# Chapter 13 As the goodness of fit for the estimated multiple regression

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Test Prep

Book Title

Essentials of Modern Business Statistics 4th (Fourth) Edition By Williams 4th Edition

Authors

J.K

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CHAPTER THIRTEEN

MULTIPLE REGRESSION

MULTIPLE CHOICE QUESTIONS

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

1. If a qualitative variable has k levels, the number of dummy variables required is

a. k − 1

b. k

c. k + 1

d. 2k

2. As the goodness of fit for the estimated multiple regression equation increases,

a. the value of the adjusted multiple coefficient of determination decreases

b. the value of the regression equation’s constant b0 decreases

c. the value of the multiple coefficient of determination increases

d. the value of the correlation coefficient increases

3. For a multiple regression model, SSR = 600 and SSE = 200. The multiple coefficient of

determination is

a. 0.333

b. 0.275

c. 0.300

d. 0.75

4. In a multiple regression analysis involving 15 independent variables and 200

observations, SST = 800 and SSE = 240. The coefficient of determination is

a. 0.300

b. 0.192

c. 0.500

d. 0.700

5. A regression model involved 5 independent variables and 126 observations. The critical

value of t for testing the significance of each of the independent variable's coefficients

will have

a. 131 degrees of freedom

b. 125 degrees of freedom

c. 130 degrees of freedom

d. 4 degrees of freedom

6. In order to test for the significance of a regression model involving 3 independent

variables and 47 observations, the numerator and denominator degrees of freedom

(respectively) for the critical value of F are

a. 47 and 3

b. 3 and 47

c. 2 and 43

d. 3 and 43

7. In regression analysis, an outlier is an observation whose

a. mean is larger than the standard deviation

b. residual is zero

c. mean is zero

d. residual is much larger than the rest of the residual values

8. A variable that cannot be measured in terms of how much or how many but instead is

assigned values to represent categories is called

a. an interaction

b. a constant variable

c. a category variable

d. a qualitative variable

9. A variable that takes on the values of 0 or 1 and is used to incorporate the effect of

qualitative variables in a regression model is called

a. an interaction

b. a constant variable

c. a dummy variable

d. None of these alternatives is correct.

10. In a multiple regression model, the error term

is assumed to be a random variable with

a mean of

a. zero

b. -1

c. 1

d. any value

11. In regression analysis, the response variable is the

a. independent variable

b. dependent variable

c. slope of the regression function

d. intercept

12. The multiple coefficient of determination is

a. MSR/MST

b. MSR/MSE

c. SSR/SST

d. SSE/SSR

13. A multiple regression model has the form

ˆ

y

= 7 + 2x1 + 9x2

As x1 increases by 1 unit (holding x2 constant), y is expected to

a. increase by 9 units

b. decrease by 9 units

c. increase by 2 units

d. decrease by 2 units

14. A multiple regression model has

a. only one independent variable

b. more than one dependent variable

c. more than one independent variable

d. at least 2 dependent variables

15. A measure of goodness of fit for the estimated regression equation is the

a. multiple coefficient of determination

b. mean square due to error

c. mean square due to regression

d. sample size

16. The numerical value of the coefficient of determination

a. is always larger than the coefficient of correlation

b. is always smaller than the coefficient of correlation

c. is negative if the coefficient of determination is negative

d. can be larger or smaller than the coefficient of correlation

17. The correct relationship between SST, SSR, and SSE is given by

a. SSR = SST + SSE

b. SSR = SST - SSE

c. SSE = SSR - SST

d. None of these alternatives is correct.

Exhibit 13-1

In a regression model involving 44 observations, the following estimated regression equation was

obtained.

ˆ

y

= 29 + 18x1 +43x2 + 87x3

For this model SSR = 600 and SSE = 400.

18. Refer to Exhibit 13-1. The coefficient of determination for the above model is

a. 0.667

b. 0.600

c. 0.336

d. 0.400

19. Refer to Exhibit 13-1. MSR for this model is

a. 200

b. 10

c. 1,000

d. 43

20. Refer to Exhibit 13-1. The computed F statistics for testing the significance of the above

model is

a. 1.500

b. 20.00

c. 0.600

d. 0.6667

21. In a multiple regression analysis SSR = 1,000 and SSE = 200. The F statistic for this

model is

a. 5.0

b. 1,200

c. 800

d. Not enough information is provided to answer this question.

