Chapter 13 – Multiple Regression
Multiple Choice
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
ANSWER:
a
POINTS:
1
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
ANSWER:
c
POINTS:
1
3. For a multiple regression model, SSR = 600 and SSE = 200. The multiple coefficient of determination is
a.
b.
c.
d.
POINTS:
1
4. In a multiple regression analysis involving 15 independent variables and 200 observations, SST = 800 and SSE = 240.
The coefficient of determination is
a.
b.
c.
d.
ANSWER:
d
POINTS:
1
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
ANSWER:
c
6. In order to test for the significance of a regression model involving 3 independent variables and 47 observations, the
Chapter 13 – Multiple Regression
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
d
1
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
d
1
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
d
1
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.
c
1
10. In a multiple regression model, the error term ε is assumed to be a random variable with a mean of
a.
zero
b.
-1
c.
1
d.
any value
a
1
11. In regression analysis, the response variable is the
a.
independent variable
b.
dependent variable
Chapter 13 – Multiple Regression
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
c
13. A multiple regression model has the form
= 7 + 2 x1 + 9 x2
As x1 increases by 1 unit (holding x2 constant), is expected to
a.
increase by 9 units
b.
decrease by 9 units
c.
increase by 2 units
d.
decrease by 2 units
c
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
c
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
a
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
Chapter 13 – Multiple Regression
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.
18. 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.
19. The ratio of MSE/MSR yields
a.
SST
b.
the F statistic
c.
SSR
d.
None of these alternatives is correct.
20. In a multiple regression model, the variance of the error term ε 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
c
21. 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
22. In multiple regression analysis, the correlation among the independent variables is termed
Chapter 13 – Multiple Regression
a.
homoscedasticity
b.
linearity
c.
multicollinearity
d.
adjusted coefficient of determination
c
23. In a multiple regression model, the values of the error term ,ε, are assumed to be
a.
zero
b.
dependent on each other
c.
independent of each other
d.
always negative
c
24. 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
25. In a multiple regression model, the error term ε 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
26. 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
c
27. 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
Chapter 13 – Multiple Regression
d.
714 degrees of freedom
c
28. 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 13-1
In a regression model involving 44 observations, the following estimated regression equation was obtained.
= 29 + 18x1 +43x2 + 87x3
For this model SSR = 600 and SSE = 400.
29. Refer to Exhibit 13–1. The coefficient of determination for the above model is
a.
b.
c.
d.
30. Refer to Exhibit 13–1. MSR for this model is
a.
b.
c.
d.
a
31. 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
Exhibit 13-2
Chapter 13 – Multiple Regression
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:
= 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
c
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
a
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
c
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
Chapter 13 – Multiple Regression
Exhibit 13-3
In a regression model involving 30 observations, the following estimated regression equation was obtained:
= 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.
b.
c.
d.
a
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
c
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 = β0 + β1x1 + β2x2 + ε
b.
E(y) = β0 + β1x1 + β2x2
41. Refer to Exhibit 13–4. Which equation describes the multiple regression model?
a.
equation a
b.
equation b
c.
equation c
d.
equation d
a
42. Refer to Exhibit 13–4. Which equation gives the estimated regression line?
a.
equation a
b.
equation b
c.
equation c
d.
equation d
c
43. Refer to Exhibit 13–4. Which equation describes the multiple regression equation?
a.
equation a
b.
equation b
c.
equation c
d.
equation 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 = β0 + β1x1 + β2x2 + β3x3 + ε
b.
E(y) = β0 + β1x1 + β2x2 + β3x3
c.
= 145.321 + 25.625x1 – 5.720x2 + 0.823x3
d.
= 48.682 + 9.15x1 + 3.575x2 + 0.183x3
c
E(y) = β0 + β1x1 + β2x2
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 y
b.
a one unit change in x1 will lead to a 25.625 unit increase in 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 β1 is significant. The test statistic equals
a.
b.
c.
d.
