Question 1

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

Accepted Answer

A) True 
 B)False

Question 2

In a regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

Accepted Answer

A) 26.00. 
B) 7.69. 
C) 3.38. 
D) 0.039.

Question 3

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.

Accepted Answer

A) True 
 B)False

Question 4

An economist wanted to develop a multiple regression model to enable him to predict the annual family expenditure on clothes. After some consideration, he developed the multiple regression model: $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ .
Where:
y = annual family clothes expenditure (in $1000s) $x _ { 1 }$ = annual household income (in $1000s) $x _ { 2 }$ = number of family members $x _ { 3 }$ = number of children under 10 years of age
The computer output is shown below.
THE REGRESSION EQUATION IS
ŷ = $1.74 + 0.091 x _ { 1 } + 0.93 x _ { 2 } + 0.26 x _ { 3 }$ $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \
\hline \text { Constant } & 1.74 & 0.630 & 2.762 \
x _ { 1 } & 0.091 & 0.025 & 3.640 \
x _ { 2 } & 0.93 & 0.290 & 3.207 \
x _ { 3 } & 0.26 & 0.180 & 1.444 \
\hline
\end{array}$$ se = 2.06, R2 = 59.6%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 288 & 96 & 22.647 \
\text { Error } & 46 & 195 & 4.239 & \
\hline \text { Total } & 49 & 483 & & \
\hline
\end{array}
\end{array}$$ What is the coefficient of determination? What does this statistic tell you?

Accepted Answer

The answer of An economist wanted to develop a multiple...

Question 5

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.

Accepted Answer

A) True 
 B)False

Question 6

A multiple regression model has the form ? = 24 - 0.001x1 + 3x2.
As x1 increases by 1 unit, holding $x _ { 2 }$ constant, the value of y is estimated to decrease by 0.001units, on average.

Accepted Answer

A) True 
 B)False

Question 7

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SST = 350 and SSE = 50. The coefficient of determination is:

Accepted Answer

A) 0.8408. 
B) 0.8571. 
C) 0.8469. 
D) 0.8529.

Question 8

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 $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

Accepted Answer

A) True 
 B)False

Question 9

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

Accepted Answer

A) 109 degrees of freedom. 
B) 8 degrees of freedom. 
C) 99 degrees of freedom. 
D) 100 degrees of freedom.

Question 10

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 4 and 20, respectively.

Accepted Answer

A) True 
 B)False

Question 11

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

Accepted Answer

A) 4 and 35. 
B) 3 and 32. 
C) 3 and 34. 
D) 4 and 30.

Question 12

In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

Accepted Answer

A) True 
 B)False

Question 13

Which of the following is not true when we add an independent variable to a multiple regression model?

Accepted Answer

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.

Question 14

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

Accepted Answer

A) F1F1F1S1F1F1F100 = F1F1F1S1F1F1F101 = F1F1F1S1F1F1F102. 
B) F1F1F1S1F1F1F100 = F1F1F1S1F1F1F101 = F1F1F1S1F1F1F102 = 0. 
C) F1F1F1S1F1F1F101 = F1F1F1S1F1F1F102 = 0. 
D) F1F1F1S1F1F1F101 = F1F1F1S1F1F1F102.

Question 15

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 ( $x _ { 1 }$ ), the cholesterol level ( $x _ { 2 }$ ), and the number of points by which the individual's blood pressure exceeded the recommended value ( $x _ { 3 }$ ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below:
THE REGRESSION EQUATION IS
ŷ = $$55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$$ $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \
\hline \text { Constant } & 55.8 & 11.8 & 4.729 \
x _ { 1 } & 1.79 & 0.44 & 4.068 \
x _ { 2 } & - 0.021 & 0.011 & - 1.909 \
x _ { 3 } & - 0.016 & 0.014 & - 1.143 \
\hline
\end{array}$$ se = 9.47 R2 = 22.5%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ What is the coefficient of determination? What does this statistic tell you?

Accepted Answer

The answer of An actuary wanted to develop a model...

Question 16

Pop-up coffee vendors have been popular in the city of Adelaide in 2013. A vendor is interested in knowing how temperature (in degrees Celsius) and number of different pastries and biscuits offered to customers impacts daily hot coffee sales revenue (in $00's).
A random sample of 6 days was taken, with the daily hot coffee sales revenue and the corresponding temperature and number of different pastries and biscuits offered on that day, noted.
Excel output for a multiple linear regression is given below: $$\begin{array} { | c | c | r | } 
\hline \text { Coffee sales revenue } & \text { Temperature } & \text { Pastries/biscuits } \
\hline 6.5 & 25 & 7 \
\hline 10 & 17 & 13 \
\hline 5.5 & 30 & 5 \
\hline 4.5 & 35 & 6 \
\hline 3.5 & 40 & 3 \
\hline 28 & 9 & 15 \
\hline
\end{array}$$ \(\begin{array}{|l|r|l|l|l|l|l|}
\hline \text { SUMMARY OUTPUT } & & & & & & \
\hline \text { Regression Statistios } & & & & & & \
\hline \text { Multiple R } & 0.87 & & & & & \
\hline \text { R Square } & 0.75 & & & & & \
\hline \text { Adjusted R Square } & 0.59 & & & & & \
\hline \text { Standard Error } & 5.95 & & & & & \
\hline \text { Observations } & 6.00 & & & & & \\hline \
\hline \text { ANOVA } & & & & & \
\hline & {d f} & SS & M S & F & \text { Significance } F \
\hline \text { Regression } & 2.00 & 322.14 & 161.07 & 4.55 & 0.12 \
\hline \text { Residual } & 3.00 & 106.20 & 35.40 & & \
\hline \text { Total } & 5.00 & 428.33 & & & \\hline \
\hline & \text { Coeffients } & \text { Standard Error } & \text { tStat } & \text { P-value } & {\text { Lower } 95 \%} & {\text { Upper } 95 \%} \
\hline \text { Intercept } & 18.68 & 37.88 & 0.49 & 0.66 & -101.88 & 139.24 \
\hline \text { Temperature } & -0.50 & 0.83 & -0.60 & 0.59 & -3.15 & 2.15 \
\hline \text { Pastries/biscuits } & 0.49 & 2.02 & 0.24 & 0.82 & -5.94 & 6.92 \
\hline
\end{array}\) Interpret the intercept. Does this make sense?