22. The ratio of MSE/MSR yields

a. SST

b. the F statistic

c. SSR

d. None of these alternatives is correct.

23. In a multiple regression model, the variance of the error term

is assumed to be

a. the same for all values of the dependent variable

b. zero

c. the same for all values of the independent variable

d. -1

24. The adjusted multiple coefficient of determination is adjusted for

a. the number of dependent variables

b. the number of independent variables

c. the number of equations

d. detrimental situations

25. In multiple regression analysis, the correlation among the independent variables is termed

a. homoscedasticity

b. linearity

c. multicollinearity

d. adjusted coefficient of determination

26. In a multiple regression model, the values of the error term ,

, are assumed to be

a. zero

b. dependent on each other

c. independent of each other

d. always negative

27. In multiple regression analysis,

a. there can be any number of dependent variables but only one independent

variable

b. there must be only one independent variable

c. the coefficient of determination must be larger than 1

d. there can be several independent variables, but only one dependent variable

28. In a multiple regression model, the error term

is assumed to

a. have a mean of 1

b. have a variance of zero

c. have a standard deviation of 1

d. be normally distributed

29. In a multiple regression analysis involving 12 independent variables and 166

observations, SSR = 878 and SSE = 122. The coefficient of determination is

a. 0.1389

b. 0.1220

c. 0.878

d. 0.7317

30. A regression analysis involved 17 independent variables and 697 observations. The

critical value of t for testing the significance of each of the independent variable's

coefficients will have

a. 696 degrees of freedom

b. 16 degrees of freedom

c. 713 degrees of freedom

d. 714 degrees of freedom

31. In order to test for the significance of a regression model involving 14 independent

variables and 255 observations, the numerator and denominator degrees of freedom

(respectively) for the critical value of F are

a. 14 and 255

b. 255 and 14

c. 13 and 240

d. 14 and 240

Exhibit 13-2

A regression model between sales (y in $1,000), unit price (x1 in dollars) and television

advertisement (x2 in dollars) resulted in the following function:

ˆ

y

= 7 – 3x1 + 5x2

For this model SSR = 3500, SSE = 1500, and the sample size is 18.

32. Refer to Exhibit 13-2. The coefficient of the unit price indicates that if the unit price is

a. increased by $1 (holding advertising constant), sales are expected to increase by

$3

b. decreased by $1 (holding advertising constant), sales are expected to decrease by

$3

c. increased by $1 (holding advertising constant), sales are expected to increase by

$4,000

d. increased by $1 (holding advertising constant), sales are expected to decrease by

$3,000

33. Refer to Exhibit 13-2. The coefficient of x2 indicates that if television advertising is

increased by $1 (holding the unit price constant), sales are expected to

a. increase by $5

b. increase by $12,000

c. increase by $5,000

d. decrease by $2,000

34. Refer to Exhibit 13-2. If we want to test for the significance of the regression model, the

critical value of F at 95% confidence is

a. 3.68

b. 3.29

c. 3.24

d. 4.54

35. Refer to Exhibit 13-2. If SSR = 600 and SSE = 300, the test statistic F is

a. 2.33

b. 0.70

c. 17.5

d. 1.75

36. Refer to Exhibit 13-2. The multiple coefficient of determination for this problem is

a. 0.4368

b. 0.6960

c. 0.3040

d. 0.2289

Exhibit 13-3

In a regression model involving 30 observations, the following estimated regression equation was

obtained:

ˆ

y

= 17 + 4x1 – 3x2 + 8x3 + 8x4

For this model SSR = 700 and SSE = 100.

