the Relation between Two Variablessite.iugaza.edu.ps/mriffi/files/2018/02/Statistics... · Ch. 4...

Ch. 4 Describing the Relation between Two Variables

4.1 Scatter Diagrams and Correlation

1 Draw and interpret scatter diagrams.

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Construct a scatter diagram for the data.

1) The data below are the final exam scores of 10 randomly selected history students and the number of hours

they studied for the exam.

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

x

y

x

y

2) The data below are the temperatures on randomly chosen days during a summer class and the number of

absences on those days.

Temperature, x

Number of absences, y

72

3

85

7

91

10

90

10

88

8

98

15

75

4

100

15

80

5

x

y

x

y

3) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly

selected adults.

Age, x

Pressure, y

38

116

41

120

45

123

48

131

51

142

53

145

57

148

61

150

65

152

x

y

x

y

4) The data below are the number of absences and the final grades of 9 randomly selected students from a

literature class.

Number of absences, x

Final grade, y

0

98

3

86

6

80

4

82

9

71

2

92

15

55

8

76

5

82

x

y

x

y

5) A manager wishes to determine the relationship between the number of miles (in hundreds of miles) the

managerʹs sales representatives travel per month and the amount of sales (in thousands of dollars) per month.

Miles traveled, x

Sales, y

2

31

3

33

10

78

7

62

8

65

15

61

3

48

1

55

11

120

x

y

x

y

6) In order for employees of a company to work in a foreign office, they must take a test in the language of the

country where they plan to work. The data below show the relationship between the number of years that

employees have studied a particular language and the grades they received on the proficiency exam.

Number of years, x

Grades on test, y

3

61

4

68

4

75

5

82

3

73

6

90

2

58

7

93

3

72

x

y

x

y

7) In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the

yield of wheat (bushels per acre).

Rainfall (in inches), x

Yield (bushels per acre), y

10.5

50.5

8.8

46.2

13.4

58.8

12.5

59.0

18.8

82.4

10.3

49.2

7.0

31.9

15.6

76.0

16.0

78.8

x

y

x

y

8) Five brands of cigarettes were tested for the amounts of tar and nicotine they contained. All measurements are

in milligrams per cigarette.

Cigarette Tar Nicotine

Brand A 16 1.2

Brand B 13 1.1

Brand C 16 1.3

Brand D 18 1.4

Brand E 6 0.6

x

y

x

y

9) The scores of nine members of a local community college womenʹs golf team in two rounds of tournament play

are listed below.

Player 1 2 3 4 5 6 7 8 9

Round 1 85 90 87 78 92 85 79 93 86

Round 2 90 87 85 84 86 78 77 91 82

x

y

x

y

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Make a scatter diagram for the data. Use the scatter diagram to describe how, if at all, the variables are related.

10) x 3 8 6 5 9 4

y 4 7 5 5 6 3

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

A) The variables appear to be

positively, linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

B) The variables do not appear to be

linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

C) The variables appear to be

negatively, linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

D) The variables do not appear to be

linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

11) x 6 4 -1 3 2 8 1y 17 19 15 18 21 17 24

x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4

A) The variables do not appear to be

linearly related.

x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4

B) The variables appear to be


x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4

C) The variables do not appear to be

linearly related.

x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4

D) The variables appear to be


x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4x-12 -8 -4 4 8 12 16 20

y28

24

20

16

12

8

4

-4

12)Subject A B C D E F G

x Time watching TV 9 5 3 8 8 6 7

y Time on Internet 14 12 8 17 18 9 18

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

A) The variables appear to be


x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

B) The variables do not appear to be

linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

C) The variables appear to be


x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

D) The variables do not appear to be

linearly related.

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2

x2 4 6 8 10 12 14 16 18 20

y20

18

16

14

12

10

8

6

4

2


Provide an appropriate response.

13) An agricultural business wants to determine if the rainfall in inches can be used to predict the yield per acre on

a wheat farm. Identify the predictor variable and the response variable.

14) A college counselor wants to determine if the number of hours spent studying for a test can be used to predict

the grades on a test. Identify the predictor variable and the response variable.


15) The variable is the variable whose value can be explained by the variable.

A) response; predictor B) response; lurking

C) lurking; response D) predictor Response

2 Describe the properties of the linear correlation coefficient.


Use the scatter diagrams shown, labeled a through f to solve the problem.

16) In which scatter diagram is r = 0.01?

a

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

b

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

c

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

d

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

e

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

f

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

A) e B) c C) f D) d

17) In which scatter diagram is r = 1?

a

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

b

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

c

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

d

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

e

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

f

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

A) b B) a C) f D) d

18) In which scatter diagram is r = -1?

a

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

b

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

c

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

d

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

e

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

f

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

A) a B) b C) f D) d

19) Which scatter diagram indicates a perfect positive correlation?

a

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

b

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

c

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

d

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

e

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

x1 2 3 4 5 6 7

y

12

10

8

6

4

2

f

x1 2 3 4 5 6

y

12

10

8

6

4

2

x1 2 3 4 5 6

y

12

10

8

6

4

2

A) b B) a C) c D) f

The scatter diagram shows the relationship between average number of years of education and births per woman of

child bearing age in selected countries. Use the scatter plot to determine whether the statement is true or false.

20) There is a strong positive correlation between years of education and births per woman.

Births per Woman

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1

Average number of years of education

of Married Women of Child-Bearing Age

A) False B) True

21) There is no correlation between years of education and births per woman.

Births per Woman

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1



A) False B) True

22) There is a strong negative correlation between years of education and births per woman.

Births per Woman

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1



A) True B) False

23) There is a causal relationship between years of education and births per woman.

Births per Woman

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1

2 4 6 8 10 12 14

10

9

8

7

6

5

4

3

2

1



A) False B) True



24) Construct a scatter diagram for the given data. Determine whether there is a positive linear correlation,

negative linear correlation, or no linear correlation.

x

y

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12



x

y

-5

11

-3

6

4

-6

1

-1

-1

3

-2

4

0

1

2

-4

3

-5

-4

8

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12



x

y

-5

11

-3

-6

4

8

1

-3

-1

-2

-2

1

0

5

2

-5

3

6

-4

7

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

27) The numbers of home runs that Mark McGwire hit in the first 13 years of his major league baseball career are

listed below. (Source: Major League Handbook) Construct a scatter diagram for the data. Is there a relationship

between the home runs and the batting averages?

Home Runs

Batting Average

33 39 22 42 9 9 39 52 58 70

.231 .235 .201 .268 .33 .252 .274 .312 .274 .299

x15 30 45 60 75

y

0.35

0.3

0.25

0.2

0.15

x15 30 45 60 75

y

0.35

0.3

0.25

0.2

0.15

28) The data below represent the numbers of absences and the final grades of 15 randomly selected students from

an astronomy class. Construct a scatter diagram for the data. Do you detect a trend?

Student Number

of Absences

Final Grade

as a Percent

1 5 79

2 6 78

3 2 86

4 12 56

5 9 75

6 5 90

7 8 78

8 15 48

9 0 92

10 1 78

11 9 81

12 3 86

13 10 75

14 3 89

15 11 65

x5 10 15

y100

90

80

70

60

50

40

x5 10 15

y100

90

80

70

60

50

40


29) A researcher determines that the linear correlation coefficient is 0.85 for a paired data set. This indicates that

there is

A) a strong positive linear correlation.

B) a strong negative linear correlation.

C) no linear correlation but that there may be some other relationship.

D) insufficient evidence to make any decision about the correlation of the data.

30) An instructor wishes to determine if there is a relationship between the number of absences from his class and

a studentʹs final grade in the course. What is the explanatory variable?

A) Absences B) Final Grade

C) The instructorʹs point scale for attendance D) Studentʹs performance on the final examination

31) A medical researcher wishes to determine if there is a relationship between the number of prescriptions written

by pediatricians and the ages of the children for whom the prescriptions are written. She surveys all the

pediatricians in a geographical region to collect her data. What is the response variable?

A) Number of prescriptions written

B) Pediatricians surveyed

C) Age of the children for whom prescriptions were written

D) Number of children for whom prescriptions were written

32) True or False: A doctor wishes to determine the relationship between a maleʹs age and that maleʹs total

cholesterol level. He tests 200 males and records each maleʹs age and that maleʹs total cholesterol level. The

males cholesterol level is the explanatory variable?

A) False B) True

33) A scatter diagram locates a point in a two dimensional plane. The diagram locates the

variable on the horizontal axis and the variable on the vertical axis.

A) explanatory; response B) response; explanatory

C) response; study D) study; explanatory

34) A history instructor has given the same pretest and the same final examination each semester. He is interested

in determining if there is a relationship between the scores of the two tests. He computes the linear correlation

coefficient and notes that it is 1.15. What does this correlation coefficient value tell the instructor?

A) The history instructor has made a computational error.

B) There is a strong positive correlation between the tests.

C) There is a strong negative correlation between the tests.

D) The correlation is something other than linear.

35) A traffic officer is compiling information about the relationship between the hour or the day and the speed over

the limit at which the motorist is ticketed. He computes a correlation coefficient of 0.12. What does this tell

the officer?

A) There is a weak positive linear correlation.

B) There is a moderate positive linear correlation.

C) There is a moderate negative linear correlation.

D) There is insufficient evidence to make any conclusions about the relationship between the variables.

3 Compute and interpret the linear correlation coefficient.



36) Calculate the linear correlation coefficient for the data below.

x

y

-11

-5 -9

-3

-2

14

-5

6

-7

3

-8

-1

-6

4

-4

8

-3

11

-10

-3

A) 0.990 B) 0.881 C) 0.819 D) 0.792


x

y

-3 19

-1 14

6 2

3 7

1 11

0 12

2 9

4 4

5 3

-2 16

A) -0.995 B) -0.671 C) -0.778 D) -0.885


x

y

-11

7

-9

-10 -2

4

-5

-7 -7

-6 -8

-3

-6

1

-4

-9 -3

2

-10

3

A) -0.104 B) -0.132 C) -0.549 D) -0.581

39) The data below are the final exam scores of 10 randomly selected calculus students and the number of hours

they slept the night before the exam. Calculate the linear correlation coefficient.

