# Chapter 14 the mean percentage of students passing the proficiency test is estimated

Document Type
Test Prep
Book Title
Authors
David M. Levine, Kathryn A. Szabat, Mark L. Berenson
Introduction to Multiple Regression 14-1
CHAPTER 14: INTRODUCTION TO MULTIPLE
REGRESSION
1. In a multiple regression problem involving two independent variables, if b1 is computed to be
+2.0, it means that
a) the relationship between X1 and Y is significant.
b) the estimated mean of Y increases by 2 units for each increase of 1 unit of X1, holding X2
constant.
c) the estimated mean of Y increases by 2 units for each increase of 1 unit of X1, without
regard to X2.
d) the estimated mean of Y is 2 when X1 equals zero.
2. The coefficient of multiple determination r2Y.12
a) measures the variation around the predicted regression equation.
b) measures the proportion of variation in Y that is explained by X1 and X2.
c) measures the proportion of variation in Y that is explained by X1 holding X2 constant.
d) will have the same sign as b1.
3. In a multiple regression model, the value of the coefficient of multiple determination
a) has to fall between -1 and +1.
b) has to fall between 0 and +1.
c) has to fall between -1 and 0.
d) can fall between any pair of real numbers.
4. In a multiple regression model, which of the following is correct regarding the value of the
r?
a) It can be negative.
b) It has to be positive.
c) It has to be larger than the coefficient of multiple determination.
d) It can be larger than 1.
14-2 Introduction to Multiple Regression
SCENARIO 14-1
A manager of a product sales group believes the number of sales made by an employee (Y) depends
on how many years that employee has been with the company (X1) and how he/she scored on a
business aptitude test (X2). A random sample of 8 employees provides the following:
Employee Y X1 X2
1 100 10 7
2 90 3 10
3 80 8 9
4 70 5 4
5 60 5 8
6 50 7 5
7 40 1 4
8 30 1 1
5. Referring to Scenario 14-1, for these data, what is the value for the regression constant, b0?
a) 0.998
b) 3.103
c) 4.698
d) 21.293
6. Referring to Scenario 14-1, for these data, what is the estimated coefficient for the variable
representing years an employee has been with the company, b1?
a) 0.998
b) 3.103
c) 4.698
d) 21.293
7. Referring to Scenario 14-1, for these data, what is the estimated coefficient for the variable
representing scores on the aptitude test, b2?
a) 0.998
b) 3.103
c) 4.698
d) 21.293
Introduction to Multiple Regression 14-3
8. Referring to Scenario 14-1, if an employee who had been with the company 5 years scored a 9 on
the aptitude test, what would his estimated expected sales be?
a) 79.09
b) 60.88
c) 55.62
d) 17.98
SCENARIO 14-2
A professor of industrial relations believes that an individual’s wage rate at a factory (Y) depends on
his performance rating (X1) and the number of economics courses the employee successfully
completed in college (X2). The professor randomly selects 6 workers and collects the following
information:
Employee Y (\$) X1 X2
1 10 3 0
2 12 1 5
3 15 8 1
4 17 5 8
5 20 7 12
6 25 10 9
9. Referring to Scenario 14-2, for these data, what is the value for the regression constant, b0?
a) 0.616
b) 1.054
c) 6.932
d) 9.103
10. Referring to Scenario 14-2, for these data, what is the estimated coefficient for performance
rating, b1?
a) 0.616
b) 1.054
c) 6.932
d) 9.103
14-4 Introduction to Multiple Regression
11. Referring to Scenario 14-2, for these data, what is the estimated coefficient for the number of
economics courses taken, b2?
a) 0.616
b) 1.054
c) 6.932
d) 9.103
12. Referring to Scenario 14-2, suppose an employee had never taken an economics course and
managed to score a 5 on his performance rating. What is his estimated expected wage rate?
a) 10.90
b) 12.20
c) 17.23
d) 25.11
13. Referring to Scenario 14-2, an employee who took 12 economics courses scores 10 on the
performance rating. What is her estimated expected wage rate?
a) 10.90
b) 12.20
c) 24.87
d) 25.70
14. The variation attributable to factors other than the relationship between the independent variables
and the explained variable in a regression analysis is represented by
a) regression sum of squares.
b) error sum of squares.
c) total sum of squares.
d) regression mean squares.
Introduction to Multiple Regression 14-5
SCENARIO 14-3
An economist is interested to see how consumption for an economy (in \$ billions) is influenced by
gross domestic product (\$ billions) and aggregate price (consumer price index). The Microsoft Excel
output of this regression is partially reproduced below.
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.991
R Square 0.982
Standard Error 0.299
Observations 10
ANOVA
df SS MS F Signif F
Regression 2 33.4163 16.7082 186.325 0.0001
Residual 7 0.6277 0.0897
Total 9 34.0440
Coeff StdError t Stat P-value
Intercept – 0.0861 0.5674 – 0.152 0.8837
GDP 0.7654 0.0574 13.340 0.0001
Price – 0.0006 0.0028 – 0.219 0.8330
15. Referring to Scenario 14-3, when the economist used a simple linear regression model with
consumption as the dependent variable and GDP as the independent variable, he obtained an r2
value of 0.971. What additional percentage of the total variation of consumption has been
explained by including aggregate prices in the multiple regression?
a) 98.2
b) 11.1
c) 2.8
d) 1.1
16. Referring to Scenario 14-3, the p-value for GDP is
a) 0.05
b) 0.01
c) 0.001
d) None of the above.
