Question 1

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 $y=$ $$55.8 + 1.79 x _ { 1 } - 0.021 x _ { 2 } - 0.016 x _ { 3 }$$ $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \
\hline \text { Constant } & 55.8 & 11.8 & 4.729 \
x _ { 1 } & 1.79 & 0.44 & 4.068 \
x _ { 2 } & - 0.021 & 0.011 & - 1.909 \
x _ { 3 } & - 0.016 & 0.014 & - 1.143 \
\hline
\end{array}$$ S = 9.47 R-Sq = 22.5%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ 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?

Accepted Answer

$H _ { 0 } : \beta _ { 1 } = 0$ . $H _ { 1 } :$ $\beta _ { 1 } \neq$ 0.
Rejection region: | t | > $t _ { 0.005,36 } \approx$ 2.724.
Test statistic: t = 4.068.
Conclusion: Reject the null hypothesis. Yes.

Question 2

Given the following statistics of a multiple regression model, can we conclude at the 5% significance level that $x _ { 1 }$ and y are linearly related?
n = 42 k = 6 $b _ { 1 } =$ -5.30 $s _ { b _ { 1 } } =$ 1.5

Accepted Answer

$H _ { 0 } : \beta _ { 1 } = 0$ . $H _ { 1 } :$ $\beta _ { 1 } \neq$ 0. Rejection region: | t | > t_0.025,35 = 2.03 Test statistic: t = - 3.53 Conclusion: Reject the null hypothesis. There is significant evidence that x₁ and y are linearly related.

Question 3

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:

Accepted Answer

A)   0.80.
B)   3.266.
C)   3.651.
D)   1.633. D

Question 4

Consider the following statistics of a multiple regression model: n = 30 k = 4 SS_y = 1500 SSE = 260. a. Determine the standard error of estimate. b. Determine the multiple coefficient of determination. c. Determine the F-statistic.

Accepted Answer

The answer of Consider the following statistics of a multiple...

Question 5

For a set of 30 data points, Excel has found the estimated multiple regression equation to be $\hat{y}$ = -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 $\beta$3 = 0, the critical region will be that the absolute value of the t statistic for $\beta$3 is greater than or equal to:

Accepted Answer

A)   2.056.
B)   2.045.
C)   1.703.
D)   1.706.

Question 6

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is positive first-order autocorrelation,
given that: the Durbin-Watson statistic d = 0.686, n = 16, k = 1 and $\alpha =$ 0.05.

Accepted Answer

The answer of Test the hypotheses: \(H _ { 0...

Question 7

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.
Describe the following scatterplots.  Scatterplot of Daily hot coffee sales revenue vs Temperature  Scatterplot of Daily hot coffee sales revenue Pastries/biscuits  Residual scatterplot of Daily hot coffee sales revenue vs fitted values

Accepted Answer

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

Question 8

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%.

Accepted Answer

A) True 
 B)False

Question 9

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is first-order autocorrelation,
given that the Durbin-Watson statistic d = 1.89, n = 28, k = 3 and $\alpha =$ 0.05.

Accepted Answer

The answer of Test the hypotheses: \(H _ { 0...

Question 10

A statistician wanted to determine whether the demographic variables of age, education and income influence the number of hours of television watched per week. A random sample of 25 adults was selected to estimate the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } + \varepsilon$ .
Where:
y = number of hours of television watched last week. $x _ { 1 }$ = age. $x _ { 2 }$ = number of years of education. $x _ { 3 }$ = income (in $1000s).
The computer output is shown below.
THE REGRESSION EQUATION IS $y =$ $22.3 + 0.41 x _ { 1 } - 0.29 x _ { 2 } - 0.12 x _ { 3 }$ $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \mathrm { T } \
\hline \text { Constant } & 22.3 & 10.7 & 2.084 \
x _ { 1 } & 0.41 & 0.19 & 2.158 \
x _ { 2 } & - 0.29 & 0.13 & - 2.231 \
x _ { 3 } & - 0.12 & 0.03 & - 4.00 \
\hline
\end{array}$$ S = 4.51 R-Sq = 34.8%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 227 & 75.667 & 3.730 \
\text { Error } & 21 & 426 & 20.286 & \
\hline \text { Total } & 24 & 653 & & \
\hline
\end{array}
\end{array}$$ What is the coefficient of determination? What does this statistic tell you?

Accepted Answer

The answer of A statistician wanted to determine whether the...

Question 11

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

Accepted Answer

A) True 
 B)False

Question 12

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

Accepted Answer

A)    ∣b2∣\left| b _ { 2 } ight|∣b2​∣  could be this large if Ho:  β2≠0\beta _ { 2 } 
eq 0β2​=0  is really true
B)    ∣b2∣\left| b _ { 2 } ight|∣b2​∣  could be this large if Ho:  β2=0\beta _ { 2 } = 0β2​=0  is really false
C)    ∣b2∣\left| b _ { 2 } ight|∣b2​∣  could be this large if Ho:  β2≠0\beta _ { 2 } 
eq 0β2​=0  is really true.
D)    ∣b2∣\left| b _ { 2 } ight|∣b2​∣  could be this large if Ho:  β2=0\beta _ { 2 } = 0β2​=0  is really true.

