Chapter 22—Multiple regression
MULTIPLE CHOICE
1. In testing the validity of a multiple regression model, a small value of the F-test statistic indicates that:
A.
most of the variation in the independent variables is explained by the variation in y.
B.
most of the variation in y is explained by the regression equation.
C.
most of the variation in y is unexplained by the regression equation.
D.
the model provides a good fit.
2. In a multiple regression analysis, if the model provides a poor fit, this indicates that:
A.
the sum of squares for error will be large.
B.
the standard error of estimate will be large.
C.
the multiple coefficient of determination will be close to zero.
D.
All of the above answers are correct.
3. In a multiple regression analysis, when there is no linear relationship between each of the independent
variables and the dependent variable, then:
A.
multiple t-tests of the individual coefficients will likely show some are significant.
B.
we will conclude erroneously that the model has some validity.
C.
the chance of erroneously concluding that the model is useful is substantially less with the
F-test than with multiple t-tests.
D.
All of the above statements are correct.
4. Which of the following statements is not true?
A.
Multicollinearity exists in virtually all multiple regression models.
B.
Multicollinearity is also called collinearity and intercorrelation.
C.
Multicollinearity is a condition that exists when the independent variables are highly
correlated with the dependent variable.
D.
Multicollinearity does not affect the F-test of the analysis of variance.
5. In a multiple regression analysis involving 30 data points, the standard error of estimate squared is
calculated as s2 = 1.5 and the sum of squares for error as SSE = 36. The number of the independent
variables must be:
A.
6.
B.
5.
C.
4.
D.
3.
6. In a multiple regression model, the standard deviation of the error variable
is assumed to be:
A.
constant for all values of the independent variables.
B.
constant for all values of the dependent variable.
C.
1.0.
D.
Not enough information is given to answer this question.
7. When the independent variables are correlated with one another in a multiple regression analysis, this
condition is called:
A.
multicollinearity.
B.
homoscedasticity.
C.
heteroscedasticity.
D.
linearity.
8. In a multiple regression model, the mean of the probability distribution of the error variable
is
assumed to be:
A.
1.0.
B.
0.0.
C.
any value greater than 1.
D.
k, where k is the number of independent variables included in the model.
9. The adjusted multiple coefficient of determination is adjusted for the:
A.
number of regression parameters including the y-intercept.
B.
number of dependent variables and the sample size.
C.
number of independent variables and the sample size.
D.
coefficient of correlation and the significance level.
10. In multiple regression analysis, the ratio MSR/MSE yields the:
A.
t-test statistic for testing each individual regression coefficient.
B.
F-test statistic for testing the validity of the regression equation.
C.
multiple coefficient of determination.
D.
adjusted multiple coefficient of determination.
11. In a multiple regression analysis involving 6 independent variables and a sample of 19 data points the
total variation in y is SSy = 900 and the amount of variation in y that is explained by the variations in
the independent variables is SSR = 600. The value of the F-test statistic for this model is:
A.
4.0.
B.
4.3.
C.
4.8.
D.
6.3.
12. In multiple regression models, the values of the error variable
are assumed to be:
A.
autocorrelated.
B.
dependent on each other.
C.
independent of each other.
D.
always positive.
13. In order to test the validity of a multiple regression model involving 5 independent variables and 30
observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for
the critical value of F are:
A.
5 and 30.
B.
6 and 29.
C.
5 and 24.
D.
6 and 25.
14. A multiple regression model involves 4 independent variables and the sample size is 45. If we want to
test the validity of the model at the 5% significance level, the critical value is:
A.
2.69.
B.
2.61.
C.
2.84.
D.
5.72.
15. An estimated multiple regression model has the form ŷ = 8 + 3x1 − 5x2 − 4x3. As x2 increases by 1 unit,
with x1 and x3 held constant, the value of y, on average, is expected to:
A.
decrease by 1 unit.
B.
increase by 5 units.