14-6 Introduction to Multiple Regression
17. Referring to Scenario 14-3, the p-value for the aggregated price index is
a) 0.05
b) 0.01
c) 0.001
d) None of the above.
18. Referring to Scenario 14-3, the p-value for the regression model as a whole is
a) 0.05
b) 0.01
c) 0.001
d) None of the above.
19. Referring to Scenario 14-3, what is the predicted consumption level for an economy with GDP
equal to \$4 billion and an aggregate price index of 150?
a) \$1.39 billion
b) \$2.89 billion
c) \$4.75 billion
d) \$9.45 billion
20. Referring to Scenario 14-3, what is the estimated mean consumption level for an economy with
GDP equal to \$4 billion and an aggregate price index of 150?
a) \$1.39 billion
b) \$2.89 billion
c) \$4.75 billion
d) \$9.45 billion
Introduction to Multiple Regression 14-7
21. Referring to Scenario 14-3, what is the estimated mean consumption level for an economy with
GDP equal to \$2 billion and an aggregate price index of 90?
a) \$1.39 billion
b) \$2.89 billion
c) \$4.75 billion
d) \$9.45 billion
22. Referring to Scenario 14-3, one economy in the sample had an aggregate consumption level of \$3
billion, a GDP of \$3.5 billion, and an aggregate price level of 125. What is the residual for this
data point?
a) \$2.52 billion
b) \$0.48 billion
c) – \$1.33 billion
d) – \$2.52 billion
23. Referring to Scenario 14-3, one economy in the sample had an aggregate consumption level of \$4
billion, a GDP of \$6 billion, and an aggregate price level of 200. What is the residual for this data
point?
a) \$4.39 billion
b) \$0.39 billion
c) – \$0.39 billion
d) – \$1.33 billion
24. Referring to Scenario 14-3, to test for the significance of the coefficient on aggregate price index,
the value of the relevant t-statistic is
a) 2.365
b) 0.143
c) – 0.219
d) – 1.960
14-8 Introduction to Multiple Regression
25. Referring to Scenario 14-3, to test for the significance of the coefficient on aggregate price index,
the p-value is
a) 0.0001
b) 0.8330
c) 0.8837
d) 0.9999
26. Referring to Scenario 14-3, to test for the significance of the coefficient on gross domestic
product, the p-value is
a) 0.0001
b) 0.8330
c) 0.8837
d) 0.9999
27. Referring to Scenario 14-3, to test whether aggregate price index has a negative impact on
consumption, the p-value is _______?
a) 0.0001
b) 0.4165
c) 0.8330
d) 0.8837
28. Referring to Scenario 14-3, to test whether aggregate price index has a positive impact on
consumption, the p-value is
a) 0.0001
b) 0.4165
c) 0.5835
d) 0.8330
Introduction to Multiple Regression 14-9
29. Referring to Scenario 14-3, to test whether gross domestic product has a positive impact on
consumption, the p-value is
a) 0.00005
b) 0.0001
c) 0.9999
d) 0.99995
SCENARIO 14-4
A real estate builder wishes to determine how house size (House) is influenced by family income
(Income) and family size (Size). House size is measured in hundreds of square feet and income is
measured in thousands of dollars. The builder randomly selected 50 families and ran the multiple
regression. Partial Microsoft Excel output is provided below:
Regression Statistics
Multiple R 0.8479
R Square 0.7189
Standard Error 17.5571
Observations 50
ANOVA
df SS MS F Significance F
Regression 37043.3236 18521.6618 0.0000
Residual 14487.7627 308.2503
Total 49 51531.0863
Coefficients Standard Error t Stat P-value
Intercept -5.5146 7.2273 -0.7630 0.4493
Income 0.4262 0.0392 10.8668 0.0000
Size 5.5437 1.6949 3.2708 0.0020
Also
()
12
|SSR X X =36400.6326 and
()
21
| 3297.7917SSR X X =
14-10 Introduction to Multiple Regression
30. Referring to Scenario 14-4, what fraction of the variability in house size is explained by income
and size of family?
a) 17.56%
b) 70.69%
c) 71.89%
d) 84.79%
31. Referring to Scenario 14-4, which of the independent variables in the model are significant at the
5% level?
a) Income only
b) Size only
c) Income and Size
d) None
32. Referring to Scenario 14-4, when the builder used a simple linear regression model with house
size (House) as the dependent variable and family size (Size) as the independent variable, he
obtained an r2 value of 1.25%. What additional percentage of the total variation in house size has
been explained by including income in the multiple regression?
a) 15.00%
b) 70.64%
c) 71.50%
d) 73.62%
33. Referring to Scenario 14-4, which of the following values for the level of significance is the
smallest for which each explanatory variable is significant individually?
a) 0.001
b) 0.010
c) 0.025
d) 0.050
Introduction to Multiple Regression 14-11
34. Referring to Scenario 14-4, which of the following values for the level of significance is the
smallest for which at least one explanatory variable is significant individually?
a) 0.005
b) 0.010
c) 0.025
d) 0.050
35. Referring to Scenario 14-4, which of the following values for the level of significance is the
smallest for which at most one explanatory variable is significant individually?
a) 0.001
b) 0.010
c) 0.025
d) 0.050
36. Referring to Scenario 14-4, which of the following values for the level of significance is the
smallest for which the regression model as a whole is significant?
a) 0.0005
b) 0.001
c) 0.01
d) 0.05
37. Referring to Scenario 14-4, what is the predicted house size (in hundreds of square feet) for an
individual earning an annual income of \$40,000 and having a family size of 4?