Hours, x

Scores, y

4

64

6

79

3

59

9

87

3

65

5

77

5

84

6

89

7

89

4

70

A) 0.847 B) 0.991 C) 0.761 D) 0.654

40) The data below are the average one-way commute times (in minutes) of selected students during a summer

literature class and the number of absences for those students for the term. Calculate the linear correlation

coefficient.

Commute time (min), x


70

8

83

12

89

15

88

15

86

13

96

20

73

9

98

20

78

10

A) 0.980 B) 0.890 C) 0.881 D) 0.819

41) The data below are the ages and annual pharmacy b ills (in dollars) of 9 randomly selected employees.

Calculate the linear correlation coefficient.

Age, x

Pharmacy bill ($), y

41

111

44

115

48

118

51

126

54

137

56

140

60

143

64

145

68

147

A) 0.960 B) 0.998 C) 0.890 D) 0.908

42) The data below are the number of hours worked (per week) and the final grades of 9 randomly selected

students from a drama class. Calculate the linear correlation coefficient.

Hours worked, x

Final Grade, y

2

90

5

78

8

72

6

74

11

63

4

84

17

47

10

68

7

74

A) -0.991 B) -0.888 C) -0.918 D) -0.899

43) A manager wishes to determine the relationship between the number of years the managerʹs sales

representatives have been with the company and their average monthly sales (in thousands of dollars).

Calculate the linear correlation coefficient.

Years with company, x

Sales, y

5

34

6

36

13

81

10

65

11

68

18

64

6

51

4

58

14

123

A) 0.632 B) 0.561 C) 0.717 D) 0.791

44) In order for a companyʹs employees to work in a foreign office, they must take a test in the language of the

country where they plan to work. The data below shows the relationship between the number of years that

employees have studied a particular language and the grades they received on the proficiency exam. Calculate

the linear correlation coefficient.

Number of years, x

Grades on test, y

5

63

6

70

6

77

7

84

5

75

8

92

4

60

9

95

5

74

A) 0.934 B) 0.911 C) 0.891 D) 0.902


yield of wheat (bushels per acre). Calculate the linear correlation coefficient.



7.8

48.5

6.1

44.2

10.7

56.8

9.8

57

16.1

80.4

7.6

47.2

4.3

29.9

12.9

74

13.3

76.8

A) 0.981 B) 0.998 C) 0.900 D) 0.899


46) Calculate the coefficient of correlation, r, letting Row 1 represent the x-values and Row 2 represent the

y-values. Now calculate the coefficient of correlation, r, letting Row 2 represent the x-values and Row 1

represent the y-values. What effect does switching the explanatory and response variables have on the linear

correlation coefficient?

Row 1

Row 2

-4 -9

-2 9

5 10

2 2

0 -1

-1 -5

1 0

3 4

4 7

-3 9

4 Determine whether a linear relation exists between two variables.


Compute the linear correlation coefficient between the two variables and determine whether a linear relation exists.

47) x 2 3 5 5 6

y 1.3 1.6 2.1 2.2 2.7

A) r = 0.983; linear relation exists B) r = 0.983; no linear relation exists

C) r = 0.883; linear relation exists D) r = 0.883; no linear relation exists

48) x -1 1 8 5 3 2 4 6 7 0y -13 -11 6 -2 -5 -9 -4 0 3 -11A) r = 0.990; linear relation exists B) r = 0.881; no linear relation exists


49) x -1 1 8 5 3 2 4 6 7 0y 18 13 1 6 10 11 8 3 2 15A) r = -0.995; linear relation exists B) r = -0.995; no linear relation exists

C) r = -0.885; no linear relation exists D) r = -0.885; linear relation exists

50) x 9 2 3 4 2 5 9 10

y 85 52 55 68 67 86 83 73


C) r = -0.708; linear relation exists D) r = 0.708; no linear relation exists

51) x 10 11 16 9 7 15 16 10

y 96 51 62 58 89 81 46 51

A) r = -0.335; no linear relation exists B) r = 0.462; linear relation exists

C) r = -0.335; linear relation exists D) r = -0.284; no linear relation exists

52) The table below shows the scores on an end-of-year project of 10 randomly selected architecture students and

the number of days each student spent working on the project.

Days, x

Score, y

4

68

6

83

3

63

9

91

3

69

5

81

5

88

6

93

7

93

4

74



53) The table below shows the ages and weights (in pounds) of 9 randomly selected tennis coaches.

Age, x

Weight (pounds), y

42

120

45

124

49

127

52

135

55

146

57

149

61

152

65

154

69

156


C) r = 0.908; no linear relation exists D) r = 0.908; linear relation exists

54) The table shows the number of days off last year and the earnings for the year (in thousands of dollars) for nine

randomly selected insurance salesmen.

Number of days off, x

Earnings for the year (thousands of dollars), y

2

88

5

76

8

70

6

72

11

61

4

82

17

45

10

66

7

72

A) r = -0.991; linear relation exists B) r = -0.991; no linear relation exists

C) r = -0.899; linear relation exists D) r = -0.899; no linear relation exists

55) A manager wishes to determine whether there is a relationship between the number of years her sales

representatives have been with the company and their average monthly sales. The table shows the years of

service for each of her sales representatives and their average monthly sales (in thousands of dollars).

Years with company, x 6 7 14 11 12 19 7 5 15

Sales , y 29 31 76 60 63 59 46 53 118

A) r = 0.632; no linear relation exists B) r = 0.632; linear relation exists


56) To investigate the relationship between yield of soybeans and the amount of fertilizer used, a researcher

divides a field into eight plots of equal size and applies a different amount of fertilizer to each plot. The table

shows the yield of soybeans and the amount of fertilizer used for each plot.

Amount of fertilizer (pounds) ,x 1 1.5 2 2.5 3 3.5 4 4.5

Yield of soybeans (pounds), y 25 21 27 28 36 35 32 34



5 Explain the difference between correlation and causation.



57) A variable that is related to either the response variable or the predictor variable or both, but which is excluded

from the analysis is a

A) lurking variable. B) random variable.

C) discrete variable. D) qualitative variable.


58) For a random sample of 100 American cities, the linear correlation coefficient between the number of robberies

last year and the number of schools in the city was found to be r = 0.725. What does this imply? Does this

suggest that building more schools in a city could lead to more robberies? Why or why not? What is a likely

lurking variable?

59) For a random sample of 30 countries, the linear correlation coefficient between the infant mortality rate and the

average number of cars per capita was found to be r = -0.717. What does this imply? Does this suggest that if

people buy more cars, this could lower the infant mortality rate? Why or why not? What is a likely lurking

variable?

60) A random sample of 200 men aged between 20 and 60 was selected from a certain city. The linear correlation

coefficient between income and blood pressure was found to be r = 0.807. What does this imply? Does this

suggest that if a man gets a salary raise his blood pressure is likely to rise? Why or why not? What are likely

lurking variables?

4.2 Least-Squares Regression

1 Find the least-squares regression line and use the line to make predictions.



1) Find the equation of the regression line for the given data. Round values to the nearest thousandth.

x

y

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

A) y^= 2.097x - 0.552 B) y

^ = 0.522x - 2.097

C) y^ = 2.097x + 0.552 D) y

^ = -0.552x + 2.097


x

y

-5

11

-3

6

4

-6

1

-1

-1

3

-2

4

0

1

2

-4

3

-5

-4

8

A) y^= -1.885x + 0.758 B) y

^ = 0.758x + 1.885 C) y

^= -0.758x - 1.885 D) y

^= 1.885x - 0.758


x

y

-5

11

-3

-6

4

8

1

-3

-1

-2

-2

1

0

5

2

-5

3

6

-4

7

A) y^= -0.206x + 2.097 B) y

^= 2.097x - 0.206 C) y

^= 0.206x - 2.097 D) y

^= -2.097x + 0.206


they slept the night before the exam. Find the equation of the regression line for the given data. What would be

the predicted score for a history student who slept 7 hours the previous night? Is this a reasonable question?

Round the regression line values to the nearest hundredth, and round the predicted score to the nearest whole

number.

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

A) y^= 5.04x + 56.11; 91; Yes, it is reasonable.

B) y^= 5.04x + 56.11; 91; No, it is not reasonable. 7 hours is well outside the scope of the model.

C) y^= -5.04x + 56.11; 21; No, it is not reasonable. 7 hours is well outside the scope of the model.

D) y^= -5.04x + 56.11; 21; Yes, it is reasonable.


they slept the night before the exam. Find the equation of the regression line for the given data. What would be

the predicted score for a history student who slept 15 hours the previous night? Is this a reasonable question?

Round your predicted score to the nearest whole number. Round the regression line values to the nearest

hundredth.

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

A) y^= 5.04x + 56.11; 132; No, it is not reasonable. 15 hours is well outside the scope of the model.

B) y^= 5.04x + 56.11; 132; Yes, it is reasonable.

C) y^= -5.04x + 56.11; -20; No, it is not reasonable.

D) y^= -5.04x + 56.11; -20; Yes, it is reasonable.

6) The data below are the average one-way commute times (in minutes) for selected students and the number of

absences for those students during the term. Find the equation of the regression line for the given data. What

would be the predicted number of absences if the commute time was 95 minutes? Is this a reasonable question?

Round the predicted number of absences to the nearest whole number. Round the regression line values to the

nearest hundredth.



72

3

85

7

91

10

90

10

88

8

98

15

75

4

100

15

80

5

A) y^ = 0.45x - 30.27; 12 absences; Yes, it is reasonable.

B) y^ = 0.45x - 30.27; 12 absences; No, it is not reasonable. 95 minutes is well outside the scope of the model.

C) y^ = 0.45x + 30.27; 73 absences; Yes, it is reasonable.

D) y^ = 0.45x + 30.27; 73 absences; No, it is not reasonable. 95 minutes is well outside the scope of the model.

7) The data below are the average one-way commute times (in minutes) for selected students and the number of

absences for those students during the term. Find the equation of the regression line for the given data. What

would be the predicted number of absences if the commute time was 40 minutes? Is this a reasonable question?