C.
decrease by 5 units.
D.
increase by 8 units.
16. A multiple regression model involves 10 independent variables and 30 observations. If we want to test
at the 5% significance level the parameter
4
, the critical value will be:
A.
2.093.
B.
1.697.
C.
2.228.
D.
1.729.
17. In a multiple regression analysis involving k independent variables and n data points, the number of
degrees of freedom associated with the sum of squares for regression is:
A.
k.
B.
n – k.
C.
k – 1.
D.
n – k – 1.
18. The problem of multicollinearity arises when the:
A.
dependent variables are highly correlated with one another.
B.
independent variables are highly correlated with one another.
C.
independent variables are highly correlated with the dependent variable.
D.
Both A and B are correct statements.
19. To test the validity of a multiple regression model, we test the null hypothesis that the regression
coefficients are all zero. We apply the:
A.
t-test.
B.
z-test.
C.
F-test.
D.
Either A or B is correct.
20. To test the validity of a multiple regression model involving 2 independent variables, the null
hypothesis is that:
A.
0 =
1 =
2.
B.
0 =
1 =
2 = 0.
C.
1 =
2 = 0.
D.
1 =
2.
21. A multiple regression analysis involving 3 independent variables and 25 data points results in a value
of 0.769 for the unadjusted multiple coefficient of determination. The adjusted multiple coefficient of
determination is:
A.
0.385.
B.
0.877.
C.
0.591.
D.
0.736.
22. If multicollinearity exists among the independent variables included in a multiple regression model,
then:
A.
regression coefficients will be difficult to interpret.
B.
the standard errors of the regression coefficients for the correlated independent variables
will increase.
C.
multiple coefficient of determination will assume a value close to zero.
D.
Both A and B are correct statements.
23. Which of the following is not true when we add an independent variable to a multiple regression
model?
A.
The adjusted coefficient of determination can assume a negative value.
B.
The unadjusted coefficient of determination always increases.
C.
The unadjusted coefficient of determination may increase or decrease.
D.
The adjusted coefficient of determination may increase.
24. A multiple regression model has the form
22110
öxbxbby ++=
. The coefficient
1
b
is interpreted as
the:
A.
change in y per unit change in
1
x
.
B.
change in y per unit change in
1
x
, holding
2
x
constant.
C.
change in y per unit change in
1
x
, when
and
2
x
values are correlated.
D.
change in the average value of y per unit change in
, holding
2
x
constant.
25. The coefficient of multiple determination ranges from:
A.
–1.0 to 1.0.
B.
0.0 to 1.0.
C.
1.0 to k, where k is the number of independent variables in the model.
D.
1.0 to n, where n is the number of observations in the dependent variable.
26. A multiple regression model has:
A.
only one independent variable.
B.
only two independent variables.
C.
more than one independent variable.
D.
more than one dependent variable.
27. For a multiple regression model with n = 35 and k = 4, the following statistics are given:
SSy = 500 and SSE = 100. The coefficient of determination is:
A.
0.82.
B.
0.80.
C.
0.77.
D.
0.20.
28. For a multiple regression model, the following statistics are given:
Total variation in y = SSY = 250, SSE = 50, k = 4, n = 20.
The coefficient of determination adjusted for degrees of freedom is:
A.
0.800.
B.
0.747.
C.
0.840.
D.
0.775.
29. An estimated multiple regression model has the form ŷ = 5.25 + 2x1 + 6x2. As x1 increases by 1 unit
while holding x2 constant, the value of y will increase by:
A.
2 units on average.
B.
2 units.
C.
7.25 units.
D.
8 units on average.
30. The graphical depiction of the equation of a multiple regression model with k independent variables
(k > 1) is referred to as:
A.
a straight line.
B.
the response variable.
C.
the response surface.
D.
a plane only when k = 3.
31. If all the points for a multiple regression model with two independent variables were on the regression
plane, then the multiple coefficient of determination would equal:
A.