38. Referring to Scenario 14-4, what annual income (in thousands of dollars) would an individual
with a family size of 4 need to attain a predicted 10,000 square foot home (House = 100)?
14-12 Introduction to Multiple Regression
39. Referring to Scenario 14-4, what annual income (in thousands of dollars) would an individual
with a family size of 9 need to attain a predicted 5,000 square foot home (House = 50)?
40. Referring to Scenario 14-4, one individual in the sample had an annual income of \$100,000 and a
family size of 10. This individual owned a home with an area of 7,000 square feet (House =
70.00). What is the residual (in hundreds of square feet) for this data point?
41. Referring to Scenario 14-4, one individual in the sample had an annual income of \$40,000 and a
family size of 1. This individual owned a home with an area of 1,000 square feet (House =
10.00). What is the residual (in hundreds of square feet) for this data point?
42. Referring to Scenario 14-4, suppose the builder wants to test whether the coefficient on Income is
significantly different from 0. What is the value of the relevant t-statistic?
a) -0.7630
b) 3.2708
c) 10.8668
d) 60.0864
Introduction to Multiple Regression 14-13
43. Referring to Scenario 14-4, at the 0.01 level of significance, what conclusion should the builder
reach regarding the inclusion of Income in the regression model?
a) Income is significant in explaining house size and should be included in the model
because its p-value is less than 0.01.
b) Income is significant in explaining house size and should be included in the model
because its p-value is more than 0.01.
c) Income is not significant in explaining house size and should not be included in the
model because its p-value is less than 0.01.
d) Income is not significant in explaining house size and should not be included in the
model because its p-value is more than 0.01.
44. Referring to Scenario 14-4, suppose the builder wants to test whether the coefficient on Size is
significantly different from 0. What is the value of the relevant t-statistic?
a) -0.7630
b) 3.2708
c) 10.8668
d) 60.0864
45. Referring to Scenario 14-4, at the 0.01 level of significance, what conclusion should the builder
draw regarding the inclusion of Size in the regression model?
a) Size is significant in explaining house size and should be included in the model because
its p-value is less than 0.01.
b) Size is significant in explaining house size and should be included in the model because
its p-value is more than 0.01.
c) Size is not significant in explaining house size and should not be included in the model
because its p-value is less than 0.01.
d) Size is not significant in explaining house size and should not be included in the model
because its p-value is more than 0.01.
46. Referring to Scenario 14-4, what is the value of the calculated F test statistic that is missing from
the output for testing whether the whole regression model is significant?
14-14 Introduction to Multiple Regression
47. Referring to Scenario 14-4, the observed value of the F-statistic is missing from the printout.
What are the degrees of freedom for this F-statistic?
a) 2 for the numerator, 47 for the denominator
b) 2 for the numerator, 49 for the denominator
c) 49 for the numerator, 47 for the denominator
d) 47 for the numerator, 49 for the denominator
48. Referring to Scenario 14-4, what are the regression degrees of freedom that are missing from the
output?
a) 2
b) 47
c) 49
d) 50
49. Referring to Scenario 14-4, what are the residual degrees of freedom that are missing from the
output?
a) 2
b) 47
c) 49
d) 50
50. Referring to Scenario 14-4 and allowing for a 1% probability of committing a type I error, what is
the decision and conclusion for the test 01 2 1
: 0 vs. : At least one 0, 1, 2
j
HH j
ββ β
== = ?
a) Do not reject H0 and conclude that the 2 independent variables taken as a group have
significant linear effects on house size.
b) Do not reject H0 and conclude that the 2 independent variables taken as a group do not
have significant linear effects on house size.
c) Reject H0 and conclude that the 2 independent variables taken as a group have significant
linear effects on house size.
d) Reject H0 and conclude that the 2 independent variables taken as a group do not have
significant linear effects on house size.
Introduction to Multiple Regression 14-15
51. Referring to Scenario 14-4, the value of the partial F test statistic is ____ for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
52. Referring to Scenario 14-4, the partial F test for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
has ____ and ____ degrees of freedom.
53. Referring to Scenario 14-4, the value of the partial F test statistic is ____ for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
54. Referring to Scenario 14-4, the partial F test for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
has ____ and ____ degrees of freedom.
55. Referring to Scenario 14-4, the coefficient of partial determination 2
12Y
r is ____.
56. Referring to Scenario 14-4, the coefficient of partial determination 2
21Y
r is ____.
57. Referring to Scenario 14-4, ____% of the variation in the house size can be explained by the
variation in the family income while holding the family size constant.
