that they are better off than their parents?

What percentage of these persons said that they are either worse off than their parents or the same? e. Draw a bar graph for the frequency distribution. f. Draw a pie chart for the percentage distribution.

3. A researcher asked 30 respondents if they felt major corporations should pay their CEOs (chief executive officers) the huge salaries that are prevalent today. (N indicates no, Y represents yes, and O stands for no opinion.)

N N Y

N Y N

Y Y N

O N N

N O Y

N N O

Y O O

N N N

O N N

N N Y

a. b. c. d.

Construct a frequency distribution table.

Calculate the relative frequencies and percentages for all categories.

What percentage of these persons is against paying huge salaries to CEOs? What percentage of these persons is either in favor of paying huge salaries to CEOs or has no opinion? e. Draw a bar graph for the relative frequency distribution. f. Draw a pie chart for the percentage distribution.

4. A researcher asks 24 mothers, who work outside their homes, whether or not they would work outside their homes if they had enough money to live comfortably. The following are the responses of these 24 mothers. (N stands for no, Y represents yes, and D means does not know.)

N N

D N

Y D

D Y

N Y

Y Y

N N

N N

D N

N D

N Y

Y D

a. Prepare a frequency distribution table. b. Calculate the relative frequencies and percentages for all categories. c. What percentage of these mothers said that they would not work outside their homes if they had enough money to live comfortably? d. What percentage of these mothers said that they would continue working outside their homes even if they had enough money to live comfortably or that they do not know? e. Draw a bar graph for the frequency distribution. f. Draw a pie chart for the percentage distribution.

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5. The following table gives the frequency distribution of the weights (in pounds) of 100 persons.

Weight (in pounds) 90 to less than 110 110 to less than 130 130 to less than 150 150 to less than 170 170 to less than 190 190 to less than 210

f 8 17 21 24 19 11

a. List the class midpoints. b. Do all classes have the same width? If yes, what is that width? c. Prepare the relative frequency and percentage distribution columns.

6. The following table gives the frequency distribution of weekly salaries (in dollars) of 200 workers.

Weekly Salaries ($) 300 to less than 400 400 to less than 500 500 to less than 600 600 to less than 700 700 to less than 800 800 to less than 900 900 to less than 1000

f 18 22 27 55 30 33 15

a. List the class midpoints. b. Do all classes have the same width? If yes, what is that width? c. Prepare the relative frequency and percentage distribution columns.

7. The following data give the amounts of electric bills (rounded to the nearest dollar) for the past one month for 30 families.

75 38 55

34 41 42

47 63 35

26 55 39

56 61 45

29 73 71

48 61 24

42 76 47

33 46 67

67 51 52

a. Construct a frequency distribution table. Take 21 as the lower limit of the first class and 10 as the width of each class. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the families has a monthly electric bill of $61 or more? d. What percentage of the families has a monthly electric bill of $40 or less? e. Draw a histogram and a polygon for the relative frequency distribution.

8. The following data give the scores of 20 students on a statistics test.

89 76

62 88

91 93

99 68

73 79

67 95

77 99

92 87

83 81

67 71

a. Construct a frequency distribution table. Take the classes as 61 – 70, 71 – 80, 81 – 90, and 91 – 100. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the students scored 81 or higher?

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d. What percentage of the students scored 80 or less? e. Draw a histogram and a polygon for the frequency distribution.

9. The following data give the ages of 21 workers working for a large company.

41 63 55

54 48 61

22 67 47

28 47 59

31 58 36

39 37 48

58 26 54

a. Construct a frequency distribution table. Take the classes as 21 – 30, 31 – 40, 41 – 50, 51 – 60, and 61 – 70. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the workers are between 31 and 50 years of age? d. What percentage of the workers are 51 years of age or older? e. Draw a histogram and a polygon for the frequency distribution.

10. The following data give the annual earnings (in thousands of dollars) of 20 families.

32.5 43.7 19.5

29.7 53.4 24.0

49.0 59.5 44.5

17.4 37.0 62.7

31.7 22.7 47.5

67.9 15.8 54.7

27.4 43.2

a. Construct a frequency distribution table. Take the classes as 10 to less than 20, 20 to less than 30, ... , and 60 to less than 70. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the families have an annual income of $50,000 or higher? d. What percentage of the families have an annual income of $40,000 to less than $60,000? e. Draw a histogram and a polygon for the relative frequency distribution.