0.
B.
1.
C.
2, since there are two independent variables.
D.
any number between 0 and 2.
32. If none of the data points for a multiple regression model with two independent variables were on the
regression plane, then the multiple coefficient of determination would be:
A.
–1.0.
B.
1.0.
C.
any number between –1 and 1, inclusive.
D.
any number greater than or equal to zero but smaller than 1.
33. For the estimated multiple regression model ŷ = 2 − 3x1 + 4x2 + 5x3, a unit increase in x3, holding x1
and x2 constant, results in:
A.
an increase of 5 units in the value of y.
B.
a decrease of 5 units in the value of y.
C.
a decrease of 5 units on average in the value of y.
D.
an increase of 5 units on average in the value of y.
34. The multiple coefficient of determination is defined as:
A.
SSE/SSY.
B.
MSE/MSR.
C.
1 – (SSE/SSY).
D.
1 – (MSE/MSR).
35. In a multiple regression model, the following statistics are given:
SSE = 100,
20.995R=
, k = 5, n = 15.
The multiple coefficient of determination adjusted for degrees of freedom is:
A.
0.955.
B.
0.992.
C.
0.930.
D.
None of the above answers is correct.
36. In a multiple regression model, the error variable is assumed to have a mean of:
A.
–1.0.
B.
0.0.
C.
1.0.
D.
any value smaller than –1.0.
37. In a multiple regression model, the probability distribution of the error variable
is assumed to be:
A.
normal.
B.
non-normal.
C.
positively skewed.
D.
negatively skewed.
38. Which of the following measures can be used to assess a multiple regression model’s fit?
A.
The sum of squares for error.
B.
The sum of squares for regression.
C.
The standard error of estimate.
D.
A single t-test.
39. For a multiple regression model:
A.
SSY = SSR – SSE.
B.
SSE = SSR – SSY.
C.
SSR = SSE – SSY.
D.
SSY = SSE + SSR.
40. In a multiple regression analysis involving 40 observations and 5 independent variables, total variation
in y = SSY = 350 and SSE = 50. The multiple coefficient of determination is:
A.
0.8408.
B.
0.8571.
C.
0.8469.
D.
0.8529.
41. In a multiple regression analysis involving 20 observations and 5 independent variables, total variation
in y = SSY = 250 and SSE = 35. The multiple coefficient of determination, adjusted for degrees of
freedom, is:
A.
0.810.
B.
0.860.
C.
0.835.
D.
0.831.
42. In testing the validity of a multiple regression model involving 8 independent variables and 79
observations, the numbers of degrees of freedom for the numerator and denominator (respectively) for
the critical value of F will be:
A.
8 and 71.
B.
7 and 79.
C.
8 and 70.
D.
7 and 70.
43. In multiple regression analysis involving 10 independent variables and 100 observations, the critical
value of t for testing individual coefficients in the model will have:
A.
100 degrees of freedom.
B.
10 degrees of freedom.
C.
89 degrees of freedom.
D.
9 degrees of freedom.
44. In a regression model involving 50 observations, the following estimated regression model was
obtained: ŷ = 10.5 + 3.2x1 + 5.8x2 + 6.5x3. For this model, SSR = 450 and SSE = 175. The value of
MSE is:
A.
9.783.
B.
58.333.
C.
150.000.
D.
3.804.
45. In a regression model involving 60 observations, the following estimated regression model was
obtained:
321 378.0679.070.04.51
öxxxy −++=
For this model, total variation in y = SSY = 119,724 and SSR = 29,029.72. The value of MSE is:
A.
1619.541.
B.
9676.572.
C.
1995.400.
D.
5020.235.
46. In a regression model involving 30 observations, the following estimated regression model was
obtained:
321 2.18.260
öxxxy −++=
.
For this model, total variation in y = SSY = 800 and SSE = 200. The value of the F–statistic for testing
the validity of this model is:
A.