58. Referring to Scenario 14-4, ____% of the variation in the house size can be explained by the
variation in the family size while holding the family income constant.
SCENARIO 14-5
A microeconomist wants to determine how corporate sales are influenced by capital and wage
spending by companies. She proceeds to randomly select 26 large corporations and record
information in millions of dollars. The Microsoft Excel output below shows results of this multiple
regression.
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.830
R Square 0.689
Standard Error 17501.643
Observations 26
ANOVA
df SS MS F Signif F
Regression 2 15579777040 7789888520 25.432 0.0001
Residual 23 7045072780 306307512
Total 25 22624849820
Coeff StdError t Stat P-value
Intercept 15800.0000 6038.2999 2.617 0.0154
Capital 0.1245 0.2045 0.609 0.5485
Wages 7.0762 1.4729 4.804 0.0001
Introduction to Multiple Regression 14-17
59. Referring to Scenario 14-5, what fraction of the variability in sales is explained by spending on
capital and wages?
a) 27.0%
b) 50.9%
c) 68.9%
d) 83.0%
60. Referring to Scenario 14-5, which of the independent variables in the model are significant at the
5% level?
a) Capital, Wages
b) Capital
c) Wages
d) None of the above
61. Referring to Scenario 14-5, when the microeconomist used a simple linear regression model with
sales as the dependent variable and wages as the independent variable, she obtained an r2 value of
0.601. What additional percentage of the total variation of sales has been explained by including
capital spending in the multiple regression?
a) 60.1%
b) 31.1%
c) 22.9%
d) 8.8%
62. Referring to Scenario 14-5, what is the p-value for Wages?
a) 0.01
b) 0.05
c) 0.0001
d) None of the above
14-18 Introduction to Multiple Regression
63. Referring to Scenario 14-5, what is the p-value for testing whether Wages have a positive impact
on corporate sales?
a) 0.01
b) 0.05
c) 0.0001
d) 0.00005
64. Referring to Scenario 14-5, what is the p-value for testing whether Wages have a negative impact
on corporate sales?
a) 0.05
b) 0.0001
c) 0.00005
d) 0.99995
65. Referring to Scenario 14-5, what is the p-value for Capital?
a) 0.01
b) 0.025
c) 0.05
d) None of the above
66. Referring to Scenario 14-5, what is the p-value for testing whether Capital has a positive
influence on corporate sales?
a) 0.025
b) 0.05
c) 0.2743
d) 0.5485
Introduction to Multiple Regression 14-19
67. Referring to Scenario 14-5, what is the p-value for testing whether Capital has a negative
influence on corporate sales?
a) 0.05
b) 0.2743
c) 0.5485
d) 0.7258
68. Referring to Scenario 14-5, which of the following values for
α
is the smallest for which the
regression model as a whole is significant?
a) 0.00005
b) 0.001
c) 0.01
d) 0.05
69. Referring to Scenario 14-5, what are the predicted sales (in millions of dollars) for a company
spending \$100 million on capital and \$100 million on wages?
a) 15,800.00
b) 16,520.07
c) 17,277.49
d) 20,455.98
70. Referring to Scenario 14-5, what are the predicted sales (in millions of dollars) for a company
spending \$500 million on capital and \$200 million on wages?
a) 15,800.00
b) 16,520.07
c) 17,277.49
d) 20,455.98
14-20 Introduction to Multiple Regression
71. Referring to Scenario 14-5, one company in the sample had sales of \$20 billion (Sales = 20,000).
This company spent \$300 million on capital and \$700 million on wages. What is the residual (in
millions of dollars) for this data point?
a) 874.55
b) 622.87
c) –790.69
d) –983.56
72. Referring to Scenario 14-5, one company in the sample had sales of \$21.439 billion (Sales =
21,439). This company spent \$300 million on capital and \$700 million on wages. What is the
residual (in millions of dollars) for this data point?
a) 790.69
b) 648.31
c) –648.31
d) –790.69
73. Referring to Scenario 14-5, suppose the microeconomist wants to test whether the coefficient on
Capital is significantly different from 0. What is the value of the relevant t-statistic?
a) 0.609
b) 2.617
c) 4.804
d) 25.432
Introduction to Multiple Regression 14-21
74. Referring to Scenario 14-5, at the 0.01 level of significance, what conclusion should the
microeconomist reach regarding the inclusion of Capital in the regression model?
a) Capital is significant in explaining corporate sales and should be included in the model
because its p-value is less than 0.01.
b) Capital is significant in explaining corporate sales and should be included in the model
because its p-value is more than 0.01.
c) Capital is not significant in explaining corporate sales and should not be included in the
model because its p-value is less than 0.01.
d) Capital is not significant in explaining corporate sales and should not be included in the
model because its p-value is more than 0.01.
75. Referring to Scenario 14-5, the observed value of the F-statistic is given on the printout as
25.432. What are the degrees of freedom for this F-statistic?
a) 25 for the numerator, 2 for the denominator
b) 2 for the numerator, 23 for the denominator
c) 23 for the numerator, 25 for the denominator
d) 2 for the numerator, 25 for the denominator
14-22 Introduction to Multiple Regression
SCENARIO 14-6
One of the most common questions of prospective house buyers pertains to the cost of heating in
dollars (Y). To provide its customers with information on that matter, a large real estate firm used the
following 2 variables to predict heating costs: the daily minimum outside temperature in degrees of
Fahrenheit ( 1
X
) and the amount of insulation in inches ( 2
). Given below is EXCEL output of the
regression model.