Round the predicted number of absences to the nearest whole number. Round the regression line values to the

nearest hundredth.



72

3

85

7

91

10

90

10

88

8

98

15

75

4

100

15

80

5

A) y^ = 0.45x - 30.27; -12 absences; No, it is not reasonable. 40 minutes is well outside the scope of the model.

B) y^ = 0.45x - 30.27; -12 absences; Yes, it is reasonable.

C) y^ = 0.45x + 30.27; 48 absences; Yes, it is reasonable.

D) y^ = 0.45x + 30.27; 48 absences; No, it is not reasonable. 40 minutes is well outside the scope of the model.

8) The data below are ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly

selected adults. Find the equation of the regression line for the given data. What would be the predicted

pressure if the age was 60? Round the predicted pressure to the nearest whole number. Round the regression

line values to the nearest hundredth.

Age, x

Pressure, y

38

116

41

120

45

123

48

131

51

142

53

145

57

148

61

150

65

152

A) y^= 1.49x + 60.46; 150 mm B) y

^= 60.46x - 1.49; 3626 mm

C) y^= 1.49x - 60.46; 29 mm D) y

^= 60.46x + 1.49; 3629 mm

9) The data below are the number of absences and the final grades of 9 randomly selected students from a

literature class. Find the equation of the regression line for the given data. What would be the predicted final

grade if a student was absent 14 times? Round the regression line values to the nearest hundredth. Round the

predicted grade to the nearest whole number.


Final grade, y

0

98

3

86

6

80

4

82

9

71

2

92

15

55

8

76

5

82

A) y^ = -2.75x + 96.14; 58 B) y

^ = 96.14x - 2.75; 1343

C) y^= -2.75x - 96.14; 134.64 D) y

^= -96.14x + 2.75; 1343

10) A manager wishes to determine the relationship between the number of miles traveled (in hundreds of miles)

by her sales representatives and their amount of sales (in thousands of dollars) per month. Find the equation of

the regression line for the given data. What would be the predicted sales if the sales representative traveled 0

miles? Is this reasonable? Why or why not? Round the regression line values to the nearest hundredth.

Miles traveled, x

Sales, y

2

31

3

33

10

78

7

62

8

65

15

61

3

48

1

55

11

120

A) y^ = 3.53x + 37.92; $37,920; No; it is not reasonable for a representative to travel 0 miles and have a positive

amount of sales.

B) y^ = 3.53x + 37.92; $3792; No; it is not reasonable for a representative to travel 0 miles and have a positive

amount of sales.

C) y^ = 3.53x + 37.92; $37,920; Yes, it is reasonable.

D) y^= 37.92x + 3.53; $3792; Yes, it is reasonable.

11) A manager wishes to determine the relationship between the number of years her sales representatives have

been employed by the firm and their amount of sales (in thousands of dollars) per month. Find the equation of

the regression line for the given data. What would be the predicted sales if the sales representative was

employed by the firm for 30 years Is this reasonable? Why or why not? Round the regression line values to the

nearest hundredth.

Years employed, x

Sales, y

2

31

3

33

10

78

7

62

8

65

15

61

3

48

1

55

11

120

A) y^ = 3.53x + 37.92; $143,820; No; it is not reasonable. 30 years of employment is well outside the scope of

the model.

B) y^ = 3.53x + 37.92; $143,820;; Yes, it is reasonable.

C) y^ = 3.53x - 37.92; $67,980; No; it is not reasonable. 30 years of employment is well outside the scope of the

model.

D) y^ = 3.53x - 37.92; $67,980; Yes; it is reasonable.

12) In order for a companyʹs employees to work in a foreign office, they must take a test in the language of the

country where they plan to work. The data below shows the relationship between the number of years that

employees have studied a particular language and the grades they received on the proficiency exam. Find the

equation of the regression line for the given data. Round the regression line values to the nearest hundredth.

Number of years, x

Grades on test, y

3

61

4

68

4

75

5

82

3

73

6

90

2

58

7

93

3

72

A) y^ = 6.91x + 46.26 B) y

^= 6.91x - 46.26 C) y

^= 46.26x - 6.91 D) y

^ = 46.26x + 6.91


yield of wheat (bushels per acre). Find the equation of the regression line for the given data. Round the

regression line values to the nearest thousandth.



10.5

50.5

8.8

46.2

13.4

58.8

12.5

59.0

18.8

82.4

10.3

49.2

7.0

31.9

15.6

76.0

16.0

78.8

A) y^ = 4.379x + 4.267 B) y

^= -4.379x + 4.267 C) y

^= 4.267x + 4.379 D) y

^ = 4.267x - 4.379


14) Find the equation of the regression line by letting Row 1 represent the x-values and Row 2 represent the

y-values. Now find the equation of the regression line letting Row 2 represent the x-values and Row 1

represent the y-values. What effect does switching the explanatory and response variables have on the

regression line?

Row 1

Row 2

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

15) Is the number of games won by a major league baseball team in a season related to the teamʹs batting average?

Data from 14 teams were collected and the summary statistics yield:

y =∑ 1,134, x∑ = 3.642, y2∑ = 93,110, x2∑ = 0.948622, and xy∑ = 295.54

Find the least squares prediction equation for predicting the number of games won, y, using a straight-line

relationship with the teamʹs batting average, x.

16) The table shows, for the years 1997-2012, the mean hourly wage for residents of the town of Pity Me and the

mean weekly rent paid by the residents.

Year Mean weekly rent

(dollars)

Mean hourly wage

(dollars)

Year Mean weekly rent

(dollars)

Mean hourly wage

(dollars)

1997 57 10.38 2005 116 28.99

1998 59 10.89 2006 113 28.63

1999 62 11.96 2007 112 36.75

2000 63 12.46 2008 86 14.55

2001 86 17.72 2009 90 17.90

2002 119 28.07 2010 90 14.67

2003 131 35.24 2011 100 17.97

2004 122 31.87 2012 115 22.23

Summary statistics yield: SSxx = 1222.2771, SSxy = 3031.7125, SSyy = 9144.9375, x = 21.2675, and

y = 95.0625. Find the least squares line that uses mean hourly wage to predict mean weekly rent. Round values

to the nearest ten-thousandth.


17) A residual is the difference between

A) the observed value of y and the predicted value of y.

B) the observed value of x and the predicted value of x.

C) the observed value of y and the predicted value of x.

D) the observed value of x and the predicted value of y.

18) The least squares regression line

A) minimizes the sum of the residuals squared.

B) maximizes the sum of the residuals squared.

C) minimizes the mean difference between the residuals squared.

D) maximizes the mean difference between the residuals squared.

19) For a given data set, the equation of the least squares regression line will always pass through

A) (x, y). B) every point in the given data set.

C) at least two point in the given data set. D) the y-intercept and the slope.

2 Interpret the slope and the y-intercept of the least-squares regression line.



20) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a

section of the county called East Meadow. One of the many variables thought to be an important predictor of

appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple

linear regression model, y^ = β0 + β1x, where y = appraised value of the house (in $thousands) and x = number

of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were

obtained:

y^= 74.80 + 17.80x

sβ = 71.24, t = 1.05 (for testing β0)

sβ = 2.63, t = 7.49 (for testing β1)

SSE = 60,775, MSE = 841, s = 29, r2 = .44

Range of the x-values: 5 - 11

Range of the y-values: 160 - 300

Give a practical interpretation of the estimate of the slope of the least squares line.

A) For each additional room in the house, we estimate the appraised value to increase $17,800.

B) For each additional room in the house, we estimate the appraised value to increase $74,800.

C) For each additional dollar of appraised value, we estimate the number of rooms in the house to increase

by 17.80 rooms.

D) For a house with 0 rooms, we estimate the appraised value to be $74,800.

21) A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a

section of the county called East Meadow. One of the many variables thought to be an important predictor of

appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple

linear regression model, y^= β0 + β1x, where y = appraised value of the house (in $thousands) and x = number

of rooms. Using data collected for a sample of n = 74 houses in East Meadow, the following results were

obtained:

y^= 74.80 + 19.72x

sβ = 71.24, t = 1.05 (for testing β0)

sβ = 2.63, t = 7.49 (for testing β1)

SSE = 60,775, MSE = 841, s = 29, r2 = 0.44

Range of the x-values: 5 - 11

Range of the y-values: 160 - 300

Give a practical interpretation of the estimate of the y-intercept of the least squares line.

A) There is no practical interpretation, since a house with 0 rooms is nonsensical.

B) For each additional room in the house, we estimate the appraised value to increase $74,800.

C) For each additional room in the house, we estimate the appraised value to increase $19,720.

D) We estimate the base appraised value for any house to be $74,800.

22) Is there a relationship between the raises administrators at State University receive and their performance on

the job? A faculty group wants to determine whether job rating (x) is a useful linear predictor of raise (y).

Consequently, the group considered the straight-line regression model, y^= β0 + β1x. Using the method of least

squares, the faculty group obtained the following prediction equation, y^= 14,000 + 2,000x. Interpret the

estimated slope of the line.

A) For a 1-point increase in an administratorʹs rating, we estimate the administratorʹs raise to increase

$2,000.

B) For a 1-point increase in an administratorʹs rating, we estimate the administratorʹs raise to decrease

$2,000.

C) For an administrator with a rating of 1.0, we estimate his/her raise to be $2,000.

D) For a $1 increase in an administratorʹs raise, we estimate the administratorʹs rating to decrease 2,000

points.

23) Is there a relationship between the raises administrators at State University receive and their performance on

the job? A faculty group wants to determine whether job rating (x) is a useful linear predictor of raise (y).

Consequently, the group considered the straight-line regression model, y^ = β0 + β1x. Using the method of least

squares, the faculty group obtained the following prediction equation, y^= 14,000 + 2,000x.

Interpret the estimated y-intercept of the line.

A) For an administrator who receives a rating of zero, we estimate his or her raise to be $14,000.

B) The base administrator raise at State University is $14,000.

C) For a 1-point increase in an administratorʹs rating, we estimate the administratorʹs raise to increase

$14,000.

D) There is no practical interpretation, since rating of 0 is nonsensical and outside the range of the sample

data.