47. Refer to Exhibit 13–5. The t value obtained from the table to test an individual parameter at the 5% level is
a.
b.
c.
d.
48. Refer to Exhibit 13–5. Carry out the test of significance for the parameter β1 at the 5% level. The null hypothesis
should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
a
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 = β0 + β1x1 + β2x2 + ε
b.
E(y) = β0 + β1x1 + β2x2
c.
= 12.924 – 3.682 x1 + 45.216 x2
d.
= 4.425 + 2.63 x1 + 12.56 x2
c
1
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 y
b.
a one unit increase in x1 will lead to a 3.682 unit decrease in 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.
c
1
51. Refer to Exhibit 13–6. We want to test whether the parameter β1 is significant. The test statistic equals
a.
-1.4
b.
1.4
c.
3.6
d.
5
a
1
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.
b.
c.
d.
d
1
53. Refer to Exhibit 13–6. Carry out the test of significance for the parameter β1 at the 1% level. The null hypothesis
should be
a.
rejected
b.
not rejected
c.
revised
d.
None of these alternatives is correct.
b
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
a
1
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
d
1
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
d
1
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.
b.
c.
d.
c
1
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.
a
Exhibit 13-7
A regression model involving 4 independent variables and a sample of 15 periods resulted in the following sum of
1
Chapter 13 – Multiple Regression
squares.
SSR = 165
SSE = 60
59. Refer to Exhibit 13–7. The coefficient of determination is
a.
0.3636
b.
0.7333
c.
0.275
d.
0.5
60. 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
61. Refer to Exhibit 13–7. The test statistic from the information provided is
a.
b.
c.
d.
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).
= 30 + 0.7x1 + 3x2
Also provided are SST = 1,200 and SSE = 384.
62. 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
63. Refer to Exhibit 13–8. The yearly income of a 24-year-old female individual is
Chapter 13 – Multiple Regression
a.
$19.80
b.
$19,800
c.
$49.80
d.
$49,800
64. 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
c
65. Refer to Exhibit 13–8. The multiple coefficient of determination is
a.
0.32
b.
0.42
c.
0.68
d.
0.50
c
66. 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
67. Refer to Exhibit 13–8. The test statistic for testing the significance of the model is
a.
b.
c.
d.
c
68. 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.
Chapter 13 – Multiple Regression
a
69. Refer to Exhibit 13–8. The estimated income of a 30-year-old male is
a.
$510,000
b.
$51,000
c.
$5,100
d.
$510
70. A multiple regression model has the form
= 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
c
71. 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
72. 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 are correct.
c
73. 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.
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
c
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
c
81. In a residual plot that does not suggest we should challenge the assumptions of our regression model, we would expect
to see
a.
a horizontal band of points centered near zero
b.
a widening band of points
c.
a band of points having a slope consistent with that of the regression equation
d.
a parabolic band of points
a
82. The difference between the observed value of the dependent variable and the value predicted by using the estimated
regression equation is the
a.
standard error
b.
residual
c.
predicted interval
d.
variance
83. The mathematical equation that explains how the dependent variable y is related to several independent variables x1,
x2, …, xp and the error term ε is
a.
a simple nonlinear regression model
b.
a multiple regression model
c.
an estimated multiple regression equation
d.
a multiple regression equation
84. A dummy variable may take
only the value 0 or 1
only the value -1 or 1
only non-negative values
any value between 0 and 1
a.
b.
c.
d.
a
85. Unlike a simple linear regression model, a multiple regression model has more than one
intercept
dependent variable
independent variable
error term
a.
b.
c.
d.
a
86. If an independent variable is added to a multiple regression model, the R2 value
a.
becomes larger or smaller depending on the statistical significance of the variable
b.
becomes larger even if the variable added is not statistically significant
c.
might or might become larger even if the variable added is statistically significant
d.
is not affected by the significance of individual variables
Subjective Short Answer
87. 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