11. The following data give the current prices (in dollars) of stocks of 25 companies.

24.75 34.50 48.50 67.50

75.75 55.25 75.50 54.00

47.25 69.50 73.50 51.50

33.25 55.00 65.25 65.75

41.50 39.50 29.25

63.75 25.75 55.25

77.50 40.25 48.50

a. Construct a frequency distribution table. Take the classes as 20 to less than 30, 30 to less than 40, ... , and 70 to less than 80. b. Calculate the relative frequencies and percentages for all classes. c. What percentage of the corporations have their stock prices less than $50? d. Draw a histogram and a polygon for the percentage distribution.

12. The following data give the number of telephones owned by a sample of 30 families.

2 1 3 a. b. c. d. e.

4 1 3

3 2 2

1 2 1

1 4 3

5 3 1

4 3 4

2 1 2

2 5 3

3 2 1

Construct a frequency distribution table using single-valued classes.

Calculate the relative frequencies and percentages for all classes.

What percentage of the families own two or three telephones?

What percentage of the families own three or more telephones?

Draw a bar graph for the frequency distribution.

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13. The following data give the number of suits owned by each of the 25 men.

1 2 0 a. b. c. d. e.

3 1 2

5 0 1

1 0 1

0 4 2

3 3 3

2 1 2

4 2

3 1

Construct a frequency distribution table using single-valued classes.

Calculate the relative frequencies and percentages for all classes.

What percentage of the persons own two or less suits?

What percentage of the persons own four or five suits?

Draw a bar graph for the relative frequency distribution.

14. The following table gives the frequency distribution of heights (in inches) of 100 persons.

Height (in inches) 60 to less than 63 63 to less than 66 66 to less than 69 69 to less than 72 72 to less than 75 75 to less than 78 a. b. c. d.

f 9 29 21 25 12 4

Construct a cumulative frequency distribution table.

Calculate the cumulative relative frequencies and cumulative percentages for all classes.

What percentage of the persons are less than 69 inches tall?

Draw an ogive for the cumulative frequency distribution.

15. The following table gives the frequency distribution of the number of hours each of the 200 students selected from a college studies per week.

Hours Studied 0 to less than 4 4 to less than 8 8 to less than 12 12 to less than 16 16 to less than 20 20 to less than 24 a. b. c. d.

f 12 29 48 62 34 15

Construct a cumulative frequency distribution table.

Calculate the cumulative relative frequencies and cumulative percentages for all classes.

What percentage of the students study for less than 16 hours a week?

Draw an ogive for the cumulative percentage.

16. The following data give the number of cars rented on each of the past 24 days at a car rental company.

12 31 9

8 22 17

18 19 29

25 32 23

33 37 31

25 28 33

15 18 25

26 30 23

Prepare a stem-and-leaf display. Arrange the leaves for each stem in increasing order.

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17. The following data give the rents (in dollars) paid per month by 21 tenants.

570 310 625

720 720 720

435 890 630

395 540 950

650 450 840

580 810 540

670 960 645

Prepare a stem-and-leaf display. Arrange the leaves for each stem in increasing order.

18. The following data give the commuting time (in minutes) from home to school for 30 college students.

22 19 36

16 8 27

11 21 8

12 28 23

23 15 37

51 43 18

42 5 13

6 19 29

31 7 17

10 14 9

Prepare a stem-and-leaf display. Arrange the leaves for each stem in increasing order.

19. Consider the following stem-and-leaf display.

1 2 3 4

29 34 25 01

45 65 74 98 46 59 73 82 36 68 88

Write the data set that corresponds to this stem-and-leaf display.

20. Consider the following stem-and-leaf display.

0-2 3-4 6-7 8-9

37••589 1368•02479 113567•2679 23366•778

Write the data set that corresponds to this stem-and-leaf display.

21. An agency asked 60 people which of the five films nominated for Best Picture in 1997 should win the Oscar. Their responses appear in the following grouped data: Response The English Patient Fargo Jerry Maguire Secrets and Lies Shine Undecided/No Opinion

f 21 9 18 3 6 3

Construct a bar graph and a pie chart to represent this data.

22. A sociologist is doing a study to try to determine the percentage of people in the U.S. who do not have any health insurance coverage. One thousand large communities participated in the study. The grouped data is:

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Percent Uninsured 0–4 5–9 10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49

f 10 95 225 354 161 59 41 25 15 15

Construct an ogive to describe this data.