37. Refer to Exhibit 13-3. The coefficient of determination for the above model is

approximately

a. -0.875

b. 0.875

c. 0.125

d. 0.144

38. Refer to Exhibit 13-3. The computed F statistic for testing the significance of the above

model is

a. 43.75

b. 0.875

c. 50.19

d. 7.00

39. Refer to Exhibit 13-3. The critical F value at 95% confidence is

a. 2.53

b. 2.69

c. 2.76

d. 2.99

40. Refer to Exhibit 13-3. The conclusion is that the

a. model is not significant

b. model is significant

c. slope of x1 is significant

d. slope of x2 is significant

Exhibit 13-4

a. y =

0 +

1x1 +

2x2 +

b. E(y) =

0 +

1x1 +

2x2

c.

ˆ

y

= bo + b1x1 + b2x2

d. E(y) =

0 +

1x1 +

2x2

41. Refer to Exhibit 13-4. Which equation describes the multiple regression model?

a

b

c

d

42. Refer to Exhibit 13-4. Which equation gives the estimated regression line?

a

b

c

d

43. Refer to Exhibit 13-4. Which equation describes the multiple regression equation?

a

b

c

d

Exhibit 13-5

Below you are given a partial Excel output based on a sample of 25 observations.

Coefficients

Standard Error

Intercept

145.321

48.682

x1

25.625

9.150

x2

-5.720

3.575

x3

0.823

0.183

44. Refer to Exhibit 13-5. The estimated regression equation is

a. y =

0 +

1x1 +

2x2 +

3x3 +

b. E(y) =

0 +

1x1 +

2x2 +

3x3

c.

ˆ

y

= 145.321 + 25.625x1 − 5.720x2 + 0.823x3

d.

ˆ

y

= 48.682 + 9.15x1 + 3.575x2 + 0.183x3

45. Refer to Exhibit 13-5. The interpretation of the coefficient on x1 is that

a. a one unit change in x1 will lead to a 25.625 unit change in

ˆ

y

b. a one unit change in x1 will lead to a 25.625 unit increase in

ˆ

y

when all other

variables are held constant

c. a one unit change in x1 will lead to a 25.625 unit increase in x2 when all other

variables are held constant

d. It is impossible to interpret the coefficient.

46. Refer to Exhibit 13-5. We want to test whether the parameter

1 is significant. The test

statistic equals

a. 0.357

b. 2.8

c. 14

d. 1.96

47. Refer to Exhibit 13-5. The t value obtained from the table to test an individual parameter

at the 5% level is

a. 2.06

b. 2.069

c. 2.074

d. 2.080

48. Refer to Exhibit 13-5. Carry out the test of significance for the parameter

1 at the 5%

level. The null hypothesis should be

a. rejected

b. not rejected

c. revised

d. None of these alternatives is correct.

Exhibit 13-6

Below you are given a partial Excel output based on a sample of 16 observations.

ANOVA

df

SS

MS

F

Regression

4,853

2,426.5

Residual

485.3

Coefficients

Standard Error

Intercept

12.924

4.425

x1

-3.682

2.630

x2

45.216

12.560

49. Refer to Exhibit 13-6. The estimated regression equation is

a. y =

0 +

1x1 +

2x2 +

b. E(y) =

0 +

1x1 +

2x2

c.

ˆ

y

= 12.924 − 3.682x1 + 45.216x2

d.

ˆ

y

= 4.425 + 2.63x1 + 12.56x2

50. Refer to Exhibit 13-6. The interpretation of the coefficient of x1 is that

a. a one unit change in x1 will lead to a 3.682 unit decrease in

ˆ

y

b. a one unit increase in x1 will lead to a 3.682 unit decrease in

ˆ

y

when all other

variables are held constant

c. a one unit increase in x1 will lead to a 3.682 unit decrease in x2 when all other

variables are held constant

d. It is impossible to interpret the coefficient.

51. Refer to Exhibit 13-6. We want to test whether the parameter

1 is significant. The test

statistic equals

a. -1.4

b. 1.4

c. 3.6

d. 5

52. Refer to Exhibit 13-6. The t value obtained from the table which is used to test an

individual parameter at the 1% level is

a. 2.65

b. 2.921

c. 2.977

d. 3.012

53. Refer to Exhibit 13-6. Carry out the test of significance for the parameter

1 at the 1%

level. The null hypothesis should be

a. rejected

b. not rejected

c. revised

d. None of these alternatives is correct.

