Chapter 2 describing data: graphs and tables



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CHAPTER 2

DESCRIBING DATA: GRAPHS AND TABLES


MULTIPLE CHOICE QUESTIONS

In the following multiple-choice questions, please circle the correct answer.


1. Data that arise from counts are called:


  1. continuous

  2. nominal

  3. counted

  4. discrete

ANSWER: d
2. A sample of a population taken at one particular point in time is categorized as:


  1. categorical

  2. discrete

  3. cross-sectional

  4. time-series

ANSWER: c
3. Gender and State are examples of which type of data?


  1. Discrete

  2. Continuous

  3. Categorical

  4. Ordinal

ANSWER: c

4. A histogram that is positively skewed is:




  1. skewed to the right

  2. skewed to the left

  3. balanced

  4. symmetric

ANSWER: a
5. A histogram that has exactly two peaks is called a distribution.


  1. unimodal

  2. bimodal

  3. skewed

  4. scatter

ANSWER: b
6. An opinion variable expressed numerically on a 1-5 scale is a(n):


  1. histogram

  2. opinion poll

  3. categorical scale

  4. Likert scale

ANSWER: d
7. A variable is classified as ordinal if:


  1. there is a natural ordering of categories

  2. there is no natural ordering of categories

  3. the data arises from continuous measurements

  4. we track the variable through a period of time

ANSWER: a
8. A population includes:


  1. only people

  2. only households

  3. all objects of interest in a particular study

  4. only machines

ANSWER: c

9. Researchers try to gain insight into the characteristics of a population by examining a __________ of the population.




  1. model

  2. sample

  3. description

  4. replica

ANSWER: b
10. Numerical variables can be subdivided into which two types?


  1. Diverse and categorical

  2. Discrete and continuous

  3. Nominal and progressive

  4. Cross-sectional and discrete

ANSWER: b
11. The tool that provides useful information about a data set by breaking it down into subpopulations is:


  1. the histogram

  2. the scatterplot

  3. the pivot table

  4. the spreadsheet

ANSWER: c
12. The tables that result from pivot tables are called:


  1. samples

  2. sub-tables

  3. specimens

  4. crosstabs

ANSWER: d
13. In order for the characteristics of a sample to be generalized to the entire population, it should be ________________ of the population.


  1. symbolic

  2. typical

  3. representative

  4. illustrative

ANSWER: c

14. A bimodal histogram is often an indication that:




  1. the data are incorrect

  2. the data are inconclusive

  3. the sample is not representative of the population

  4. the data come from two or more distinct populations

ANSWER: d
15. The Literary Digest fiasco of 1936 is an example of:


  1. a sample that is not representative of its population

  2. an incomplete population

  3. an inconclusive data set

  4. a symmetric histogram

ANSWER: a
16. A _________________ indicates how many observations fall into various categories.


  1. Likert scale

  2. frequency table

  3. sample table

  4. tabulation scale

ANSWER: b
17. A _________________ is the graphical analog of a frequency table.


  1. histogram

  2. graph

  3. diagram

  4. table

ANSWER: a
18. A histogram that has a single peak and looks approximately the same to the left and right of the peak is:


  1. bimodal

  2. symmetric

  3. balanced

  4. proportional

ANSWER: b
19. An observation is a:


  1. variable

  2. field

  3. member of a population or sample

  4. characteristic of a population

ANSWER: c
20. If data is stored in a database package, which of the following terms are typically used?


  1. Fields and records

  2. Cases and columns

  3. Variables and samples

  4. Variables and observations

ANSWER: a
21. A scatterplot allows one to see:


  1. whether there is any relationship between two variables

  2. what type of relationship there is between two variables

  3. both a and b

  4. neither a nor b

ANSWER: c
22. A time series plot is essentially a:


  1. histogram

  2. scatterplot

  3. diagram

  4. figure

ANSWER: b
23. The commonly observed shapes of histograms are:

I. symmetric II. asymmetric III. bimodal

IV. polymodal V. positively skewed VI. negatively skewed


  1. I, II, III and IV

  2. III, IV, V, and VI

  3. II, III, IV, and V

  4. I, III, V, and VI

ANSWER: d

24. When we look at a time series plot, we usually look for which two things?




  1. “Is there an observable trend?” and “Is there a seasonal pattern?”

  2. “Is there an observable trend” and “Can we make predictions?”

  3. “Is the sample representative?” and “Is there a seasonal pattern?”