24) A large national bank charges local companies for using its services. A bank official reported the results of a

regression analysis designed to predict the bankʹs charges (y), measured in dollars per month, for services

rendered to local companies. One independent variable used to predict service charge to a company is the

companyʹs sales revenue (x), measured in millions of dollars. Data for 21 companies who use the bankʹs

services were used to fit the model, y^= β0 + β1x. The results of the simple linear regression are provided below.

y^= 2,700 + 20x, s = 65, 2-tailed p-value = 0.064 (for testing β1)

Interpret the estimate of β0, the y-intercept of the line.

A) There is no practical interpretation since a sales revenue of $0 is a nonsensical value.

B) All companies will be charged at least $2,700 by the bank.

C) About 95% of the observed service charges fall within $2,700 of the least squares line.

D) For every $1 million increase in sales revenue, we expect a service charge to increase $2,700.

25) Civil engineers often use the straight-line equation, y^= β0 + β1x, to model the relationship between the mean

shear strength of masonry joints and precompression stress, x. To test this theory, a series of stress tests were

performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied

for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The

stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the

regression analysis.

Triplet Test 1 2 3 4 5 6 7

Shear Strength (tons), y 1.00 2.18 2.24 2.41 2.59 2.82 3.06

Precomp. Stress (tons), x 0 0.60 1.20 1.33 1.43 1.75 1.75

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Prob > F

Model 1 2.39555 2.39555 47.732 0.0010

Error 5 0.25094 0.05019

C Total 6 2.64649

Root MSE 0.22403 R-square 0.9052

Dep Mean 2.32857 Adj R-sq 0.8862

C.V. 9.62073

Parameter Estimates

Parameter Standard T for HO:

Variable DF Estimate Error Parameter=0 Prob > |T|

INTERCEP 1 1.191930 0.18503093 6.442 0.0013

X 1 0.987157 0.14288331 6.909 0.0010

Give a practical interpretation of the estimate of the slope of the least squares line.

A) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase

by 0.987 ton.

B) For a triplet test with a precompression stress of 1 ton, we estimate the shear strength of the joint to be

0.987 ton.

C) For every 0.987 ton increase in precompression stress, we estimate the shear strength of the joint to

increase by 1 ton.

D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be

1.19 tons.

26) Civil engineers often use the straight-line equation, y^ = β0 + β1x, to model the relationship between the mean

shear strength of masonry joints and precompression stress, x. To test this theory, a series of stress tests were

performed on solid bricks arranged in triplets and joined with mortar. The precompression stress was varied

for each triplet and the ultimate shear load just before failure (called the shear strength) was recorded. The

stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS printout of the

regression analysis.


Shear Strength, y

(tons)

1.00 2.18 2.24 2.41 2.59 2.82 3.06

Precomp. Stress, x

(tons)

0 0.60 1.20 1.33 1.43 1.75 1.75


Sum of Mean


Model 1 2.39555 2.39555 47.732 0.0010

Error 5 0.25094 0.05019

C Total 6 2.64649



C.V. 9.62073

Parameter Estimates



INTERCEP 1 1.191930 0.18503093 6.442 0.0013

X 1 0.987157 0.14288331 6.909 0.0010

Give a practical interpretation of the estimate of the y-intercept of the least squares line.

A) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to be

1.19 tons.

B) For every 1 ton increase in precompression stress, we estimate the shear strength of the joint to increase

by 0.987 ton.

C) There is no practical interpretation since a triplet test with a precompression stress of 0 tons is outside the

range of the sample data.

D) For a triplet test with a precompression stress of 0 tons, we estimate the shear strength of the joint to

increase 1.19 tons.

27) Each year a nationally recognized publication conducts its ʺSurvey of Americaʹs Best Graduate and

Professional Schools.ʺ An academic advisor wants to predict the typical starting salary of a graduate at a top

business school using GMAT score of the school as a predictor variable. Total GMAT scores range from 200 to

800. A simple linear regression of SALARY versus GMAT using 25 data points shown below.

β0^ = -92040 β

^1 = 228 s = 3213 R

2 = 0.66 r = 0.81 df = 23 t = 6.67

Give a practical interpretation of β0^ = -92040.

A) The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the

sample data.

B) We expect to predict SALARY to within 2(92040) = $184,080 of its true value using GMAT in a

straight-line model.

C) We estimate SALARY to decrease $92,040 for every 1-point increase in GMAT.

D) We estimate the base SALARY of graduates of a top business school to be $-92,040.



business school using GMAT score of the school as a predictor variable. Total GMAT scores range from 200 to

800. A simple linear regression of SALARY versus GMAT using 25 data points shown below.

β0^ = -92040 β

^1 = 228 s = 3213 R

2 = 0.66 r = 0.81 df = 23 t = 6.67

Give a practical interpretation of β1^ = 228.

A) We estimate SALARY to increase $228 for every 1-point increase in GMAT.

B) We expect to predict SALARY to within 2(228) = $456 of its true value using GMAT in a straight-line

model.

C) We estimate GMAT to increase 228 points for every $1 increase in SALARY.

D) The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the

sample data.

29) A real estate magazine reported the results of a regression analysis designed to predict the price (y), measured

in dollars, of residential properties recently sold in a northern Virginia subdivision. One independent variable

used to predict sale price is GLA, gross living area (x), measured in square feet. Data for 157 properties were

used to fit the model, y^= β0 + β1x. The results of the simple linear regression are provided below.

y^ = 96,600 + 22.5x s = 6500 r2 = -0.77 t = 6.1 (for testing β1)

Interpret the estimate of β0, the y-intercept of the line.

A) There is no practical interpretation, since a gross living area of 0 is a nonsensical value.

B) All residential properties in Virginia will sell for at least $96,600.

C) About 95% of the observed sale prices fall within $96,600 of the least squares line.

D) For every 1-sq ft. increase in GLA, we expect a propertyʹs sale price to increase $96,600.


30) In a comprehensive road test on all new car models, one variable measured is the time it takes a car to

accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is conducted on a

random sample of 129 new cars.

TIME60: y = Elapsed time (in seconds) from 0 mph to 60 mph

MAX: x1 = Maximum speed attained (miles per hour)

Initially, the simple linear model E(y) = β0 + β1x1 was fit to the data. Computer printouts for the analysis are

given below:

UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME60

PREDICTOR

VARIABLES COEFFICIENT STD ERROR STUDENTʹS T P

CONSTANT 18.7171 0.63708 29.38 0.0000

MAX -0.08365 0.00491 -17.05 0.0000

R-SQUARED 0.6960 RESID. MEAN SQUARE (MSE) 1.28695

ADJUSTED R-SQUARED 0.6937 STANDARD DEVIATION 1.13444

SOURCE DF SS MS F P

REGRESSION 1 374.285 374.285 290.83 0.0000

RESIDUAL 127 163.443 1.28695

TOTAL 128 537.728

CASES INCLUDED 129 MISSING CASES 0

Find and interpret the estimate b1 in the printout above.

31) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the

15 minute period following the addition of food. The data showing the weekly number of grunts and the age

of the warthog (in days) are listed below:

Week Number of Grunts Age (days)

1 82 117

2 60 133

3 31 147

4 36 152

5 55 159

6 32 166

7 54 175

8 9 181

9 12 187

a. Write the equation of a straight-line model relating number of grunts (y) to age (x).

b. Give the least squares prediction equation.

c. Give a practical interpretation of the value of β0^ if possible.

d. Give a practical interpretation of the value of β1^ if possible.


32) Given the following least squares prediction equation, y^ = -173 + 74x, we estimate y to by

with each 1-unit increase in x.

A) increase; 74 B) decrease; 74 C) decrease; 173 D) increase; 173

33) Given the equation of a regression line is y^ = 3x - 4, what is the best predicted value for y given x = 2?

A) 2 B) 10 C) 5 D) 1

34) Given the equation of a regression line is y^= -3.5x+ 1.5, what is the best predicted value for y given x = 9.9?

A) -33.15 B) -36.15 C) 36.15 D) 33.15

35) Use the regression equation to predict the value of y for x = 0.2.

x

y

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

A) -0.133 B) 0.971 C) 1.987 D) 2.207

36) Use the regression equation to predict the value of y for x = 3.3.

x

y

-5

11

-3

6

4

-6

1

-1

-1

3

-2

4

0

1

2

-4

3

-5

-4

8

A) -5.462 B) 6.979 C) 0.616 D) 4.386

37) The data below are the final exam scores of 10 randomly selected chemistry students and the number of hours

they slept the night before the exam. What is the best predicted value for y given x = 8?

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

A) 96 B) 95 C) 94 D) 97

38) The data below are the temperatures on randomly chosen days during the summer in one city and the number

of employee absences on those days for a company located in the same city. What is the best predicted value

for y given x = 84?

Temperature, x


72

3

85

7

91

10

90

10

88

8

98

15

75

4

100

15

80

5

A) 7 B) 8 C) 9 D) 10

39) The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly

selected adults. What is the best predicted value for y given x = 66?

Age, x

Pressure, y

38

116

41

120

45

123

48

131

51

142

53

145

57

148

61

150

65

152

A) 159 B) 161 C) 157 D) 155

40) The data below are the number of absences and the salaries (in thousands of dollars) of 9 randomly selected

employees from an engineering firm. What is the best predicted value for y given x = 10?

.Number of absences, x

Salary, y

0

98

3

86

6

80

4

82

9

71

2

92

15

55

8

76

5

82

A) 69 B) 70 C) 71 D) 68

41) In order for a companyʹs employees to work for the foreign office, they must take a test in the language of the

country where they plan to work. The data below show the relationship between the number of years that

employees have studied a particular language and the grades they received on the proficiency exam. What is

the best predicted value for y given x = 2?

Number of years, x

Grades on test, y

3

61

4

68

4

75

5

82

3

73

6

90

2

58

7

93

3

72

A) 60 B) 58 C) 56 D) 62


yield of wheat (bushels per acre). Which is the best predicted value for y given x = 17.6?