23. A research company asked 500 people: “What was the last grade of school you completed?”

The grouped data is as follows: Education Level Grade School Some High School High School Grad Some College College Grad Some Postgraduate

Number of Respondents 10 30 185 145 65 65

Construct a pie chart for this data. What percentage of the people did not take any postgraduate courses?

24. A small company is analyzing its monthly phone bills. Over the past two years, the billing amounts were: 205.06 198.45 214.25 224.11 220.90 210.85 180.20 196.40 193.22 220.06 199.23 206.75 215.12 222.68 211.77 206.58 201.18 216.88 221.47 202.44 210.40 179.00 188.30 188.00 Construct a histogram to describe this data using the following classes: 176.00 – 185.99, 186.00 –195.99, 196.00 – 205.99, 206.00 – 215.99, 216.00 – 225.99.

25. The following grouped data represents the responses of two thousand grocery shoppers to the question: “How much do you spend each week at the grocery store?”

Spent Weekly Under 40.00 40.00 – 49.99 50.00 – 59.99 60.00 – 69.99 70.00 – 79.99 80.00 – 99.99 100.00 – 119.99 120.00 and over

Number of Responses 260 240 260 360 340 180 200 160 21

What are the relative frequency and the cumulative frequency of the 80.00 – 99.99 class?

Construct a histogram for this grouped data.

26. To complete a demographic study, three hundred people supplied their annual income amount.

The percentages of responses appear grouped as follows: Annual Income ($) Under 20,000 20,000 – 29,999 30,000 – 39,999 40,000 – 49,999 50,000 – 59,999 60,000 and over

Percent of Responses 22 21 23 12 10 12

How many people in the sample make between $30,000 and $39,999 per year? Construct the polygon for this data.

27. The general manager of a baseball team is analyzing its success in the past 17 years. The team’s win totals (excluding strike years) are: Year 1990 1991 1992 1993 1994 1995 1996 1997

Games Won 89 95 70 69 73 88 93 101

Year 1998 1999 2000 2001 2002 2003 2004

Games Won 95 88 89 82 73 74 85

You are to construct a histogram for this data. The histogram is to have a low value of 66, a high value of 105, and 4 equally-sized classes. What will be the boundaries and midpoints of the four classes?

28. When rolling a fair, six-sided die, a person has an equal chance of rolling a one, two, three, four, five, or six. Therefore, if the die is rolled a large number of times (say 1,000,000), each number should come up about the same number of times as each of the others. If you were to construct a histogram for 1,000,000 rolls of a fair, six-sided die, with six classes (one for each number on the die), what approximate shape would you expect the histogram to have?

29. Construct a stem-and-leaf display for the following raw data, which represents the issue ages (age at policy issue) of ten randomly-selected life insurance policies.

12

9

0

14

35

23

49

25

14

11

30. If a histogram of major league baseball player salaries is constructed, what shape will the histogram have?

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Solutions 1. Ungrouped data contain information on each individual member of a sample or a population, whereas a frequency distribution of classes displays grouped data and shows the number of values that are within each class.

2. a. and b.

Category B W S

f 10 5 5

Relative Frequency .50 .25 .25

Percentage 50 25 25

c. 50% d. 50% e.

Financial Position 12

Frequency

10 8 6 4 2 0 B

W

S

f.

Financial Position

S 25% B 50% W 25%

3. a. and b.

Category N Y O

f 17 7 6

Relative Frequency .567 .233 .200

c. 56.7% 23

Percentage 56.7 23.3 20.0

d. 43.3% e.

Frequency

Huge Salary Opinion 18 16 14 12 10 8 6 4 2 0 N

Y

O

f.

Huge Salary Opinion

O 20%

N 57%

Y 23%

4. a. and b.

Category N Y D

f 11 7 6

Relative Frequency .458 .292 .250

Percentage 45.8 29.2 25.0

c. 45.8% d. 4.2% e.

Would Work Outside Home 12

Frequency

10 8 6 4 2 0 N

Y

24

D

f.

Would Work Outside Home

D 25% N 46% Y 29%

5. a. The class midpoints are: 100, 120, 140, 160, 180, and 200 b. All classes have the same width, which is 20. c.

Relative Weight (in pounds) f Frequency Percentage 90 to less than 110 8 .08 8 110 to less than 130 17 .17 17 130 to less than 150 21 .21 21 150 to less than 170 24 .24 24 170 to less than 190 19 .19 19 190 to less than 210 11 .11 11 6. a. The class midpoints are: 350, 450, 550, 650, 750, 850, and 950 b. All classes have the same width, which is 100. c.