  4. “Is there an observable trend?” and “Is the trend symmetric?”

ANSWER: a

TEST QUESTIONS

25. The students at small community college in Iowa apply to study either English or business. Some administrators at the college are concerned that women are being discriminated against in being allowed admittance, particularly in the business program. Below, you will find two pivot tables that show the percentage of students admitted by gender to the English program and the business school. The data has also been presented graphically. What do the data and graphs indicate?




English program

Gender

No

Yes

Total

Female

46.0%

54.0%

100%

Male

60.8%

39.2%

100%

Total

53.5%

46.5%

100%

Business school

Gender

No

Yes

Total

Female

69.2%

30.8%

100%

Male

64.1%

35.9%

100%

Total

65.4%

34.6%

100%





ANSWER:

These data indicate that a smaller percentage of women are being admitted to the business program. Only 30.8% of women are being admitted to the business program compared to 35.9% for men. However, it is also important to note that only 34.6% of all applicants (women and men) are admitted to the business program compared to 46.5% for the English program. Maybe the males should say something about being discriminated against in being admitted to the English program.


26. A sample of 30 schools produced the pivot table shown below for the average percentage of students graduating from high school. Use this table to determine how the type of school (public or Catholic) that students attend affects their chance of graduating from high school.




ANSWER:

The percentages in the right column suggest that if we look at all schools, the rate of graduation is much higher in Catholic schools than in public schools. But a look at the breakdowns in the three ethnic group columns shows that this difference is due primarily to schools that are black and Latino. There isn't much difference in graduation rates between Catholic and public schools that are white.


27. The data below represents monthly sales for two years of beanbag animals at a local retail store (Month 1 represents January and Month 12 represents December). Given the time series plot below, do you see any obvious patterns in the data? Explain.




ANSWER:

This is a representation of seasonal data. There seems to be a small increase in months 3, 4, and 5 and a large increase at the end of the year. The sales of this item seem to peak in December and have a significant drop off in January.


28. A data set from a sample of 399 Michigan families was collected. The characteristics of the data include family size (large or small), number of cars owned by family (1, 2, 3, or 4), and whether family owns a foreign car. Excel produced the pivot table shown below.



Use this pivot table to determine how family size and number of cars owned influence the likelihood that a family owns a foreign car.


ANSWER:

The pivot table shows that the more cars a family owns, the more likely it is that they own a foreign car (makes sense!). Also, the percentage of large families who own a foreign car is larger than the similar percentage of small families (36.0% versus 10.4%).


29. An operations management professor is interested in how her students performed on her midterm exam. The data and histogram are presented below. The histogram represents the distribution of exam scores (where the maximum score is 100) for 50 students.


Student

Score

Student

Score

Student

Score

Student

Score

1

72

14

78

27

83

40

84

2

95

15

72

28

87

41

78

3

83

16

73

29

82

42

84

4

85

17

75

30

72

43

71

5

65

18

79

31

97

44

77

6

82

19

80

32

88

45

78

7

84

20

87

33

70

46

70

8

74

21

80

34

83

47

86

9

74

22

91

35

92

48

67

10

82

23

74

36

94

49

91

11

81

24

69

37

63

50

78

12

75

25

88

38

83







13

78

26

84

39

81






B
ased on this histogram, how would you characterize the students’ performance on this exam?


ANSWER:

Exam scores are fairly normally distributed. Majority of scores (76%) are between 70 and 90 points, while 12% of scores are above 90 and 12% of scores are 70 or below.

30. A health magazine reported that a man’s weight at birth has a significant impact on the chance that the man will suffer a heart attack during his life. A statistician analyzed a data set for a sample of 798 men, and produced the pivot table and histogram shown below. Determine how birth weight influences the chances that a man will have a heart attack.










ANSWER:

The above pivot table shows counts (as percentages of row) of heart attack versus birth weight, where birth weight has been grouped into categories. The percentages in each category with heart attacks have then been plotted versus weight at birth as shown in the histogram. It appears that the likelihood of a heart attack is greatest for light babies, and then decreases steadily, but increases slightly for the heaviest babies.


31. The table shown below contains information technology (IT) investment as a percentage of total investment for eight countries during the 1990s. It also contains the average annual percentage change in employment during the 1990s. Explain how these data shed light on the question of whether IT investment creates or costs jobs.