Regression Statistics
Multiple R 0.5270
R Square 0.2778
Standard Error 40.9107
Observations 20
ANOVA
df SS MS F Significance F
Regression 2 10943.0190 5471.5095 3.2691 0.0629
Residual 17 28452.6027 1673.6825
Total 19 39395.6218
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 448.2925 90.7853 4.9379 0.0001 256.7522 639.8328
Temperature -2.7621 1.2371 -2.2327 0.0393 -5.3721 -0.1520
Insulation -15.9408 10.0638 -1.5840 0.1316 -37.1736 5.2919
Also
()
12
|SSR X X =8343.3572 and
()
21
|SSR X X =4199.2672
76. Referring to Scenario 14-6, the estimated value of the regression parameter 1
β
in means that
a) holding the effect of the amount of insulation constant, an estimated expected \$1 increase
in heating costs is associated with a decrease in the daily minimum outside temperature
by 2.76 degrees.
b) holding the effect of the amount of insulation constant, a 1 degree increase in the daily
minimum outside temperature results in a decrease in heating costs by \$2.76.
c) holding the effect of the amount of insulation constant, a 1 degree increase in the daily
minimum outside temperature results in an estimated decrease in mean heating costs by
\$2.76.
d) holding the effect of the amount of insulation constant, a 1% increase in the daily
minimum outside temperature results in an estimated decrease in mean heating costs by
2.76%.
Introduction to Multiple Regression 14-23
77. Referring to Scenario 14-6, what can we say about the regression model?
a) The model explains 17.12% of the variability of heating costs; after correcting for the
degrees of freedom, the model explains 27.78% of the sample variability of heating costs.
b) The model explains 19.28% of the variability of heating costs; after correcting for the
degrees of freedom, the model explains 27.78% of the sample variability of heating costs.
c) The model explains 27.78% of the variability of heating costs; after correcting for the
degrees of freedom, the model explains 19.28% of the sample variability of heating costs.
d) The model explains 19.28% of the variability of heating costs; after correcting for the
degrees of freedom, the model explains 17.12% of the sample variability of heating costs.
78. Referring to Scenario 14-6, what is your decision and conclusion for the test
02 12
: 0 vs. : 0HH
ββ
=≠
at the
α
= 0.01 level of significance?
a) Do not reject H0 and conclude that the amount of insulation has a linear effect on heating
costs.
b) Reject H0 and conclude that the amount of insulation does not have a linear effect on
heating costs.
c) Reject H0 and conclude that the amount of insulation has a linear effect on heating costs.
d) Do not reject H0 and conclude that the amount of insulation does not have a linear effect
on heating costs.
79. Referring to Scenario 14-6, what is the 95% confidence interval for the expected change in
heating costs as a result of a 1 degree Fahrenheit change in the daily minimum outside
temperature?
a) [256.7522, 639.8328]
b) [204.7854, 497.1733]
c) [5.3721, 0.1520]
d) [37.1736, 5.2919]
14-24 Introduction to Multiple Regression
80. Referring to Scenario 14-6 and allowing for a 1% probability of committing a type I error, what is
the decision and conclusion for the test 01 2 1
: 0 vs. : At least one 0, 1, 2
j
HH j
ββ β
== = ?
a) Do not reject H0 and conclude that the 2 independent variables taken as a group have
significant linear effects on heating costs.
b) Do not reject H0 and conclude that the 2 independent variables taken as a group do not
have significant linear effects on heating costs.
c) Reject H0 and conclude that the 2 independent variables taken as a group have significant
linear effects on heating costs.
d) Reject H0 and conclude that the 2 independent variables taken as a group do not have
significant linear effects on heating costs.
81. Referring to Scenario 14-6, the value of the partial F test statistic is ____ for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
82. Referring to Scenario 14-6, the partial F test for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
has ____ and ____ degrees of freedom.
83. Referring to Scenario 14-6, the value of the partial F test statistic is ____ for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
Introduction to Multiple Regression 14-25
84. Referring to Scenario 14-6, the partial F test for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
has ____ and ____ degrees of freedom.
85. Referring to Scenario 14-6, the coefficient of partial determination 2
12Y
r is ____.
86. Referring to Scenario 14-6, the coefficient of partial determination 2
21Y
r is ____.
87. Referring to Scenario 14-6, ____% of the variation in heating cost can be explained by the
variation in minimum outside temperature while holding the amount of insulation constant.
88. Referring to Scenario 14-6, ____% of the variation in heating cost can be explained by the
variation in the amount of insulation while holding the minimum outside temperature constant.
89. True or False: The interpretation of the slope is different in a multiple linear regression model as
compared to a simple linear regression model.
14-26 Introduction to Multiple Regression
90. True or False: The coefficient of multiple determination r2Y.12 measures the proportion of
variation in Y that is explained by X1 and X2.
91. True or False: When an additional explanatory variable is introduced into a multiple regression
model, the coefficient of multiple determination will never decrease.