26.00.
B.
7.69.
C.
3.38.
D.
0.039.
47. Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression
model. In the case of
2
b
, this represents the probability that:
A.
0
2=b
.
B.
0
2=
.
C.
|| 2
b
could be this large if
0
2=
.
D.
|| 2
b
could be this large if
0
2
.
48. In testing the validity of a multiple regression model in which there are four independent variables, the
null hypothesis is:
A.
1: 43210 ====
H
.
B.
432100 :
====H
.
C.
0: 43210 ====
H
.
D.
0: 432100 ====
H
.
49. For a set of 30 data points, Excel has found the estimated multiple regression equation to be
ŷ = −8.61 + 22x1 + 7x2 + 28x3, and has listed the t statistic for testing the significance of each
regression coefficient. Using the 5% significance level for testing whether
3 = 0, the critical region
will be that the absolute value of the t statistic for
3 is greater than or equal to:
A.
2.056.
B.
2.045.
C.
1.703.
D.
1.706.
50. For the multiple regression model
321 10152575
öxxxy +−+=
, if
2
x
were to increase by 5, holding
1
x
and
3
x
constant, the value of y would:
A.
increase by 5.
B.
increase by 75.
C.
decrease on average by 5.
D.
decrease on average by 75.
51. In a multiple regression analysis, there are 20 data points and 4 independent variables, and the sum of
the squared differences between observed and predicted values of y is 180. The multiple standard error
of estimate will be:
A.
6.708.
B.
3.464.
C.
9.000.
D.
3.000.
52. A multiple regression equation includes 5 independent variables, and the coefficient of determination
is 0.81. The percentage of the variation in y that is explained by the regression equation is:
A.
81%.
B.
90%.
C.
86%.
D.
about 16%.
53. A multiple regression analysis that includes 4 independent variables results in a sum of squares for
regression of 1200 and a sum of squares for error of 800. The multiple coefficient of determination
will be:
A.
0.667.
B.
0.600.
C.
0.400.
D.
0.200.
54. A multiple regression analysis that includes 20 data points and 4 independent variables results in total
variation in y = SSY = 200 and SSR = 160. The multiple standard error of estimate will be:
A.
0.80.
B.
3.266.
C.
3.651.
D.
1.633.
55. In a multiple regression analysis involving 25 data points and 5 independent variables, the sum of
squares terms are calculated as: total variation in y = SSY = 500, SSR = 300, and SSE = 200. In testing
the validity of the regression model, the F-value of the test statistic will be:
A.
5.70.
B.
2.50.
C.
1.50.
D.
0.176.
TRUE/FALSE
1.
2. Multicollinearity is a situation in which the independent variables are highly correlated with the
dependent variable.
3. In multiple regression, the descriptor ‘multiple’ refers to more than one independent variable.
4. For each x term in the multiple regression equation, the corresponding
is referred to as a partial
regression coefficient.
5. In reference to the equation
1 2
ö0.80 0.12 0.08y x x= − + +
, the value –0.80 is the
y
intercept.
6.
7. In a multiple regression problem involving 24 observations and three independent variables, the
estimated regression equation is
1 2 3
72 3.2 1.5
ö
y x x x+= + −
. For this model, SST = 800 and SSE =
245. The value of the F-statistic for testing the significance of this model is 15.102.
8. In order to test the significance of a multiple regression model involving 4 independent variables and
25 observations, the number of degrees of freedom for the numerator and denominator, respectively,
for the critical value of F are 3 and 21, respectively.
9. A multiple regression the coefficient of determination is 0.81. The percentage of the variation in
y
that is explained by the regression equation is 81%.
10. In a multiple regression analysis involving 4 independent variables and 30 data points, the number of
degrees of freedom associated with the sum of squares for error, SSE, is 25.
11. The adjusted multiple coefficient of determination is adjusted for the number of independent variables
and the sample size.