Country

% IT

% Change

Netherlands

2.5%

1.6%

Italy

4.1%

2.2%

Germany

4.5%

2.0%

France

5.5%

1.8%

Canada

8.3%

2.7%

Japan

8.3%

2.7%

Britain

8.3%

3.3%

U.S.

12.4%

3.7%


for eight countries during the 1990s









ANSWER:



QUESTIONS 32 THROUGH 35 ARE BASED ON THE FOLLOWING INFORMATION:
A real estate agent has gathered information on 40 houses that were recently sold in a local community. The data below represents the following variables: the selling price of each house (in thousands of dollars), the appraised value of each house (in thousands of dollars), the size of the house (in hundreds of square feet), and the number of bedrooms.

















House

Value

Price

Sq. Footage

# of Bedrooms

1

121.87

119.37

20.5

4

2

122.78

130.39

15.9

3

3

144.35

135.70

18.6

3

4

116.20

126.30

12.1

2

5

139.49

137.08

17.1

3

6

144.80

139.53

17.2

3

7

107.06

114.34

16.7

3

8

147.47

140.04

16.5

3

9

135.12

136.01

16.1

2

10

140.24

140.93

15.7

3

11

129.89

132.42

16.5

4

12

121.14

118.30

16.4

3

13

157.79

155.55

22.7

5

14

135.57

128.50

19.7

4

15

151.99

143.36

18.2

3

16

120.53

119.65

16.5

3

17

118.64

122.57

14.7

2

18

149.51

145.27

18.5

4

19

146.86

149.73

21.7

4

20

152.84

156.13

19.6

3

21

122.27

126.72

19.2

4

22

145.71

141.13

18.5

3

23

138.38

136.53

14.3

2

24

109.46

118.04

13.9

2

25

144.68

153.70

21.3

4

26

133.27

126.31

18.9

4

27

133.27

134.02

16.4

2

28

150.38

141.56

20.7

3

29

135.26

142.96

18.1

3

30

112.60

118.53

14.6

2

31

114.23

121.59

14.1

2

32

153.24

146.40

21.9

4

33

125.89

141.25

15.8

3

34

135.62

130.73

18.4

3

35

121.45

115.97

16.3

2

36

132.45

125.21

15.8

3

37

135.83

130.37

17.6

3

38

125.76

119.75

18.1

4

39

125.84

120.93

17.1

3

40

135.32

126.80

18.9

3

32. Indicate whether each of the four variables is continuous or discrete.
ANSWER:

Value – continuous

Price – continuous

Square Footage – continuous

Number of Bedrooms – discrete
33. The histograms for both the appraised values and selling prices are presented below. In what ways are the two distributions similar? In what ways are they different?






ANSWER:

Both distributions seem to center around $130,000 - $140,000. The selling price appears to be slightly higher than the appraisal value.


34. The following scatterplot compares the selling price and the appraised value.


Is there a linear relationship between these two variables? If so, how would you characterize the relationship?


ANSWER:

Yes, there is a linear relationship. Correlation value = 0.877 represents a rather strong relationship. You can also see from the scatterplot, that there is a positive relationship between the selling price and the appraisal value.


35. The two scatterplots below use the same home sales data presented above. The first chart shows the relationship between the size of the home and the selling price. The second chart examines the relationship between the number of bedrooms in the home and its selling price. Which of these two variables (the size of the home or the number of bedrooms) seems to have the stronger relationship with the home’s selling price? Justify your answer.





ANSWER:

The relationship between selling price and house size (in square feet) seems to be a stronger relationship. The correlation value is higher for house size (0.657 to 0.452). The house size and the number of bedrooms seem to be closely related, but the house size variable seems to offer more information. The number of bedrooms is a discrete variable.



QUESTIONS 36 THROUGH 42 ARE BASED ON THE FOLLOWING INFORMATION:
A recent survey data collected from 1000 randomly selected Internet users. The characteristics of the users include their gender, age, education, marital status and annual income. Using Excel, the following pivot tables were produced.













36. What percentage of these Internet users are men under the age of 30?
ANSWER:

Approximately 19% of these Internet users are men under the age of 30.


37. What percentage of these Internet users are single with no formal education beyond high school?
ANSWER:

Approximately 16% of these Internet users are single with no formal education beyond high school.


38. What percentage of these Internet users are currently employed?
ANSWER:

Approximately 77% of these Internet users are currently employed.


39. What is the average salary of the employed Internet users in this sample?
ANSWER:

The average salary of employed Internet users in this sample is about $60,564.