92. True or False: When an additional explanatory variable is introduced into a multiple regression
rcan never decrease.
93. True or False: When an explanatory variable is dropped from a multiple regression model, the
coefficient of multiple determination can increase.
94. True or False: When an explanatory variable is dropped from a multiple regression model, the
rcan increase.
95. True or False: The slopes in a multiple regression model are called net regression coefficients.
Introduction to Multiple Regression 14-27
96. True or False: In calculating the standard error of the estimate, SYX =MSE , there are nk – 1
degrees of freedom, where n is the sample size and k represents the number of independent
variables in the model.
97. True or False: The total sum of squares (SST) in a regression model will never be greater than the
regression sum of squares (SSR).
98. True or False: The coefficient of multiple determination measures the proportion of the total
variation in the dependent variable that is explained by the set of independent variables.
99. True or False: The coefficient of multiple determination is calculated by taking the ratio of the
regression sum of squares over the total sum of squares (SSR/SST) and subtracting that value from
1.
100. True or False: In a particular model, the sum of the squared residuals was 847. If the model
had 5 independent variables, and the data set contained 40 points, the value of the standard error
of the estimate is 24.911.
101. True or False: A multiple regression is called “multiple” because it has several data points.
14-28 Introduction to Multiple Regression
102. True or False: A multiple regression is called “multiple” because it has several explanatory
variables.
103. True or False: If you have taken into account all relevant explanatory factors, the residuals
from a multiple regression model should be random.
104. True or False: Multiple regression is the process of using several independent variables to
predict a number of dependent variables.
105. True or False: You have just computed a regression model in which the value of coefficient of
multiple determination is 0.57. To determine if this indicates that the independent variables
explain a significant portion of the variation in the dependent variable, you would perform an F-
test.
106. True or False: From the coefficient of multiple determination, you cannot detect the strength
of the relationship between Y and any individual independent variable.
107. True or False: Consider a regression in which b2 = – 1.5 and the standard error of this
coefficient equals 0.3. To determine whether X2 is a significant explanatory variable, you would
compute an observed t-value of – 5.0.
Introduction to Multiple Regression 14-29
108. True or False: A regression had the following results: SST = 82.55, SSE = 29.85. It can be said
that 73.4% of the variation in the dependent variable is explained by the independent variables in
the regression.
109. True or False: A regression had the following results: SST = 82.55, SSE = 29.85. It can be said
that 63.84% of the variation in the dependent variable is explained by the independent variables
in the regression.
110. True or False: A regression had the following results: SST = 102.55, SSE = 82.04. It can be
said that 90.0% of the variation in the dependent variable is explained by the independent
variables in the regression.
111. True or False: A regression had the following results: SST = 102.55, SSE = 82.04. It can be
said that 20.0% of the variation in the dependent variable is explained by the independent
variables in the regression.
14-30 Introduction to Multiple Regression
SCENARIO 14-7
The department head of the accounting department wanted to see if she could predict the GPA of
students using the number of course units (credits) and total SAT scores of each. She takes a sample
of students and generates the following Microsoft Excel output:
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.916
R Square 0.839
Standard Error 0.24685
Observations 6
ANOVA
df SS MS F Signif F
Regression 2 0.95219 0.47610 7.813 0.0646
Residual 3 0.18281 0.06094
Total 5 1.13500
Coeff StdError t Stat P-value
Intercept 4.593897 1.13374542 4.052 0.0271
Units – 0.247270 0. 06268485 – 3.945 0.0290
SAT Total 0.001443 0.00101241 1.425 0.2494
112. Referring to Scenario 14-7, the estimate of the unit change in the mean of Y per unit change in
X1, holding X2 constant, is ________.
113. Referring to Scenario 14-7, the net regression coefficient of X2 is ________.
114. Referring to Scenario 14-7, the predicted GPA for a student carrying 15 course units and who
has a total SAT of 1,100 is ________.
Introduction to Multiple Regression 14-31
115. Referring to Scenario 14-7, the value of the coefficient of multiple determination, r2Y.12, is
________.
116. Referring to Scenario 14-7, the value of the adjusted coefficient of multiple determination, r2adj,
is ________.
117. Referring to Scenario 14-7, the department head wants to test H0:
β
1=
β
2=0. The appropriate
alternative hypothesis is ________.
118. Referring to Scenario 14-7, the department head wants to test H0:
β
1=
β
2=0. The critical
value of the F test for a level of significance of 0.05 is ________.
119. Referring to Scenario 14-7, the department head wants to test H0:
β
1=
β
2=0. The value of
the F-test statistic is ________.
120. Referring to Scenario 14-7, the department head wants to test H0:
β
1=
β
2=0. The p-value of
the test is ________.
14-32 Introduction to Multiple Regression
121. True or False: Referring to Scenario 14-7, the department head wants to test H0:
β
1=
β
2=0.
At a level of significance of 0.05, the null hypothesis is rejected.
122. Referring to Scenario 14-7, the department head wants to use a t test to test for the significance
of the coefficient of X1. For a level of significance of 0.05, the critical values of the test are
________.
123. Referring to Scenario 14-7, the department head wants to use a t test to test for the significance
of the coefficient of X1. The value of the test statistic is ________.