10.5

50.5

8.8

46.2

13.4

58.8

12.5

59.0

18.8

82.4

10.3

49.2

7.0

31.9

15.6

76.0

16.0

78.8

A) 81.3 B) 81.6 C) 81.1 D) 81.8


43) A calculus instructor is interested in finding the strength of a relationship between the final exam grades of

students enrolled in Calculus I and Calculus II at his college. The data (in percentages) are listed below.

Calculus I

Calculus II

88

81

78

80

62

55

75

78

95

90

91

90

83

81

86

80

98

100

a) Graph a scatter diagram of the data.

b) Find an equation of the regression line.

c) Predict a Calculus II exam score for a student who receives an 80 in Calculus I.


44) In one area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of

wheat (bushels per acre). The data for a 9 year period is as follows:

Rain Fall, x 13.1 11.4 16.0 15.1 21.4 12.9 9.6 18.2 18.6

Yield, y 48.5 44.2 56.8 80.4 47.2 29.9 74.0 74.0 76.8

The equation of the line of least squares is given as y^= -9.12 + 4.38x. How many bushels of wheat per acre can

be predicted if it is expected that there will be 17 inches of rain?

A) 65.34 B) 5.96 C) 61.18 D) 52.06

45) In an area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of


Rain Fall, x 13.1 11.4 16.0 15.1 21.4 12.9 9.6 18.2 18.6

Yield, y 48.5 44.2 56.8 80.4 47.2 29.9 74.0 74.0 76.8

The equation of the line of least squares is given as y^= -9.12 + 4.38x. What would be the expected number of

inches of rain if the yield is 60 bushels of wheat per acre?

A) 15.78 B) 253.68 C) 11.62 D) 64.74

46) In an area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of


Rain Fall, x 13.1 11.4 16.0 15.1 21.4 12.9 9.6 18.2 18.6

Yield, y 48.5 44.2 56.8 80.4 47.2 29.9 74.0 74.0 76.8

The equation of the line of least squares is given as y^ = -9.12 + 4.38x. How many bushels of wheat per acre can

be predicted if it is expected that there will be 30 inches of rain?

A) Cannot be certain of the result because 30 inches of rain exceeds the observed data.

B) 122.28

C) 140.52

D) 8.93

3 Compute the sum of squared residuals.



47) The regression line for the given data is y^ = 2.097x - 0.552. Determine the residual of a data point for which x =

0 and y = -1.

x

y

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

A) -0.448 B) -1.552 C) -0.552 D) 2.649

48) The regression line for the given data is y^ = -1.885x + 0.758. Determine the residual of a data point for which x

= 2 and y = -4.x

y

-5

11

-3

6

4

-6

1

-1

-1

3

-2

4

0

1

2

-4

3

-5

-4

8

A) -0.988 B) -7.012 C) -3.012 D) -6.298

49) The regression line for the given data is y^ = -0.206x + 2.097. Determine the residual of a data point for which x

= -5 and y = 11.

x

y

-5

11

-3

-6

4

8

1

-3

-1

-2

-2

1

0

5

2

-5

3

6

-4

7

A) 7.873 B) 14.127 C) 3.127 D) -4.831

50) The regression line for the given data is y^ = 5.044x + 56.11. Determine the residual of a data point for which x =

5 and y = 80.

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

A) -1.33 B) 161.33 C) 81.33 D) -454.63

51) The regression line for the given data is y^ = 0.449x - 30.27. Determine the residual of a data point for which x =

98 and y = 15.

Temperature, x


72

3

85

7

91

10

90

10

88

8

98

15

75

4

100

15

80

5

A) 1.268 B) 28.732 C) 13.732 D) 121.535


45 and y = 123.

Age, x

Pressure, y

38

116

41

120

45

123

48

131

51

142

53

145

57

148

61

150

65

152

A) -4.42 B) 250.42 C) 127.42 D) -198.484

53) The regression line for the given data is y^ = -2.75x + 96.14. Determine the residual of a data point for which x =

9 and y = 71.


Final grade, y

0

98

3

86

6

80

4

82

9

71

2

92

15

55

8

76

5

82

A) -0.39 B) 142.39 C) 71.39 D) 108.11


8 and y = 65.

Years employed, x

Sales, y

2

31

3

33

10

78

7

62

8

65

15

61

3

48

1

55

11

120

A) -1.16 B) 131.16 C) 66.16 D) -259.37

55) The regression line for the given data is y^ = 6.91x + 46.26. Determine the residual of a data point for which x = 3

and y = 72.

Number of years, x

Grades on test, y

3

61

4

68

4

75

5

82

3

73

6

90

2

58

7

93

3

72

A) 5.01 B) 138.99 C) 66.99 D) -540.78


16 and y = 78.8.



10.5

50.5

8.8

46.2

13.4

58.8

12.5

59.0

18.8

82.4

10.3

49.2

7.0

31.9

15.6

76.0

16.0

78.8

A) 4.469 B) 153.131 C) 74.331 D) -333.3322

57) Compute the sum of the squared residuals of the least-squares line for the given data.

x

y

-5

-10

-3

-8

4

9

1

1

-1

-2

-2

-6

0

-1

2

3

3

6

-4

-8

A) 7.624 B) 1.036 C) 2.097 D) 0

58) The data below are the final exam scores of 10 randomly selected statistics students and the number of hours

they slept the night before the exam. Compute the sum of the squared residuals of the least-squares line for the

given data.

Hours, x

Scores, y

3

65

5

80

2

60

8

88

2

66

4

78

4

85

5

90

6

90

3

71

A) 318.038 B) 804.062 C) 1122.1 D) 39.755


yield of wheat (bushels per acre). Compute the sum of the squared residuals of the least-squares line for the

given data.

Rain fall (in inches), x


10.5

50.5

8.8

46.2

13.4

58.8

12.5

59.0

18.8

82.4

10.3

49.2

7.0

31.9

15.6

76.0

16.0

78.8

A) 87.192 B) 2207.628 C) 4.379 D) 0

60) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in a 15

minute time period following the addition of food. The data showing the weekly number of grunts and the age

of the warthog (in days) are listed below. Compute the sum of the squared residuals of the least squared line

for the given data.

Week Number of

Grunts

Age (days)

1 90 125

2 68 141

3 39 155

4 44 160

5 63 167

6 40 174

7 62 183

8 17 189

9 20 195

A) 5533.53 B) 188.84 C) 74.39 D) 13.74

61) The data below are the ages and systolic blood pressure (measured in Millimeters of mercury) of 9 randomly

selected adults.

Age, x Pressure, y

38 116

41 12.

45 123

48 131

51 142

53 145

57 148

61 150

65 152

A) 123.63 B) 1.41 C) 1.99 D) 11.11

62) A calculus instructor is interested the performance of his students from Calculus I that go on to Calculus II.

Their final grades in each course (in percent) are given below. Compute the sum of the squared residuals of the

least squared line for the given data.

Calculus I 88 78 62 75 95 91 83 86 98

Calculus II 81 80 55 78 90 90 81 80 100

A) 130.14 B) 30.85 C) 11.41 D) 1075.9

4.3 Diagnostics on the Least-Squares Regression Line

1 Compute and interpret the coefficient of determination.


Choose the coefficient of determination that matches the scatterplot. Assume that the scales on the horizontal and

vertical axes are the same.

1) Response

Explanatory

A) R2 = 0.77 B) R2 = 0.38 C) R2 = 0.96 D) R2 = 0.51

2) Response

Explanatory

A) R2 = 0.43 B) R2 = -0.43 C) R2 = 0.82 D) R2 = 0.12

3) Response

Explanatory

A) R2 = 0.097 B) R2 = -0.31 C) R2 = 0.76 D) R2 = 0.41

Use the linear correlation coefficient given to determine the coefficient of determination, R2.

4) r = 0.16

A) R2 = 2.56% B) R2 = 40.00% C) R2 = 4.00% D) R2 = 0.256%

5) r = -0.71

A) R2 = 50.41% B) R2 = 84.26% C) R2 = -50.41% D) R2 = -84.26%



6) Calculate the coefficient of determination to the nearest thousandth, given that the linear correlation coefficient,

r, is 0.837. What does this tell you about the explained variation and the unexplained variation of the data

about the regression line?

7) Calculate the coefficient of determination to the nearest thousandth, given that the linear correlation coefficient,

r, is -0.625. What does this tell you about the explained variation and the unexplained variation of the data

about the regression line?

8) Calculate the coefficient of determination, given that the linear correlation coefficient, r, is 1. What does this tell

you about the explained variation and the unexplained variation of the data about the regression line?

9) In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the

15 minute period following the addition of food. The data showing the weekly number of grunts and the age of

the warthog (in days) are listed below. Find and interpret the value of R2. Round R2 to the nearest thousandth.

Number of Grunts Age (days)

81 116

59 132

30 146

35 151

54 158

31 165

53 174

8 180

11 186


10) In a comprehensive road test on all new car models, one variable measured is the time it takes a car to

accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is conducted on a

random sample of 129 new cars.

TIME60: y = Elapsed time (in seconds) from 0 mph to 60 mph

MAX: x1 = Maximum speed attained (miles per hour)

Initially, the simple linear model E(y) = β0 + β1x1 was fit to the data. Computer printouts for the analysis are

given below:

UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME60

PREDICTOR


CONSTANT 18.7171 0.63708 29.38 0.0000

MAX -0.08365 0.00491 -17.05 0.0000



SOURCE DF SS MS F P

REGRESSION 1 374.285 374.285 290.83 0.0000

RESIDUAL 127 163.443 1.28695

TOTAL 128 537.728

CASES INCLUDED 129 MISSING CASES 0

Approximately what percentage, rounded to the nearest whole percent, of the sample variation in acceleration

time can be explained by the simple linear model?

A) 70% B) 0% C) -17% D) 8%

11) A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect

drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following

two variables:

MANHRS: y = Number of man-hours required to erect the drum

PRESSURE: x1= Boiler design pressure (pounds per square inch, i.e., psi)

Initially, the simple linear model E(y) = β0 + β1x1 was fit to the data. A printout for the analysis appears below:

UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS

PREDICTOR


CONSTANT 1.88059 0.58380 3.22 0.0028

PRESSURE 0.00321 0.00163 2.17 0.0300



SOURCE DF SS MS F P

REGRESSION 1 111.008 111.008 5.19 0.0300

RESIDUAL 34 144.656 4.25160

TOTAL 35 255.665

Give a practical interpretation of the coefficient of determination, R2. Express R2 to the nearest whole percent.