54. Refer to Exhibit 13-6. The degrees of freedom for the sum of squares explained by the

regression (SSR) are

a. 2

b. 3

c. 13

d. 15

55. Refer to Exhibit 13-6. The sum of squares due to error (SSE) equals

a. 37.33

b. 485.3

c. 4,853

d. 6,308.9

56. Refer to Exhibit 13-6. The test statistic used to determine if there is a relationship among

the variables equals

a. -1.4

b. 0.2

c. 0.77

d. 5

57. Refer to Exhibit 13-6. The F value obtained from the table used to test if there is a

relationship among the variables at the 5% level equals

a. 3.41

b. 3.63

c. 3.81

d. 19.41

58. Refer to Exhibit 13-6. Carry out the test to determine if there is a relationship among the

variables at the 5% level. The null hypothesis should

a. be rejected

b. not be rejected

c. revised

d. None of these alternatives is correct.

59. A regression model in which more than one independent variable is used to predict the

dependent variable is called

a. a simple linear regression model

b. a multiple regression model

c. an independent model

d. None of these alternatives is correct.

60. A term used to describe the case when the independent variables in a multiple regression

model are correlated is

a. regression

b. correlation

c. multicollinearity

d. None of the alternative answers is correct.

61. A variable that cannot be measured in numerical terms is called

a. a nonmeasurable random variable

b. a constant variable

c. a dependent variable

d. a qualitative variable

62. A multiple regression model has the form

ˆ

y

= 5 + 6x + 7w

As x increases by 1 unit (holding w constant), y is expected to

a. increase by 11 units

b. decrease by 11 units

c. increase by 6 units

d. decrease by 6 units

Exhibit 13-7

A regression model involving 4 independent variables and a sample of 15 periods resulted in the

following sum of squares.

SSR = 165

63. Refer to Exhibit 13-7. The coefficient of determination is

a. 0.3636

b. 0.7333

c. 0.275

d. 0.5

64. Refer to Exhibit 13-7. If we want to test for the significance of the model at 95%

confidence, the critical F value (from the table) is

a. 3.06

b. 3.48

c. 3.34

d. 3.11

65. Refer to Exhibit 13-7. The test statistic from the information provided is

a. 2.110

b. 3.480

c. 4.710

d. 6.875

Exhibit 13-8

The following estimated regression model was developed relating yearly income (y in $1,000s) of

30 individuals with their age (x1) and their gender (x2) (0 if male and 1 if female).

ˆ

y

= 30 + 0.7x1 + 3x2

Also provided are SST = 1,200 and SSE = 384.

66. Refer to Exhibit 13-8. From the above function, it can be said that the expected yearly

income of

a. males is $3 more than females

b. females is $3 more than males

c. males is $3,000 more than females

d. females is $3,000 more than males

67. Refer to Exhibit 13-8. The yearly income of a 24-year-old female individual is

a. $19.80

b. $19,800

c. $49.80

d. $49,800

68. Refer to Exhibit 13-8. The yearly income of a 24-year-old male individual is

a. $13.80

b. $13,800

c. $46,800

d. $49,800

69. Refer to Exhibit 13-8. The multiple coefficient of determination is

a. 0.32

b. 0.42

c. 0.68

d. 0.50

70. Refer to Exhibit 13-8. If we want to test for the significance of the model, the critical

value of F at a 5% significance level is

a. 3.33

b. 3.35

c. 3.34

d. 2.96

71. Refer to Exhibit 13-8. The test statistic for testing the significance of the model is

a. 0.73

b. 1.47

c. 28.69

d. 5.22

72. Refer to Exhibit 13-8. The model

a. is significant

b. is not significant

c. would be significant is the sample size was larger than 30

d. None of these alternatives is correct.