12. A multiple regression analysis that includes 25 data points and 4 independent variables produces SST
= 400 and SSR = 300. The multiple standard error of estimate will be 5.
13. A multiple regression model has the form
22110
öxbxbby ++=
. The coefficient
1
b
is interpreted as
the change in
y
per unit change in
.
14. In order to test the significance of a multiple regression model involving 4 independent variables and
30 observations, the number of degrees of freedom for the numerator and denominator for the critical
value of F are 4 and 26, respectively.
15. In a multiple regression analysis involving 50 observations and 5 independent variables, SST = 475
and SSE = 71.25. The multiple coefficient of determination is 0.85.
16. 16. A multiple regression model has the form
1 2
6.75 2.25 3.5
ö
y x x+= +
. As
1
x
increases
by 1 unit, holding
2
x
constant, the value of y will increase by 9 units.
17. For the multiple regression model
1 2 3
40 15 10 5
ö
y x x x+ − +=
, if
2
x
were to increase by 5 units,
holding
1
x
and
3
x
constant, the value of
y
would decrease by 50 units, on average.
18. A multiple regression model involves 40 observations and 4 independent variables produces
SST = 100 000 and SSR = 82,500. The value of MSE is 500.
19. In testing the significance of a multiple regression model in which there are three independent
variables, the null hypothesis is
0 1 2 3
:H
 
= =
.
20. In reference to the equation
1 2
ö1.86 0.51 0.60y x x= − +
, the value 0.60 is the change in
y
per unit
change in
, regardless of the value of
1
x
.
21. In regression analysis, we judge the magnitude of the standard error of estimate relative to the values
of the dependent variable, and particularly to the mean of y.
22. Excel and Minitab print a second
2
R
statistic, called the coefficient of determination adjusted for
degrees of freedom, which has been adjusted to take into account the sample size and the number of
independent variables.
23. In multiple regression, the standard error of estimate is defined by
/( )SSE n ks
−=
, where n is the
sample size and k is the number of independent variables.
24. In regression analysis, the total variation in the dependent variable y, measured by
2
( )
i
y y
−
, can
be decomposed into two parts: the explained variation, measured by SSR, and the unexplained
variation, measured by SSE.
25. In a multiple regression, a large value of the test statistic F indicates that most of the variation in y is
explained by the regression equation, and that the model is useful; while a small value of F indicates
that most of the variation in y is unexplained by the regression equation, and that the model is useless.
26. In multiple regression, when the response surface (the graphical depiction of the regression equation)
hits every single point, the sum of squares for error SSE = 0, the standard error of estimate
s
= 0, and
the coefficient of determination
2
R
= 1.
27. In multiple regression with k independent variables, the t-tests of the individual coefficients allow us to
determine whether
0
i
(for i = 1, 2, …, k), which tells us whether a linear relationship exists
between
i
x
and y.
28. In multiple regression, and because of a commonly occurring problem called multicollinearity, the
t-tests of the individual coefficients may indicate that some independent variables are not linearly
related to the dependent variable, when in fact they are.
29. In multiple regression, the problem of multicollinearity affects the t-tests of the individual coefficients
as well as the F-test in the analysis of variance for regression, since the F-test combines these t-tests
into a single test.
30. The most commonly used method to remedy non-normality or heteroscedasticity in regression analysis
is to transform the dependent variable. The most commonly used transformations are
log provided ( 0)y y y
= 
,
2
y y
=
,
provided( 0)y y y
= 
, and
1y y
=
.
SHORT ANSWER
1. Consider the following statistics of a multiple regression model:
n = 70, k = 5, SSy = 1000 and SSE = 250.
a. Determine the standard error of estimate.
b. Determine the multiple coefficient of determination.
c. Determine the F-statistic.
2. Given the following statistics of a multiple regression model, can we conclude at the 5% significance
level that
and y are linearly related?
n = 41, k = 5,
=
1
b
–6.31,
=
1
b
s
2.98.