Accepted Answer

The answer of Pop-up coffee vendors have been popular in...

Question 17

In a multiple regression model, the probability distribution of the error variable F1F1F1S1F1F1F10 is assumed to be:

Accepted Answer

A) normal. 
B) non-normal. 
C) positively skewed. 
D) negatively skewed.

Question 18

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\hat { y }$ = 72 + 3.2x1 + 1.5 x2 - x3. For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

Accepted Answer

A) True 
 B)False

Question 19

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.

Accepted Answer

A) True 
 B)False

Question 20

In multiple regression, the standard error of estimate is defined by $S _ { \varepsilon } = \sqrt { SSE / ( n - k ) }$ , where n is the sample size and k is the number of independent variables.

Accepted Answer

A) True 
 B)False

Exam 16: Multiple Regression

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

In a regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

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.

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.

A multiple regression model has the form ? = 24 - 0.001x1 + 3x2. As x1 increases by 1 unit, holding $x _ { 2 }$ constant, the value of y is estimated to decrease by 0.001units, on average.

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SST = 350 and SSE = 50. The coefficient of determination is:

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 $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

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 4 and 20, respectively.

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

Which of the following is not true when we add an independent variable to a multiple regression model?

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

In a multiple regression model, the probability distribution of the error variable  is assumed to be:

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\hat { y }$ = 72 + 3.2x1 + 1.5 x2 - x3. For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

In multiple regression, the standard error of estimate is defined by $S _ { \varepsilon } = \sqrt { SSE / ( n - k ) }$ , where n is the sample size and k is the number of independent variables.

Exam 16: Multiple Regression

For each x term in the multiple regression equation, the corresponding β\betaβ is referred to as a partial regression coefficient or slope of the independent variable.

In a regression model involving 30 observations, the following estimated regression model was obtained: y^\hat { y }y^​ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

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.

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.

A multiple regression model has the form ? = 24 - 0.001x1 + 3x2. As x1 increases by 1 unit, holding x2x _ { 2 }x2​ constant, the value of y is estimated to decrease by 0.001units, on average.

In a multiple regression analysis involving 40 observations and 5 independent variables, total variation in y = SST = 350 and SSE = 50. The coefficient of determination is:

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 SεS _ { \varepsilon }Sε​ = 0, and the coefficient of determination R2R ^ { 2 }R2 = 1.

In multiple regression analysis involving 9 independent variables and 110 observations, the critical value of t for testing individual coefficients in the model will have:

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 4 and 20, respectively.

In order to test the validity of a multiple regression model involving 4 independent variables and 35 observations, the numbers of degrees of freedom for the numerator and denominator, respectively, for the critical value of F are:

In multiple regression, the Durbin-Watson test is used to determine if there is autocorrelation in the regression model

Which of the following is not true when we add an independent variable to a multiple regression model?

To test the validity of a multiple regression model involving 2 independent variables, the null hypothesis is that:

In a multiple regression model, the probability distribution of the error variable  is assumed to be:

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is y^\hat { y }y^​ = 72 + 3.2x1 + 1.5 x2 - x3. For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

In multiple regression, the standard error of estimate is defined by Sε=SSE/(n−k)S _ { \varepsilon } = \sqrt { SSE / ( n - k ) }Sε​=SSE/(n−k)​ , where n is the sample size and k is the number of independent variables.

For each x term in the multiple regression equation, the corresponding $\beta$ is referred to as a partial regression coefficient or slope of the independent variable.

In a regression model involving 30 observations, the following estimated regression model was obtained: $\hat { y }$ = 60 + 2.8x1 + 1.2 x2 - x3. For this model, total variation in y = SST = 800 and SSE = 200. The value of the F-statistic for testing the validity of this model is:

A multiple regression model has the form ? = 24 - 0.001x1 + 3x2. As x1 increases by 1 unit, holding $x _ { 2 }$ constant, the value of y is estimated to decrease by 0.001units, on average.

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 $S _ { \varepsilon }$ = 0, and the coefficient of determination $R ^ { 2 }$ = 1.

In a multiple regression problem involving 24 observations and three independent variables, the estimated regression equation is $\hat { y }$ = 72 + 3.2x1 + 1.5 x2 - x3. For this model, SST = 800 and SSE = 245. The value of the F-statistic for testing the significance of this model is 15.102.

In multiple regression, the standard error of estimate is defined by $S _ { \varepsilon } = \sqrt { SSE / ( n - k ) }$ , where n is the sample size and k is the number of independent variables.