40. What percentage of these Internet users are married with formal education beyond high school?
ANSWER:

Approximately 37% of these Internet users are married with formal education beyond high school?


41. What percentage of these Internet users are married?
ANSWER:

Approximately 69% of these Internet users are married.


42. What percentage of these Internet users are in the 58-71 age group?
ANSWER:

Approximately 9% of these Internet users are in the 58-71 age group.



QUESTIONS 43 THROUGH 45 ARE BASED ON THE FOLLOWING INFORMATION:
Below you will find current annual salary data and related information for 30 employees at Gamma Technologies, Inc. These data include each selected employees gender (0 = male, 1 = female), age, number of years of relevant work experience prior to employment at Gamma, number of years of employment at Gamma, the number of years of post-secondary education, and annual salary.

Gamma Technologies, Inc. Employee Salary Structure























Gender

Age

Prior

Experience



Gamma

Experience



Education

Annual

Salary


1

39

5

12

4

$37,500

0

44

12

8

6

$50,912

0

24

0

2

4

$29,356

1

25

2

1

4

$27,750

0

56

5

24

8

$97,844

1

41

9

10

4

$48,442

1

33

6

2

6

$40,207

0

37

11

6

4

$42,331

1

51

12

16

6

$87,489

0

23

0

1

4

$26,118

1

36

5

5

6

$40,025

0

58

9

22

4

$88,763

0

31

1

1

6

$35,829

1

21

0

1

2

$17,784

0

47

5

16

4

$54,199

1

35

3

7

4

$36,932

1

52

12

14

8

$93,278

0

29

3

3

2

$22,100

1

42

11

7

4

$49,987

0

60

10

21

4

$85,471

1

50

8

13

4

$52,220

1

33

1

2

6

$36,109

0

26

0

5

2

$23,105

0

38

6

6

6

$39,455

1

44

7

12

4

$49,861

0

25

0

3

4

$30,327

1

37

8

5

4

$31,008

0

53

13

13

6

$90,874

0

46

7

18

4

$57,966

1

20

0

1

0

$16,500

43. Indicate the type of data for each of the six variables included in this set.
ANSWER:

Gender – categorical, nominal

Age – numerical, continuous

Prior experience – numerical, discrete

Gamma experience – numerical, discrete

Education – numerical, discrete

Annual salary – numerical, continuous
4
4. Based on the histogram shown below, how would you describe the age distribution

for these data?


ANSWER:

The age distribution is skewed slightly to the right. Largest grouping is in the 30-40 range. This means that most workers are above the age of 30 years and only one worker is 20 years old or younger.


45. Based on the histogram shown below, how would you describe the salary

distribution for these data?







ANSWER:

The salary distribution is skewed to the right. There appears to be several workers who are being paid substantially more than the others. If you eliminate those above $80,000, the salaries are fairly normally distributed around $35,000.




QUESTIONS 46 THROUGH 55 ARE BASED ON THE FOLLOWING INFORMATION:
A sample of 150 students at a State University was taken after the final business statistics exam to ask them whether they went partying the weekend before the final or spent the weekend studying, and whether they did well or poorly on the final. The following table contains the result.






Did Well in Exam

Did Poorly in Exam

Studying for Exam

60

15

Went Partying

22

53

46. Of those in the sample who went parting the weekend before the final exam, what percentage of them did well in the exam?


ANSWER:

22 out of 75, or 29.33%


47. Of those in the sample who did well on the final exam, what percentage of them went partying the weekend before the exam?
ANSWER:

22 out of 82, or 26.83%


48. What percentage of the students in the sample went partying the weekend before the final exam and did well in the exam?
ANSWER:

22 out of 150, or 14.67%


49. What percentage of the students in the sample spent the weekend studying and did well in the final exam?
ANSWER:

60 out of 150, or 40%

50. What percentage of the students in the sample went partying the weekend before the final exam and did poorly on the exam?
ANSWER:

53 out of 150, or 35.33%


51. If the sample is a good representation of the population, what percentage of the students in the population should we expect to spend the weekend studying and do poorly on the final exam?
ANSWER:

15 out of 150, or 10%


52. If the sample is a good representation of the population, what percentage of those who spent the weekend studying should we expect to do poorly on the final exam?
ANSWER:

15 out of 75, or 20%


53. If the sample is a good representation of the population, what percentage of those who did poorly on the final exam should we expect to have spent the weekend studying?
ANSWER:

15 out of 68, or 22.06%


54. Of those in the sample who went parting the weekend before the final exam, what percentage of them did poorly in the exam?
ANSWER:

53 out of 75, or 70.67%


55. Of those in the sample who did well in the final exam, what percentage of them spent the weekend before the exam studying?
ANSWER:

60 out of 82, or 73.17%



QUESTIONS 56 THROUGH 63 ARE BASED ON THE FOLLOWING INFORMATION:
The histogram below represents scores achieved by 250 job applicants on a personality profile.