3. Test the hypotheses:
:
0
H
There is no first-order autocorrelation
:
1
H
There is negative first-order autocorrelation,
given that the Durbin–Watson statistic d = 2.50, n = 40, k = 3 and
=
0.05.
4. A statistician estimated the multiple regression model
+++= 22110 xxy
, with 45
observations. The computer output is shown below. However, because of a printer malfunction, some
of the results are not shown. These are indicated by the boldface letters a to l. Fill in the missing
results (up to three decimal places).
Predictor
Coef
StDev
T
Constant
2.794
a
6.404
1
x
b
0.007
-0.025
2
x
0.383
0.072
c
S = d R-Sq = e.
:
1
H
ANALYSIS OF VARIANCE
Source of
Variation
df
SS
MS
F
Regression
f
i
j
l
Error
g
11.884
k
Total
h
26.887
5. Test the hypotheses:
:
0
H
There is no first-order autocorrelation
:
1
H
There is positive first-order autocorrelation,
given that: the Durbin–Watson statistic d = 0.686, n = 16, k = 1 and
=
0.05.
6. The computer output for the multiple regression model
+++= 22110 xxy
is shown below.
However, because of a printer malfunction some of the results are not shown. These are indicated by
the boldface letters a to i. Fill in the missing results (up to three decimal places).
Predictor
Coef
StDev
T
Constant
a
0.120
3.18
0.068
b
3.38
2
x
0.024
0.010
c
S = d R-Sq = e.
ANALYSIS OF VARIANCE
Source of Variation
df
SS
MS
F
Regression
2
7.382
g
i
Error
22
f
h
Total
24
7.530
7. Test the hypotheses:
:
0
H
There is no first-order autocorrelation
:
1
H
There is first-order autocorrelation,
given that the Durbin–Watson statistic d = 1.89, n = 28, k = 3 and
=
0.05.
8. An actuary wanted to develop a model to predict how long individuals will live. After consulting a
number of physicians, she collected the age at death (y), the average number of hours of exercise per
week (
), the cholesterol level (
2
x
), and the number of points by which the individual’s blood
pressure exceeded the recommended value (
3
x
). A random sample of 40 individuals was selected. The
computer output of the multiple regression model is shown below:
THE REGRESSION EQUATION IS
=y
321 016.0021.079.18.55 xxx −−+
Predictor
Coef
StDev
T
Constant
55.8
11.8
4.729
1.79
0.44
4.068
2
x
–0.021
0.011
–1.909
3
x
–0.016
0.014
–1.143
S = 9.47 R-Sq = 22.5%.
ANALYSIS OF VARIANCE
Source of Variation
df
SS
MS
F
Regression
3
936
312
3.477
Error
36
3230
89.722
Total
39
4166
Is there enough evidence at the 10% significance level to infer that the model is useful in predicting
length of life?
9. An actuary wanted to develop a model to predict how long individuals will live. After consulting a
number of physicians, she collected the age at death (y), the average number of hours of exercise per
week (
), the cholesterol level (
2
x
), and the number of points by which the individual’s blood
pressure exceeded the recommended value (
3
x
). A random sample of 40 individuals was selected. The
computer output of the multiple regression model is shown below:
THE REGRESSION EQUATION IS
=y
321 016.0021.079.18.55 xxx −−+
Predictor
Coef
StDev
T
Constant
55.8
11.8
4.729
1.79
0.44
4.068
2
x
–0.021
0.011
–1.909
3
x
–0.016
0.014
–1.143
S = 9.47 R-Sq = 22.5%.
ANALYSIS OF VARIANCE
Source of Variation
df
SS
MS
F
Regression
3
936
312
3.477
Error
36
3230
89.722
Total
39
4166
Is there enough evidence at the 1% significance level to infer that the average number of hours of
exercise per week and the age at death are linearly related?