A) About 43% of the sample variation in number of man-hours can be explained by the simple linear model.

B) y^ = 1.88 + 0.00321x will be correct 43% of the time.

C) Man hours needed to erect drums will be associated with boiler design pressure 43% of the time.

D) About 2.06% of the sample variation in number of man-hours can be explained by the simple linear

model.

12) Civil engineers often use the straight-line equation, E(y) = β0 + β1x, to model the relationship between the

mean shear strength E(y) of masonry joints and precompression stress, x. To test this theory, a series of stress

tests were performed on solid bricks arranged in triplets and joined with mortar. The precompression stress

was varied for each triplet and the ultimate shear load just before failure (called the shear strength) was

recorded. The stress results for n = 7 triplet tests is shown in the accompanying table followed by a SAS

printout of the regression analysis.


Shear Strength (tons), y 1.00 2.18 2.24 2.41 2.59 2.82 3.06

Precomp. Stress (tons), x 0 0.60 1.20 1.33 1.43 1.75 1.75


Sum of Mean


Model 1 2.39555 2.39555 47.732 0.0010

Error 5 0.25094 0.05019

C Total 6 2.64649



C.V. 9.62073

Parameter Estimates



INTERCEP 1 1.191930 0.18503093 6.442 0.0013

X 1 0.987157 0.14288331 6.909 0.0010

Give a practical interpretation of R2, the coefficient of determination for the least squares model. Express R2 to

the nearest whole percent.

A) About 91% of the total variation in the sample of y-values can be explained by (or attributed to) the linear

relationship between shear strength and precompression stress.

B) In repeated sampling, approximately 91% of all similarly constructed regression lines will accurately

predict shear strength.

C) We expect to predict the shear strength of a triplet test to within about .91 ton of its true value.

D) We expect about 91% of the observed shear strength values to lie on the least squares line.

13) The dean of the Business School at a small Florida college wishes to determine whether the grade -point

average (GPA) of a graduating student can be used to predict the graduateʹs starting salary. More specifically,

the dean wants to know whether higher GPAʹs lead to higher starting salaries. Records for 23 of last yearʹs

Business School graduates are selected at random, and data on GPA (x) and starting salary (y, in $thousands)

for each graduate were used to fit the model, E(y) = β0 + β1x. The results of the simple linear regression are

provided below.

y^ = 4.25 + 2.75x, SSxy = 5.15, SSxx = 1.87

SSyy = 15.17, SSE = 1.0075

Range of the x-values: 2.23 - 3.85

Range of the y-values: 9.3 - 15.6

Calculate the value of R2, the coefficient of determination.

A) 0.934 B) 0.661 C) 0.872 D) 0.339



business school using GMAT score of the school as a predictor variable. A simple linear regression of SALARY

versus GMAT using 25 data points shown below.

b0 = -92040 b1 = 228 s = 3213 R2 = 0.66 r = 0.81 df = 23 t = 6.67

Give a practical interpretation of R2 = 0.66.

A) 66% of the sample variation in SALARY can be explained by using GMAT in a straight -line model.

B) 66% of the differences in SALARY are caused by differences in GMAT scores.

C) We estimate SALARY to increase $.66 for every 1-point increase in GMAT.

D) We can predict SALARY correctly 66% of the time using GMAT in a straight-line model.

15) A real estate magazine reported the results of a regression analysis designed to predict the price (y), measured

in dollars, of residential properties recently sold in a northern Virginia subdivision. One independent variable

used to predict sale price is GLA, gross living area (x), measured in square feet. Data for 157 properties were

used to fit the model, E(y) = β0 + β1x. The results of the simple linear regression are provided below.

y = 96,600 + 22.5x s = 6500 R2 = 0.77 t = 6.1 (for testing β1)

Interpret the value of the coefficient of determination, R2.

A) 77% of the total variation in the sample sale prices can be attributed to the linear relationship between

GLA (x) and (y).

B) GLA (x) is linearly related to sale price (y) 77% of the time.

C) 77% of the observed sale prices (yʹs) will fall within 2 standard deviations of the least squares line.

D) There is a moderately strong positive correlation between sale price (y) and GLA (x).


16) A company keeps extensive records on its new salespeople on the premise that sales should increase with

experience. A random sample of seven new salespeople produced the data on experience and sales shown in

the table.

Months on Job Monthly Sales

y ($ thousands)

2 2.4

4 7.0

8 11.3

12 15.0

1 0.8

5 3.7

9 12.0

Summary statistics yield SSxx = 94.8571, SSxy = 124.7571, SSyy = 176.5171, x = 5.8571, and y = 7.4571. Find and

interpret the coefficient of determination. Round R2 to the nearest hundredth of a percent.

17) To investigate the relationship between yield of potatoes, y, and level of fertilizer application, x, an

experimenter divides a field into eight plots of equal size and applies differing amounts of fertilizer to each.

The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data

are as follows:

x 1 1.5 2 2.5 3 3.5 4 4.5

y 25 31 27 28 36 35 32 34

Summary statistics yield SSxx = 10.5, SSyy = 112, and SSxy = 25. Calculate the coefficient of determination

rounded to the nearest ten-thousandth.


18) The coefficient of correlation between x and y is r = 0.59. Calculate the coefficient of determination R2. Round

R2 to the nearest hundredth.

A) 0.35 B) 0.59 C) 0.41 D) 0.65


19) The coefficient of determination for a straight-line model relating selling price y to manufacturing cost x for a

particular item is R2 = 0.83. Interpret this value.


20) The measures the percentage of total variation in the response variable that is explained

by the least squares regression line.

A) coefficient of determination B) coefficient of linear correlation

C) sum of the residuals squared D) slope of the regression line

21) If the coefficient of determination is close to 1, then

A) the least squares regression line equation explains most of the variation in the response variable.

B) the least squares regression line equation has no explanatory value.

C) the sum of the square residuals is large compared to the total variation.

D) the linear correlation coefficient is close to zero.

22) The coefficient of determination is the of the linear correlation coefficient.

A) square B) square root C) opposite D) reciprocal

2 Perform residual analysis on a regression model.


Analyze the residual lot below. Does it violate any of the conditions for an adequate linear model?

23)

A) No, the plot of residuals is random.

B) Yes, there is a discernable pattern in the residuals.

C) Yes, the residuals do not display constant error variance.

24)

A) Yes, the residuals do not display constant error variance.

B) Yes, there is a discernable pattern in the residuals.

C) No, the plot of residuals is random.

25)

A) Yes, there is a discernable pattern in the residuals.

B) Yes, the residuals do not display constant error variance.

C) No, the plot of residuals is random.


26) True or False: Residual analysis cannot be used to check for outliers.

A) False B) True

27) True or False: If a residual plot shows an almost straight line then a linear model is appropriate.

A) False B) True

28) To determine if there are outliers in a least squares regression modelʹs data set, we could construct a boxplot of

the

A) residuals. B) response variables.

C) predictor variables. D) lurking variables.

3 Identify influential observations.


A scatter diagram is given with one of the points labeled ʺA.ʺ In addition, there are two least -squares regression lines

drawn. The solid line excludes the point A. The dashed line includes the point A. Based on the graph, is the point A

influential?

29)

x1 2 3 4 5 6 7 8 9 10

y10

9

8

7

6

5

4

3

2

1

A

x1 2 3 4 5 6 7 8 9 10

y10

9

8

7

6

5

4

3

2

1

A

A) yes B) no

30)

x1 2 3 4 5 6 7 8 9 10

y10

9

8

7

6

5

4

3

2

1

A

x1 2 3 4 5 6 7 8 9 10

y10

9

8

7

6

5

4

3

2

1

A

A) no B) yes


31) An influential observation is an observation that significantly affects the value of the

A) the slope of the least squares regression line. B) the mean of the response variable.

C) the median of the predictor variable. D) the median of the response variable.

32) What effect will an influential observation have upon the graph of the least squares regression line?

A) It will pull the graph toward the observation.

B) It will have no effect.

C) It will push the graph away from the observation.

D) It will lower the value of the correlation coefficient to make further analysis meaningless.

4.4 Contingency Tables and Association

1 Compute the marginal distribution of a variable.



1) The following data represent the living situation of newlyweds in a large metropolitan area and their annual

household income. Find the marginal frequency for newlyweds who own their own home.

<$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000

Own home 31 52 202 355 524

Rent home 67 66 52 23 11

Live w/family 89 69 30 4 2

A) 1164 B) 31 C) 524 D) 202


household income. Find the marginal frequency for newlyweds who make between $35,000 and $50,000 per

year.

<$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000

Own home 31 52 202 355 524

Rent home 67 66 52 23 11


A) 284 B) 30 C) 202 D) 52


household income. What percent of people who make between $35,000 and $50,000 per year own their own

home? Round to the nearest tenth of a percent.

<$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000

Own home 31 52 202 355 524

Rent home 67 66 52 23 11


A) 71.1% B) 18.3% C) 10.6% D) 17.4%


household income. What percent of people who own their own home make between $35,000 and $50,000 per

year? Round to the nearest tenth of a percent.