124. Referring to Scenario 14-7, the department head wants to use a t test to test for the
significance of the coefficient of X1. The p-value of the test is ________.
125. True or False: Referring to Scenario 14-7, the department head wants to use a t test to test for
the significance of the coefficient of X1. At a level of significance of 0.05, the department head
would decide that
β
10.
126. Referring to Scenario 14-7, the department head decided to construct a 95% confidence interval
for
β
1. The confidence interval is from ________ to ________.
Introduction to Multiple Regression 14-33
SCENARIO 14-8
A financial analyst wanted to examine the relationship between salary (in \$1,000) and 2 variables:
age (X1 = Age) and experience in the field (X2 = Exper). He took a sample of 20 employees and
obtained the following Microsoft Excel output:
Regression Statistics
Multiple R 0.8535
R Square 0.7284
Standard Error 10.5630
Observations 20
ANOVA
df SS MS F Significance F
Regression 2 5086.5764 2543.2882 22.7941 0.0000
Residual 17 1896.8050 111.5768
Total 19 6983.3814
Coefficients Standard Error t Stat P-value Lower 95%
Upper
95%
Intercept 1.5740 9.2723 0.1698 0.8672 -17.9888 21.1368
Age 1.3045 0.1956 6.6678 0.0000 0.8917 1.7173
Exper -0.1478 0.1944 -0.7604 0.4574 -0.5580 0.2624
Also the sum of squares due to the regression for the model that includes only Age is 5022.0654 while the
sum of squares due to the regression for the model that includes only Exper is 125.9848.
127. Referring to Scenario 14-8, the estimate of the unit change in the mean of Y per unit change in
X1, taking into account the effects of the other variable, is ________.
128. Referring to Scenario 14-8, the estimated change in the mean salary (in \$1,000) when an
employee is a year older holding experience constant is ________.
14-34 Introduction to Multiple Regression
129. Referring to Scenario 14-8, the estimated change in the mean salary (in \$1,000) for an
employee who has one additional year of experience holding age constant is ________.
130. Referring to Scenario 14-8, the predicted salary (in \$1,000) for a 35-year-old person with 10
years of experience is ________.
131. Referring to Scenario 14-8, the value of the coefficient of multiple determination is ________.
132. Referring to Scenario 14-8, the value of the adjusted coefficient of multiple determination is
________.
133. Referring to Scenario 14-8, the analyst wants to use an F-test to test 01 2
: 0H
ββ
==
. The
appropriate alternative hypothesis is ________.
134. Referring to Scenario 14-8, the critical value of an F test on the entire regression for a level of
significance of 0.01 is ________.
Introduction to Multiple Regression 14-35
135. Referring to Scenario 14-8, the value of the F-statistic for testing the significance of the entire
regression is ________.
136. Referring to Scenario 14-8, the p-value of the F test for the significance of the entire regression
is ________.
137. True or False: Referring to Scenario 14-8, the F test for the significance of the entire regression
performed at a level of significance of 0.01 leads to a rejection of the null hypothesis.
138. Referring to Scenario 14-8, the analyst wants to use a t test to test for the significance of the
coefficient of X2. For a level of significance of 0.01, the critical values of the test are ________.
139. Referring to Scenario 14-8, the analyst wants to use a t test to test for the significance of the
coefficient of X2. The value of the test statistic is ________.
140. Referring to Scenario 14-8, the analyst wants to use a t test to test for the significance of the
coefficient of X2. The p-value of the test is ________.
14-36 Introduction to Multiple Regression
141. True or False: Referring to Scenario 14-8, the analyst wants to use a t test to test for the
significance of the coefficient of X2. At a level of significance of 0.01, the department head would
decide that 20
β
.
142. Referring to Scenario 14-8, the analyst decided to construct a 95% confidence interval for 2
β
.
The confidence interval is from ________ to ________.
143. Referring to Scenario 14-8, the value of the partial F test statistic is ____ for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
144. Referring to Scenario 14-8, the partial F test for
H0 : Variable X1 does not significantly improve the model after variable X2 has been included
H1 : Variable X1 significantly improves the model after variable X2 has been included
has ____ and ____ degrees of freedom.
145. Referring to Scenario 14-8, the value of the partial F test statistic is ____ for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
Introduction to Multiple Regression 14-37
146. Referring to Scenario 14-8, the partial F test for
H0 : Variable X2 does not significantly improve the model after variable X1 has been included
H1 : Variable X2 significantly improves the model after variable X1 has been included
has ____ and ____ degrees of freedom.
147. Referring to Scenario 14-8, the coefficient of partial determination 2
12Y
r is ____.
148. Referring to Scenario 14-8, the coefficient of partial determination 2
21
Y
r is ____.
149. Referring to Scenario 14-8, ____% of the variation in salary can be explained by the variation
in age while holding experience constant.
150. Referring to Scenario 14-8, ____% of the variation in salary can be explained by the variation
in experience while holding age constant.
14-38 Introduction to Multiple Regression
SCENARIO 14-9
You decide to predict gasoline prices in different cities and towns in the United States for your term
project. Your dependent variable is price of gasoline per gallon and your explanatory variables are
per capita income and the number of firms that manufacture automobile parts in and around the city.
