CHAPTER 15—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
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
= 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
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 15-1
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.
18. Refer to Exhibit 15-1. 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 15-1. MSR for this model is
a.
200
b.
10
c.
1,000
d.
43
20. Refer to Exhibit 15-1. 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 15-2
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:
= 7 – 3x1 + 5x2
For this model SSR = 3500, SSE = 1500, and the sample size is 18.
32. Refer to Exhibit 15-2. 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 15-2. 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 15-2. 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 15-2. 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 15-2. The multiple coefficient of determination for this problem is
a.
0.4368
b.
0.6960
c.
0.3040
d.
0.2289
Exhibit 15-3
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 15-3. 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 15-3. 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 15-3. The critical F value at 95% confidence is
a.
2.53
b.
2.69
c.
2.76
d.
2.99
40. Refer to Exhibit 15-3. 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 15-4
a.
y = 0 + 1x1 + 2x2 +
b.
E(y) = 0 + 1x1 + 2x2
c.
= bo + b1 x1 + b2 x2
d.
E(y) = 0 + 1x1 + 2x2
41. Refer to Exhibit 15-4. Which equation describes the multiple regression model?
a.
equation a
b.
equation b
c.
equation c
d.
equation d
42. Refer to Exhibit 15-4. Which equation gives the estimated regression line?
a.
equation a
b.
equation b
c.
equation c
d.
equation d
43. Refer to Exhibit 15-4. Which equation describes the multiple regression equation?
a.
equation a
b.
equation b
c.
equation c
d.
equation d
Exhibit 15-5
Below you are given a partial Excel output based on a sample of 25 observations.
Standard Error
Intercept
48.682
x1
9.150
x2
3.575
x3
0.183
44. Refer to Exhibit 15-5. The estimated regression equation is
a.
y = 0 + 1x1 + 2x2 + 3x3 +
b.
E(y) = 0 + 1x1 + 2x2 + 3x3
c.
= 145.321 + 25.625x1 – 5.720x2 + 0.823x3
d.
= 48.682 + 9.15x1 + 3.575x2 + 0.183x3
45. Refer to Exhibit 15-5. 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 15-5. 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 15-5. 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 15-5. 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 15-6
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 15-6. The estimated regression equation is
a.
y = 0 + 1x1 + 2x2 +
b.
E(y) = 0 + 1x1 + 2x2
c.
= 12.924 – 3.682 x1 + 45.216 x2
d.
= 4.425 + 2.63 x1 + 12.56 x2
50. Refer to Exhibit 15-6. 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.
51. Refer to Exhibit 15-6. 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 15-6. 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 15-6. 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 15-6. 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 15-6. 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 15-6. 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 15-6. 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 15-6. 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 are 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
= 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 15-7
A regression model involving 4 independent variables and a sample of 15 periods resulted in the
following sum of squares.
SSR = 165
SSE = 60
63. Refer to Exhibit 15-7. The coefficient of determination is
a.
0.3636
b.
0.7333
c.
0.275
d.
0.5
64. Refer to Exhibit 15-7. 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 15-7. The test statistic from the information provided is
a.
2.110
b.
3.480
c.
4.710
d.
6.875
Exhibit 15-8
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.
66. Refer to Exhibit 15-8. 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 15-8. 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 15-8. 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 15-8. The multiple coefficient of determination is
a.
0.32
b.
0.42
c.
0.68
d.
0.50
70. Refer to Exhibit 15-8. 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 15-8. 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 15-8. 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 15-8. 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
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
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 is
a.
a simple nonlinear regression model
b.
a multiple regression model
c.
an estimated multiple regression equation
d.
a multiple regression equation
84. The least squares criterion is
a. min
−
ii
xy
2
()
b. min
−
i
yy
2
()
c. min
−
ii
yy
2
ˆ
()
d. min
ˆ
()
ii
yy−
PROBLEM
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.
a.
0.4286
b.
c.
d.
Male’s income is lower than female’s by $510
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.
a.
5
b.
40
c.
1.93
c.
0.9378
d.
e.
g.