<$20,000 $20-35,000 $35-50,000 $50-75,000 >$75,000

Own home 31 52 202 355 524

Rent home 67 66 52 23 11


A) 17.4% B) 4.5% C) 30.5% D) 71.1%

5) Construct a frequency marginal distribution for the given contingency table.

x1 x2 x3

y1 25 40 40

y2 75 50 55

A)

x1 x2 x3 Marginal Distribution

y1 25 40 40 105

y2 75 50 55 180

Marginal Distribution 100 90 95 285

B)


y1 25 40 40 105

y2 75 50 55 180


C)


y1 25 40 40 105

y2 75 50 55 180


D)


y1 25 40 40 100

y2 75 50 55 90


6) Construct a relative frequency marginal distribution for the given contingency table. Round valuese to the

nearest thousandth.

x1 x2 x3

y1 20 25 10

y2 40 35 35

A)

x1 x2 x3

Relative Frequency

Marginal Distribution

y1 20 25 10 0.333

y2 40 35 35 0.667

Relative Frequency

Marginal Distribution 0.364 0.364 0.273 1

B)

x1 x2 x3

Relative Frequency


y1 20 25 10 0.333

y2 40 35 35 0.667

Relative Frequency


C)

x1 x2 x3

Relative Frequency


y1 20 25 10 0.55

y2 40 35 35 1.10

Relative Frequency


D)

Relative Frequency

Marginal Distribution x1 x2 x3

y1 0.121 0.152 0.061

y2 0.242 0.212 0.212

7) Construct a conditional distribution by x for the given contingency table. Round valuese to the nearest

thousandth.

x1 x2 x3

y1 30 40 20

y2 50 65 65

A)

x1 x2 x3

y1 0.375 0.381 0.235

y2 0.625 0.619 0.765

Total 1 1 1

B)

x1 x2 x3

y1 0.375 0.381 0.235

y2 0.278 0.361 0.361

Total 1 1 1

C)

x1 x2 x3

y1 0.111 0.148 0.074

y2 0.185 0.241 0.241

D)

x1 x2 x3 Total

y1 0.333 0.444 0.222 1

y2 0.278 0.361 0.361 1

8) A contingency table relates

A) two categories of data.

B) the difference in the means of two random variables.

C) a particular response with order in which that response should be applied.

D) only continuous random variables.

9) To eliminate the effects of either the row or the column variables in a contingency table, a

distribution is created.

A) marginal B) normalized C) χ2 D) Studentʹs t

2 Use the conditional distribution to identify association among categorical data.



10) The data below show the age and favorite type of music of 779 randomly selected people. Use α = 0.05. What, if

any, association exists between favorite music and age? Discuss the association.

Age Country Rock Pop Classical

15 - 21 21 45 90 33

21 - 30 60 55 42 48

30 - 40 65 47 31 57

40 - 50 68 39 25 53


household income. What, if any, association exists between living situation and household income? Discuss the

association.

< $20,000 $20-35,000 $35-50,000 $50-75,000 > $75,000

Own home 31 52 202 355 524

Rent home 67 66 52 23 11


3 Explain Simpsonʹs Paradox.



12) Researchers conducted a study to determine which of two different treatments, A or B, is more effective in the

treatment of atherosclerosis. The results of their experiment are given in the table. (a) Which treatment appears

to be more effective? Why?

Treatment A Treatment B

Effective 420 435

Not effective 130 140

The data in the table do not take into account the seriousness of the case. The data shown in the next table

show the effectiveness of each treatment for both mild and advanced cases of atherosclerosis.

Mild Advanced

atherosclerosis atherosclerosis

Treatment A Treatment B Treatment A Treatment B

Effective 310 95 110 340

Not effective 80 20 50 120

(b) Determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment A.

Determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment B.

(c) Repeat part (b) for advanced cases of atherosclerosis to create a conditional distribution of effectiveness by

treatment for each category of the disease.

(d) Write a short report detailing and explaining your findings.

13) A company encourages applications from minority groups who they feel are under-represented in the

company. The table shows the number of applications that were accepted last year from people belonging to

minority groups and the number of applications that were accepted from people not belonging to minority

groups. Only applications from well qualified applicants are included in the analysis. (a) Does the acceptance

rate appear to be higher for those belonging to minority groups or for those not belonging to minority groups ?

Why?

Minority Not minority

Accepted 70 79

Rejected 460 500

The data in the table do not take into account the department of the company. The data shown in the next table

show the number of applications accepted from each group within each department.

Department A Department B Department C

Minority Not minority Minority Not minority Minority Not minority

Accepted 27 10 22 34 21 35

Rejected 260 110 80 150 120 240

(b) Determine the proportion of minority applications that were accepted within department A. Determine the

proportion of non-minority applications that were accepted within department A.

(c) Repeat part (b) for departments B and C to create a conditional distribution of acceptance rate by group for

each department of the company.

(d) Write a short report detailing and explaining your findings.

4.5 Nonlinear Regression: Transformations (online)

1 Convert between exponential and logarithmic expressions.


Change the exponential expression to an equivalent expression involving a logarithm.

1) 23 = 8

A) log28 = 3 B) log82 = 3 C) log38 = 2 D) log23 = 8

2) 63 = x

A) log6x = 3 B) logx6 = 3 C) log3x = 6 D) log63 = x

3) 83 = y

A) log8y = 3 B) log3y = 8 C) logy8 = 3 D) logy3 = 8

4) 6x = 216

A) log6216 = x B) logx216 = 6 C) log2166 = x D) log216x = 6

5) 3x = 311

A) log3311 = x B) log3113 = x C) log311x = 3 D) log3x = 311

Change the logarithmic expression to an equivalent expression involving an exponent.

6) log5125 = 3

A) 53 = 125 B) 35 = 125 C) 5125 = 3 D) 1253 = 5

7) log5x = 3

A) 53 = x B) 35 = x C) 5x = 3 D) x3 = 5

8) logb256 = 4

A) b4 = 256 B) 4b = 256 C) 2564 = b D) 256b = 4

9) log381 = x

A) 3x = 81 B) x3 = 81 C) 81x = 3 D) 813 = x

2 Simplify logarithmic expressions.


Write the expression as a sum of logs. Express powers as factors.

10) log6 xy

A) log6 x + log6 y B) log6 x - log6 y C) log3 x + log3 y D) log3 x - log3 y

11) log7x5

A) 5log7x B) 7log5x C) 7 log x D) 5log7x5

12) logbyz8

A) logby + 8logbz B) 8logby + 8logbz C) 8logbyz D) logby + logb8z

13) logby9z3

A) 9logby + 3logbz B) 27logb yz C) logb (yz)27 D) logb 27yz

Use a calculator to evaluate the expression. Round your answer to three decimal places.

14) log 100

A) 2.000 B) 4.605 C) 2.004 D) 1.996

15) log 3.21

A) 0.507 B) 1.166 C) 0.520 D) 0.493

16) 101.7

A) 50.119 B) 63.096 C) 5.474 D) 79.433

17) 100.7029

A) 5.045 B) 0.198 C) 0.495 D) 1.176

3 Use logarithmic transformations to linearize exponential relations.



18) The following data represent the bacteria population in a laboratory experiment. The researchers suspect that

the population is growing exponentially. Determine the logarithm of the y-values so that Y = log y.

Day, x Population, y

0 1762

1 4803

2 13,006

3 35,456

4 96,445

5 262,326


19) The following data represent the bacteria population in a laboratory experiment. The researchers suspect that

the population is growing exponentially. Find the least-squares regression line of the transformed data by

determining the logarithm of the y-values so that Y = log y.

Day, x Population, y

0 1952

1 5319

2 14410

3 39279

4 106845

5 290613

A) Y^ = 3.291 + 0.435x B) Y

^= 3.458 + 1.151x

C) Y^ = -7.537 + 2.301x D) Y

^ = -50,222.143 + 50,650.057x

4 Use logarithmic transformations to linearize power relations.



20) The following data represent the periods (in seconds) of simple pendulums of various lengths (in feet).

Determine the logarithm of both the x- and y-values so that X = log x and Y = log y.

Length, x Period, y

1 1.1

2 1.6

3 1.9

4 2.2

5 2.5

6 2.7


21) The following data represent the periods (in seconds) of simple pendulums of various lengths (in feet). Find the

power equation of best fit.

Length, x Period, y

1 1.3

2 1.9

3 2.3

4 2.7

5 3.0

6 3.3

A) y^ = 0.1171 + 0.5176x B) y

^ = -0.2258 + 1.9309x

C) y^ = 0.0948 + 0.0768x D) y

^ = 1.0467 + 0.3914x

22) The following data represent the infection rate for a particular disease in a third world country x years after a

vaccine became widely available. Determine the ʺbestʺ model to describe the relationship between years past

and infection rate.

Years After Vaccine, x Infection Rate (%), y

1 32

2 11

3 5

4 2

5 0.5

A) exponential model: y^= 1.955 - 0.435x B) power model: y

^= 1.648 - 2.402x

C) linear model: y^= 31.7 - 7.2x D) linear model: y

^= 31.7 + 7.2x

23) The following data represent the compound yield in grams for a chemical reaction for various temperatures of

the reaction. Determine the ʺbestʺ model to describe the relationship between temperature and compound

yield.

Temperature (° C), x Compound Yield (grams), y

50 4

60 12

70 14

80 21

90 26

A) linear model: y^ = -21.7 + 0.53x B) exponential model: y

^= -0.195 + 0.019x

C) power model: y^= -4.392 + 2.999x D) power model: y

^= 4.392 + 2.999x

24) The following data represent the height (relative to the ground) of a projectile shot into the air after x seconds

of travel. Determine the ʺbestʺ model to describe the relationship between travel time and height.