Relative Weekly Salaries ($) f Frequency Percentage 300 to less than 400 18 .090 9.0 400 to less than 500 22 .110 11.0 500 to less than 600 27 .135 13.5 600 to less than 700 55 .275 27.5 700 to less than 800 30 .150 15.0 800 to less than 900 33 .165 16.5 900 to less than 1000 15 .075 7.5 7. a. and b.

Classes 21 to 30 31 to 40 41 to 50 51 to 60 61 to 70 71 to 80

Relative Frequency .100 .167 .267 .167 .167 .133

f 3 5 8 5 5 4

c. 30.0% 25

Percentage 10.0 16.7 26.7 16.7 16.7 13.3

d. 26.7% e.

Electric Bills ($)

Relative Frequency

0.30 0.25 0.20 0.15 0.10 0.05 0.00 21 to 30

31 to 40

41 to 50

51 to 60

61 to 70

71 to 80

Electric Bills

Relative Frequency

0.30 0.25 0.20 0.15 0.10 0.05 0.00 21 to 30

31 to 40

41 to 50

51 to 60

61 to 70

71 to 80

8. a. and b.

Classes 61 to 70 71 to 80 81 to 90 91 to 100

Relative Frequency .20 .25 .25 .30

f 4 5 5 6

Percentage 20 25 25 30

c. 55% d. 45% e.

Statistics Test Scores 7

Frequency

6 5 4 3 2 1 0 61 to 70

71 to 80

26

81 to 90

91 to 100

Statistics Test Scores 7 6 Frequency

5 4 3 2 1 0 61 to 70

71 to 80

81 to 90

91 to 100

9. a. and b.

Classes 21 to 30 31 to 40 41 to 50 51 to 60 61 to 70

Relative Frequency .143 .190 .238 .286 .143

f 3 4 5 6 3

Percentage 14.3 19.0 23.8 28.6 14.3

c. 42.8% d. 42.9% e.

Ages of Workers 7

Frequency

6 5 4 3 2 1 0 21 to 30

31 to 40

41 to 50

51 to 60

61 to 70

Ages of Workers 7

Frequency

6 5 4 3 2 1 0 21 to 30

31 to 40

41 to 50

27

51 to 60

61 to 70

10. a. and b.

Annual Earnings (in $000s) 10 to less than 20 20 to less than 30 30 to less than 40 40 to less than 50 50 to less than 60 60 to less than 70

f 3 4 3 5 3 2

Relative Frequency .15 .20 .15 .25 .15 .10

Percentage 15 20 15 25 15 10

c. 25% d. 40% e.

Annual Earnings ($000s)

Relative Frequency

0.30 0.25 0.20 0.15 0.10 0.05 0.00 10 to less 20 to less 30 to less 40 to less 50 to less 60 to less than 20 than 30 than 40 than 50 than 60 than 70

Annual Earnings ($000s)

Relative Frequency

0.30 0.25 0.20 0.15 0.10 0.05 0.00 10 to less than 20

20 to less than 30

30 to less than 40

40 to less than 50

50 to less than 60

60 to less than 70

11. a. and b.

Stock Prices ($) 20 to less than 30 30 to less than 40 40 to less than 50 50 to less than 60 60 to less than 70 70 to less than 80

f 3 4 5 5 5 4

c. 44% 28

Relative Frequency .12 .12 .20 .20 .20 .16

Percentage 12 12 20 20 20 16

d.

Stock Prices ($)

Percentage

25 20 15 10 5 0 20 to less 30 to less 40 to less 50 to less 60 to less 70 to less than 30 than 40 than 50 than 60 than 70 than 80

Stock Prices ($) 25 Percentage

20 15 10 5 0 20 to less than 30

30 to less than 40

40 to less than 50

50 to less than 60

60 to less than 70

70 to less than 80

12. a. and b.

Number of Telephones Owned 1 2 3 4 5

Relative Frequency .267 .267 .267 .133 .067

f 8 8 8 4 2

Percentage 26.7 26.7 26.7 13.3 6.7

c. 53.4% d. 46.7% e.

Frequency

Number of Telephones Owned 9 8 7 6 5 4 3 2 1 0 1

2

3

29

4

5

13. a. and b.

Number of Suits Owned 0 1 2 3 4 5

Relative Frequency .16 .28 .24 .20 .08 .04

f 4 7 6 5 2 1

Percentage 16 28 24 20 8 4

c. 68% d. 12% e.