Relative Frequency

56. What percentage of the job applicants scored between 30 and 40?


ANSWER:

10%
57. What percentage of the job applicants scored below 60?


ANSWER:

90%
58. How many job applicants scored between 10 and 30?


ANSWER:

100


59. How many job applicants scored above 50?
ANSWER:

50

60. Seventy percent of the job applicants scored above what value?


ANSWER:

20
61. Half of the job applicants scored below what value?


ANSWER:

30
62. What percentage of the applicants scored at most 50?


ANSWER:

80%
63. How many applicants scored between 10 and 50?


ANSWER:

175


TRUE / FALSE QUESTIONS


64. A frequency table indicates how many observations fall within each category, and a histogram is its graphical analog.



ANSWER: T
65. Individual observations within each category may be found in a frequency table.

ANSWER: F

66. In the term “frequency table,” frequency refers to the number of data values falling within each category.



ANSWER: T
67. A frequency table is a listing of the individual observations arranged in ascending or descending order.

ANSWER: F
68. Both ordinal and nominal variables are categorical.

ANSWER: T
69. A variable is some characteristic of a population, while data are the observed values of a variable based on a sample.

ANSWER: F
70. Age, height, and weight are examples of numerical data.

ANSWER: T
71. Categorical variables can be coded numerically or left uncoded.
ANSWER: T
72. Data can be categorized as cross-sectional or time series.
ANSWER: T
73. All nominal data may be treated as ordinal data.
ANSWER: F
74. Four different shapes of histograms are commonly observed: symmetric, positively skewed, negatively skewed, and bimodal.
ANSWER: T
75. A histogram is skewed to the right (or positively skewed) if it has a single peak and the values of the distribution extend much further to the left of the peak than to the right of the peak.
ANSWER: F
76. A histogram is skewed to the left (or negatively skewed) if it has a single peak and the values of the distribution extend much further to the right of the peak than to the left of the peak.
ANSWER: F
77. A histogram is said to be symmetric if it has a single peak and looks approximately the same to the left and right of the peak.
ANSWER: T
78. A skewed histogram is one with a long tail extending either to the right or left. The former is called negatively skewed, and the later is called positively skewed.
ANSWER: F
79. Some histograms have two or more peaks. This is often an indication that the data come from two or more distinct populations.
ANSWER: T
80. A bimodal histogram is one with two peaks equal in height.

ANSWER: F
81. Creating a histogram can be a tedious task, but an add-in such as StatPro makes it relatively easy. However, you must be prepared to specify the categories.
ANSWER: T
82. The scatterplot is a graphical technique used to describe the relationship between two numerical variables.
ANSWER: T

83. If we draw a straight line through the points in a scatterplot and most of the points fall close to the line, we say that there is a strong positive linear relationship between the two variables.



ANSWER: F
84. Time series data are often graphically depicted on a line chart, which is a plot of the variable of interest over time.

ANSWER: T
85. Statisticians often refer to the pivot tables as contingency tables or crosstabs.

ANSWER: T
86. Numerical variables usually represent membership in groups or categories.
ANSWER: F
87. The time required to drive from Iowa City to Lansing is an example of a discrete random variable.
ANSWER: F
88. The number of car insurance policy holders is an example of a discrete random variable
ANSWER: T
89. A population includes all elements or objects of interest in a study, whereas a sample is a subset of the population used to gain insights into the characteristics of the population.

ANSWER: T
90. A variable (or field) is an attribute, or measurement, on members of a population, whereas an observation (or case or record) is a list of all variable values for a single member of a population.
ANSWER: T
91. A variable is usually listed in a row; an observation is usually listed in a column.
ANSWER: F

92. Phone numbers, Social Security numbers, and zip codes are examples of numerical variables.


ANSWER: F
93. Cross-sectional data are data on a population at a distinct point in time, whereas time series data are data collected across time.
ANSWER: T




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