73. Refer to Exhibit 13-8. The estimated income of a 30-year-old male is

a. $51,000

b. $5,100

c. $510

d. $51

74. For a multiple regression model, SST = 200 and SSE = 50. The multiple coefficient of

determination is

a. 0.25

b. 4.00

c. 250

d. 0.75

75. In a multiple regression analysis involving 10 independent variables and 81 observations,

SST = 120 and SSE = 42. The coefficient of determination is

a. 0.81

b. 0.11

c. 0.35

d. 0.65

76. A regression model involved 18 independent variables and 200 observations. The critical

value of t for testing the significance of each of the independent variable's coefficients

will have

a. 18 degrees of freedom

b. 200 degrees of freedom

c. 199 degrees of freedom

d. 181 degrees of freedom

77. In order to test for the significance of a regression model involving 8 independent

variables and 121 observations, the numerator and denominator degrees of freedom

(respectively) for the critical value of F are

a. 8 and 121

b. 7 and 120

c. 8 and 112

d. 7 and 112

78. In a multiple regression analysis involving 5 independent variables and 30 observations,

SSR = 360 and SSE = 40. The coefficient of determination is

a. 0.80

b. 0.90

c. 0.25

d. 0.15

79. A regression analysis involved 6 independent variables and 27 observations. The critical

value of t for testing the significance of each of the independent variable's coefficients

will have

a. 27 degrees of freedom

b. 26 degrees of freedom

c. 21 degrees of freedom

d. 20 degrees of freedom

80. In order to test for the significance of a regression model involving 4 independent

variables and 36 observations, the numerator and denominator degrees of freedom

(respectively) for the critical value of F are

a. 4 and 36

b. 3 and 35

c. 4 and 31

d. 4 and 32

PROBLEMS

1. Multiple regression analysis was used to study how an individual's income (y in

thousands of dollars) is influenced by age (x1 in years), level of education (x2 ranging

from 1 to 5), and the person's gender (x3 where 0 =female and 1=male). The following is

a partial result of Excel output that was used on a sample of 20 individuals.

ANOVA

df

SS

MS

F

Regression

84

Residual

112

Coefficients

Standard Error

x1

0.6251

0.094

x2

0.9210

0.190

x3

-0.510

0.920

a. Compute the coefficient of determination.

b. Perform a t test and determine whether or not the coefficient of the variable "level of

education" (i.e., x2) is significantly different from zero. Let = 0.05.

c. At = 0.05, perform an F test and determine whether or not the regression model is

significant.

d. As you note the coefficient of x3 is -0.510. Fully interpret the meaning of this

coefficient.

2. A multiple regression analysis between yearly income (y in $1,000s), college grade point

average (x1), age of the individuals (x2), and the gender of the individual (x3; zero

representing female and one representing male) was performed on a sample of 10 people,

and the following results were obtained using Excel.

ANOVA

df

SS

MS

F

Regression

360.59

Residual

23.91

Coefficients

Standard Error

Intercept

4.0928

1.4400

x1

10.0230

1.6512

x2

0.1020

0.1225

x3

-4.4811

1.4400

a. Write the regression equation for the above.

b. Interpret the meaning of the coefficient of x3.

c. Compute the coefficient of determination.

d. Is the coefficient of x1 significant? Use = 0.05.

e. Is the coefficient of x2 significant? Use = 0.05.

f. Is the coefficient of x3 significant? Use = 0.05.

g. Perform an F test and determine whether or not the model is significant.

3. The following results were obtained from a multiple regression analysis.

Source of

Variation

Degrees of

Freedom

Sum of

Squares

Mean

Square

F

Regression

900

Error

35

Total

39

4,980

a. How many independent variables were involved in this model?

b. How many observations were involved?

c. Determine the F statistic.

4. Shown below is a partial Excel output from a regression analysis.

ANOVA

df

SS

MS

F

Regression

60

Residual

Total

19

140

Coefficients

Standard Error

Intercept

10.00

2.00

x1

-2.00

1.50

x2

6.00

2.00

x3

-4.00

1.00

a. Use the above results and write the regression equation.

b. Compute the coefficient of determination and fully interpret its meaning.

c. Is the regression model significant? Perform an F test and let = 0.05.

d. At = 0.05, test to see if there is a relation between x1 and y.

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