Travel Time (seconds), x Height (feet), y

8 890

9 869

10 798

11 708

12 571

A) power model: y^ = 3.930 - 1.056x B) exponential model: y

^ = 3.354 - 0.047x

C) linear model: y^ = 1566.2 - 79.9x D) exponential model: y

^ = 3.354 + 0.047x

Ch. 4 Describing the Relation between Two VariablesAnswer Key

4.1 Scatter Diagrams and Correlation1 Draw and interpret scatter diagrams.

1)

Exam

Scores

x1 2 3 4 5 6 7 8

y

95

90

85

80

75

70

65

60

x1 2 3 4 5 6 7 8

y

95

90

85

80

75

70

65

60

Hours Studied

2)

Number

of

Absences

x70 75 80 85 90 95 100

y

16

14

12

10

8

6

4

2

x70 75 80 85 90 95 100

y

16

14

12

10

8

6

4

2

Temperature

3)

Blood

Pressure

(mm of mercury)

x35 40 45 50 55 60 65

y155

150

145

140

135

130

125

120

115

x35 40 45 50 55 60 65

y155

150

145

140

135

130

125

120

115

Age

4)

Final

Grade

x2 4 6 8 10 12 14 16

y

100

90

80

70

60

50

x2 4 6 8 10 12 14 16

y

100

90

80

70

60

50

Number of Absences

5)

Sales

(in thousands)

x2 4 6 8 10 12 14 16

y120

110

100

90

80

70

60

50

40

30

x2 4 6 8 10 12 14 16

y120

110

100

90

80

70

60

50

40

30

Miles traveled (in hundreds)

6)

Final

Grade

x1 2 3 4 5 6 7

y

100

90

80

70

60

50

x1 2 3 4 5 6 7

y

100

90

80

70

60

50

Number of years studied

7)

Yield

(bushels)

x6 8 10 12 14 16 18 20

y

90

80

70

60

50

40

30

x6 8 10 12 14 16 18 20

y

90

80

70

60

50

40

30

Rainfall (inches)

8)

Nicotine

(mg)

x6 9 12 15 18

y

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

x6 9 12 15 18

y

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

Tar (mg)

9)

Round 2

x75 80 85 90 95

y

95

90

85

80

75

x75 80 85 90 95

y

95

90

85

80

75

Round 1

10) A

11) A

12) A

13) predictor variable: rainfall in inches; response variable: yield per acre

14) predictor variable: hours studying; response variable: grades on the test

15) A

2 Describe the properties of the linear correlation coefficient.

16) A

17) A

18) A

19) A

20) A

21) A

22) A

23) A

24) There appears to be a positive linear correlation.

x-5 -4 -3 -2 -1 1 2 3 4 5

y

10

8

6

4

2

-2

-4

-6

-8

-10

x-5 -4 -3 -2 -1 1 2 3 4 5

y

10

8

6

4

2

-2

-4

-6

-8

-10

25) There appears to be a negative linear correlation.

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

26) There appears to be no linear correlation.

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

y12

10

8

6

4

2

-2

-4

-6

-8

-10

-12

27) In general, there appears to be a relationship between the home runs and batting averages. As the number of home

runs increased, the batting averages increased.

Batting

Average

x15 30 45 60 75

y

0.35

0.3

0.25

0.2

0.15

x15 30 45 60 75

y

0.35

0.3

0.25

0.2

0.15

Home Runs

28) There appears to be a trend in the data. As the number of absences increases, the final grade decreases.

Final

Grade

(%)

x5 10 15

y100

90

80

70

60

50

40

x5 10 15

y100

90

80

70

60

50

40

Number of Absences

29) A

30) A

31) A

32) A

33) A

34) A

35) A

3 Compute and interpret the linear correlation coefficient.

36) A

37) A

38) A

39) A

40) A

41) A

42) A

43) A

44) A

45) A

46) The linear correlation coefficient remains unchanged.

4 Determine whether a linear relation exists between two variables.

47) A

48) A

49) A

50) A

51) A

52) A

53) A

54) A

55) A

56) A

5 Explain the difference between correlation and causation.

57) A

58) A positive correlation exists between the number of schools in a city and the number of robberies but this is an

example of correlation not causation. Building more schools is unlikely to lead to an increase in robberies. A likely

lurking variable is the population of the city and this lurking variable accounts for the positive correlation. Larger

cities tend to have both more schools and more robberies.

59) A negative correlation exists between the infant mortality rate and the number of cars per capita but this is an example

of correlation not causation. If people buy more cars, this is unlikely to lead to a decrease in the infant mortality rate.

A likely lurking variable is wealth and this lurking variable accounts for the negative correlation. More affluent

countries tend to have both more cars per capita and lower infant mortality rates.

60) A positive correlation exists between income and blood pressure but this is an example of correlation not causation.

An increase in salary is unlikely to lead to an increase in blood pressure. Age and level of job stress are possible

lurking variables and these lurking variables account for the positive correlation. Older men tend to have both higher

blood pressures and higher incomes. Also men in high stress jobs tend to have both higher blood pressures and higher

incomes.

4.2 Least-Squares Regression1 Find the least-squares regression line and use the line to make predictions.

1) A

2) A

3) A

4) A

5) A

6) A

7) A

8) A

9) A

10) A

11) A

12) A

13) A

14) The regression lines are not necessarily the same.

15) SSxx = x2∑ - x∑ 2

n = 0.948622 -

(3.642)2

14 = 0.00118171

SSxy = xy∑ - x∑ y∑n

= 295.54 - (3.642)(1,134)

14 = 0.538

y = y∑n =

1,134

14 = 81

x = x∑n =

3.642

14 = 0.26014

β^1 =

SSxy

SSxx =

0.538

0.00118171 = 455.27

β0^ = y - β

^1x = 81 - 455.27(0.26014) = -37.434

The least squares equation is y^= -37.434 + 455.27x.

16) β^1 =

SSxy

SSxx =

3031.7125

1222.2771 = 2.4804

β0^ = y - β

^1x = 95.0625 - 2.4804(21.2675) = 42.3106

The least squares prediction equation is y^ = 42.3106 + 2.4804x.

17) A

18) A

19) A

2 Interpret the slope and the y-intercept of the least-squares regression line.

20) A

21) A

22) A

23) A

24) A

25) A

26) A

27) A

28) A

29) A

30) b1 = -0.08365. For every 1 mile per hour increase in the maximum attained speed of a new car, we estimate the

elapsed 0 to 60 acceleration time to decrease by .08365 seconds.

31) a. E(y) = β0 + β1x

b. y^= β

^0 + β

^1x = 170.24 - 0.8195x

c. We would expect approximately 170 grunts after feeding a warthog that was just born. However, since the value 0

in outside the range of the original data set, this estimate is highly unreliable.

d. For each additional day, we estimate the number of grunts will decrease by 0.8195.

32) A

33) A

34) A

35) A

36) A

37) A

38) A

39) A

40) A

41) A

42) A

43) a)

Calculus II

x50 60 70 80 90 100

y

100

90

80

70

60

50

x50 60 70 80 90 100

y

100

90

80

70

60

50

Calculus I

b) y^= 1.044x - 5.990

c) When x = 80, y^ = 78.

44) A

45) A

46) A

3 Compute the sum of squared residuals.

47) A

48) A

49) A

50) A

51) A

52) A

53) A

54) A

55) A

56) A

57) A

58) A

59) A

60) A

61) A

62) A

4.3 Diagnostics on the Least-Squares Regression Line1 Compute and interpret the coefficient of determination.

1) A

2) A

3) A

4) A

5) A

6) The coefficient of determination, R2, = 0.701. That is, 70.1% of the variation is explained and 29.9% of the variation is

unexplained.

7) The coefficient of determination, R2, = 0.391. That is, 39.1% of the variation is explained and 60.9% of the variation is

unexplained.

8) The coefficient of determination, R2, = 1. That is, 100% of the variation is explained and there is no variation that is

unexplained.

9) R2 = 0.627; Approximately 62.7% of the variation in the number of grunts is explained by age.

10) A

11) A

12) A

13) A

14) A

15) A

16) R2 = 92.96% of the variation in the sample monthly sales values about their mean can be explained by using months

on the job in a linear model.

17) R2 = 0.5315

18) A

19) The model explains 83% of sample variation in cost.

20) A

21) A

22) A

2 Perform residual analysis on a regression model.

23) A

24) A

25) A

26) A

27) A

28) A

3 Identify influential observations.

29) A

30) A

31) A

32) A

4.4 Contingency Tables and Association1 Compute the marginal distribution of a variable.

1) A

2) A

3) A

4) A

5) A

6) A

7) A

8) A

9) A

2 Use the conditional distribution to identify association among categorical data.

10) The proportion who prefers country music increases as age increases. The proportion who prefers rock is roughly

constant as age increases. The proportion who prefers pop decreases as age increases. The proportion who prefers

classical increases as age increases.

11) The proportion of home owners increases as the household income increases. The proportions of renters and of those

living with family decrease as household income increases.

3 Explain Simpsonʹs Paradox.

12) (a) Treatment A appears to be more effective:

treatment A was effective in 76.4% of cases

treatment B was effective in 75.7% of cases.

(b) Mild cases: treatment A was effective in 79.5% of cases


(c) Advanced cases: treatment A was effective in 68.8% of cases


(d) Within each category of disease, B has a higher rate of effectiveness and yet overall it has a lower rate of

effectiveness.

The initial analysis failed to take into account the lurking variable - seriousness of the case.

Treatment B is used more often in the more serious cases where success rates are lower for both methods.

13) (a) The acceptance rate appears to be higher for those not belonging to minority groups:

13.2% of applications from people belonging to minority groups were accepted

13.6% of applications from people not belonging to minority groups were accepted

(b) Department A: minority applications: 9.4% accepted

non-minority applications: 8.3% accepted

(c) Department B: minority applications: 21.6% accepted

non-minority applications: 18.5% accepted

Department C: minority applications: 14.9% accepted

non-minority applications: 12.7% accepted.

(d) Within each department, the rate of acceptance is higher for people belonging to minority groups and yet overall

the acceptance rate is higher for people not belonging to minority groups.

The initial analysis failed to take into account the lurking variable - the department of the company. In department A,

there are more applications from minorities and this department has the lowest acceptance rates. Department B has

the fewest applications from minorities and this department has the highest acceptance rates.

4.5 Nonlinear Regression: Transformations (online)1 Convert between exponential and logarithmic expressions.

1) A

2) A

3) A

4) A

5) A

6) A

7) A

8) A

9) A

2 Simplify logarithmic expressions.

10) A

11) A

12) A

13) A

14) A

15) A

16) A

17) A

3 Use logarithmic transformations to linearize exponential relations.

18)

Day, x Population, y Log y

0 1762 3.246

1 4803 3.6815

2 13,006 4.1141

3 35,456 4.5497

4 96,445 4.9843

5 262,326 5.4188

19) A

4 Use logarithmic transformations to linearize power relations.

20)

Length, x Log x Period, y Log y

1 0 1.1 0.0414

2 0.3010 1.6 0.2041

3 0.4771 1.9 0.2788

4 0.6021 2.2 0.3424

5 0.6990 2.5 0.3979

6 0.7782 2.7 0.4314

21) A

22) A

23) A

24) A

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