Number of Suits Owned

Relative Frequency

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

5

14. a. and b.

Height (in inches) 60 to less than 63 63 to less than 66 66 to less than 69 69 to less than 72 72 to less than 75 75 to less than 78

Cumulative Frequency 9 38 59 84 96 100

Cumulative Relative Frequency .09 .38 .59 .84 .96 1.00

Cumulative Percentage 9 38 59 84 96 100

c. 59% d.

Cumulative Frequency

Height in Inches 120 100 80 60 40 20 0 60 to less 63 to less 66 to less 69 to less 72 to less 75 to less than 63 than 66 than 69 than 72 than 75 than 78

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15. a. and b.

Cumulative Frequency 12 41 89 151 185 200

Hours Studied 0 to less than 4 0 to less than 8 0 to less than 12 0 to less than 16 0 to less than 20 0 to less than 24

Cumulative Relative Frequency .060 .205 .445 .755 .925 1.00

Cumulative Percentage 6.0 20.5 44.5 75.5 92.5 100.0

c. 75.5% d.

Cumulative Percentage

Hours Studied 120.0 100.0 80.0 60.0 40.0 20.0 0.0 0 to less than 4

0 to less than 8

0 to less than 12

0 to less than 16

16. 0 1 2 3

89 285987 556289353 3127013

0 1 2 3

89 257889 233555689 0112337

17. 3 4 5 6 7 8 9

95 10 35 50 70 80 40 40 50 70 25 30 45 20 20 20 90 10 40 60 50

3 4 5 6 7 8 9

10 95 35 50 40 40 70 80 25 30 45 50 70 20 20 20 10 40 90 50 60

18. 0 1 2 3 4 5

685789 61209594837 2318739 167 23 1

0 1 2 3 4 5

567889 01234567899 1233789 167 23 1

19.

129 359

145 373

234 382

265 401

274 436

298 468 31

325 488

0 to less than 20

346

0 to less than 24

20.

3 44 77

7 47 79

25 49 82

28 61 83

29 61 83

31 63 86

33 65 86

36 66 97

38 67 97

40 72 98

42 76

21.

Favorite Film to Win Oscar 25

Frequency

20 15 10 5 0 TEP

F

JM

S&L

S

U/NO

Favorite Film to Win Oscar U/NO 5%

S 10% TEP 35%

S&L 5% JM 30%

F 15%

22.

Percentage Insured

Cumulative Frequency

1200 1000 800 600 400 200 0 0–4

5–9

10–14 15–19 20–24 25–29 30–34 35–39 40–44 45–49

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23. 87% of the people did not take any postgraduate classes Last Grade of School Completed

SP 13%

GSSHS 2% 6%

CG 13%

HSG 37%

SC 29%

24. The frequencies for the classes are: 2, 3, 6, 7, and 6.

Frequency

Monthly Phone Bills ($) 8 7 6 5 4 3 2 1 0 176.00 – 185.99

186.00 –195.99

196.00 – 205.99

206.00 – 215.99

216.00 – 225.99

25. The relative frequency of the 80.00 – 99.99 class is .09 and the cumulative frequency is 1640.

400 350 300 250 200 150 100 50 0

Un de r

40 .0 0 40 .0 0– 49 .9 9 50 .0 0– 59 .9 9 60 .0 0– 69 .9 9 70 .0 0– 79 .9 9 80 .0 0– 99 .9 10 9 0.

00 –1 19 12 .9 9 0.

00 an d ov er

Frequency

Weekly Grocery Store Expenditures ($)

26. 300 × .23 = 69 people make between $30,000 and $39,999 per year.

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Annual Incomes ($)

Percentage

25 20 15 10 5 0 Under 20,000

20,000 – 29,999

30,000 – 39,999

40,000 – 49,999

50,000 – 59,999

60,000 and over

27. The class boundaries and midpoints will be: 65.5 and 75.5, midpoint is 70.5 75.5 and 85.5, midpoint is 80.5 85.5 and 95.5, midpoint is 90.5 95.5 and 105.5, midpoint is 100.5 28. One would expect the histogram to be rectangular (or uniform).

29. 0 1 2 3 4

0 9 1 2 4 4 3 5 5 9

30. Only a few of the players have extremely high salaries while most of the other players’ salaries are considerably less. That pattern produces a right-skewed histogram.

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