You collected data of 32 cities and obtained a regression sum of squares SSR= 122.8821. Your
computed value of standard error of the estimate is 1.9549.
151. Referring to Scenario 14-9, what is the value of the coefficient of multiple determination?
152. Referring to Scenario 14-9, the value of adjusted 2
r is
153. Referring to Scenario 14-9, if the variable that measures the number of firms that manufacture
automobile parts in and around the city is removed from the multiple regression model, which of
the following would be true?
r will definitely increase.
r cannot increase.
c) The coefficient of multiple determination will not increase.
d) The coefficient of multiple determination will definitely increase.
Introduction to Multiple Regression 14-39
SCENARIO 14-10
You worked as an intern at We Always Win Car Insurance Company last summer. You notice that
individual car insurance premiums depend very much on the age of the individual and the number of
traffic tickets received by the individual. You performed a regression analysis in EXCEL and
obtained the following partial information:
Regression Statistics
Multiple R 0.8546
R Square 0.7303
Standard Error 226.7502
Observations 15
ANOVA
df SS MS F Significance F
Regression 2 835284.6500 16.2457 0.0004
Residual 12 616987.8200
Total 2287557.1200
Coefficients Standard Error t Stat P-value Lower 99% Upper 99%
Intercept 821.2617 161.9391 5.0714 0.0003 326.6124 1315.9111
Age -1.4061 2.5988 -0.5411 0.5984 -9.3444 6.5321
Tickets 243.4401 43.2470 5.6291 0.0001 111.3406 375.5396
154. Referring to Scenario 14-10, the proportion of the total variability in insurance premiums that
can be explained by AGE and TICKETS is _________.
155. Referring to Scenario 14-10, the proportion of the total variability in insurance premiums that
can be explained by AGE and TICKETS after adjusting for the number of observations and the
number independent variables is _________.
156. Referring to Scenario 14-10, the standard error of the estimate is _________.
14-40 Introduction to Multiple Regression
157. Referring to Scenario 14-10, the estimated mean change in insurance premiums for every 2
158. Referring to Scenario 14-10, the 99% confidence interval for the change in mean insurance
premiums of a person who has become 1 year older (i.e., the slope coefficient for AGE) is
-1.4061 ± _______.
159. Referring to Scenario 14-10, the total degrees of freedom that are missing in the ANOVA table
should be ______.
160. Referring to Scenario 14-10, the regression sum of squares that is missing in the ANOVA table
should be ______.
161. Referring to Scenario 14-10, the residual mean squares (MSE) that are missing in the ANOVA
table should be _____.
Introduction to Multiple Regression 14-41
162. Referring to Scenario 14-10, to test the significance of the multiple regression model, what is
the form of the null hypothesis?
a) 01
:0H
β
=
b) 02
:0H
β
=
c) 01 2
:0H
ββ
==
d) 00 1 2
:0H
βββ
===
163. Referring to Scenario 14-10, to test the significance of the multiple regression model, the value
of the test statistic is ______.
164. Referring to Scenario 14-10, to test the significance of the multiple regression model, the p-
value of the test statistic in the sample is ______.
165. Referring to Scenario 14-10, to test the significance of the multiple regression model, what are
the degrees of freedom?
166. True or False: Referring to Scenario 14-10, to test the significance of the multiple regression
model, the null hypothesis should be rejected while allowing for 1% probability of committing a
type I error.
14-42 Introduction to Multiple Regression
167. True or False: Referring to Scenario 14-10, the multiple regression model is significant at a
10% level of significance.
168. If a categorical independent variable contains 2 categories, then _________ dummy variable(s)
will be needed to uniquely represent these categories.
a) 1
b) 2
c) 3
d) 4
169. If a categorical independent variable contains 4 categories, then _________ dummy variable(s)
will be needed to uniquely represent these categories.
a) 1
b) 2
c) 3
d) 4
170. A dummy variable is used as an independent variable in a regression model when
a) the variable involved is numerical.
b) the variable involved is categorical.
c) a curvilinear relationship is suspected.
d) when 2 independent variables interact.
Introduction to Multiple Regression 14-43
171. An interaction term in a multiple regression model may be used when
a) the coefficient of determination is small.
b) there is a curvilinear relationship between the dependent and independent variables.
c) neither one of 2 independent variables contribute significantly to the regression model.
d) the relationship between X1 and Y changes for differing values of X2.
172. To explain personal consumption (CONS) measured in dollars, data is collected for
INC: personal income in dollars
CRDTLIM: \$1 plus the credit limit in dollars available to the individual
APR: mean annualized percentage interest rate for borrowing for the individual
ADVT: per person advertising expenditure in dollars by manufacturers in the city where
the individual lives
SEX: gender of the individual; 1 if female, 0 if male
A regression analysis was performed with CONS as the dependent variable and CRDTLIM, APR,
ADVT, and GENDER as the independent variables. The estimated model was
l
2.28 - 0.29 CRDTLIM 5.77 APR 2.35 ADVT 0.39 SEXY=+++
What is the correct interpretation for the estimated coefficient for GENDER?
a) Holding the effect of the other independent variables constant, mean personal
consumption for females is estimated to be \$0.39 higher than males.
b) Holding the effect of the