Question 13

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:

Accepted Answer

A)   0.667.
B)   0.600.
C)   0.400.
D)   0.200.

Question 14

Given the multiple linear regression equation, ŷ = b0 + b1x1 + b2x2, the value of b2 is the estimated average increase in y for a one unit increase in x2, whilst holding x1 constant.

Accepted Answer

A) True 
 B)False

Question 15

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 16

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 17

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:

Accepted Answer

A)   0.82.
B)   0.80.
C)   0.77.
D)   0.20.

Question 18

A statistician estimated the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \varepsilon$ , 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). $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \text { T } \
\hline \text { Constant } & \mathbf { 2 . 7 9 4 } & \boldsymbol { a } & 6.404 \
x _ { 1 } & \boldsymbol { b } & 0.007 & - 0.025 \
x _ { 2 } & 0.383 & 0.072 & c \
\hline
\end{array}$$ S = d R-Sq = e. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \begin{array} { l } 
\text { Source of } \
\text { Variation }
\end{array} & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & f & i & j & l \
\text { Error } & \boldsymbol { g } & 11.884 & \boldsymbol { k } & \
\hline \text { Total } & h & 26.887 & & \
\hline
\end{array}
\end{array}$$

Accepted Answer

The answer of A statistician estimated the multiple regression model...

Question 19

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:

Accepted Answer

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.

Question 20

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

Accepted Answer

A)    H0:β1=β2=β3=β4=1H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 1H0​:β1​=β2​=β3​=β4​=1  .
B)    H0:β0=β1=β2=β3=β4H _ { 0 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 }H0​:β0​=β1​=β2​=β3​=β4​  .
C)    H0:β1=β2=β3=β4=0H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0H0​:β1​=β2​=β3​=β4​=0  .
D)    H0:β0=β1=β2=β3=β4≠0H _ { 0 } : \beta _ { 0 } = \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } 
eq 0H0​:β0​=β1​=β2​=β3​=β4​=0  .

Exam 19: Multiple Regression

Given the following statistics of a multiple regression model, can we conclude at the 5% significance level that $x _ { 1 }$ and y are linearly related? n = 42 k = 6 $b _ { 1 } =$ -5.30 $s _ { b _ { 1 } } =$ 1.5

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:

Consider the following statistics of a multiple regression model: n = 30 k = 4 SS_y = 1500 SSE = 260. a. Determine the standard error of estimate. b. Determine the multiple coefficient of determination. c. Determine the F-statistic.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is positive first-order autocorrelation, given that: the Durbin-Watson statistic d = 0.686, n = 16, k = 1 and $\alpha =$ 0.05.

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%.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is first-order autocorrelation, given that the Durbin-Watson statistic d = 1.89, n = 28, k = 3 and $\alpha =$ 0.05.

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

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

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:

Given the multiple linear regression equation, ŷ = b₀+ b₁x₁ + b₂x₂, the value of b₂ is the estimated average increase in y for a one unit increase in x₂, whilst holding x₁ constant.

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.

For a multiple regression model with n = 35 and k = 4, the following statistics are given: SS_y = 500 and SSE = 100. The coefficient of determination is:

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:

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

Exam 19: Multiple Regression

Given the following statistics of a multiple regression model, can we conclude at the 5% significance level that x1x _ { 1 }x1​ and y are linearly related? n = 42 k = 6 b1=b _ { 1 } =b1​= -5.30 sb1=s _ { b _ { 1 } } =sb1​​= 1.5

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:

Consider the following statistics of a multiple regression model: n = 30 k = 4 SSy = 1500 SSE = 260. a. Determine the standard error of estimate. b. Determine the multiple coefficient of determination. c. Determine the F-statistic.

Test the hypotheses: H0:H _ { 0 } :H0​: There is no first-order autocorrelation H1:H _ { 1 } :H1​: There is positive first-order autocorrelation, given that: the Durbin-Watson statistic d = 0.686, n = 16, k = 1 and α=\alpha =α= 0.05.

A multiple regression the coefficient of determination is 0.81. The percentage of the variation in yyy that is explained by the regression equation is 81%.

Test the hypotheses: H0:H _ { 0 } :H0​: There is no first-order autocorrelation H1:H _ { 1 } :H1​: There is first-order autocorrelation, given that the Durbin-Watson statistic d = 1.89, n = 28, k = 3 and α=\alpha =α= 0.05.

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

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of b2b _ { 2 }b2​ , this represents the probability that:

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:

Given the multiple linear regression equation, ŷ = b0 + b1x1 + b2x2, the value of b2 is the estimated average increase in y for a one unit increase in x2, whilst holding x1 constant.

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.

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:

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:

In testing the validity of a multiple regression model in which there are four independent variables, the null hypothesis is:

Given the following statistics of a multiple regression model, can we conclude at the 5% significance level that $x _ { 1 }$ and y are linearly related? n = 42 k = 6 $b _ { 1 } =$ -5.30 $s _ { b _ { 1 } } =$ 1.5

Consider the following statistics of a multiple regression model: n = 30 k = 4 SS_y = 1500 SSE = 260. a. Determine the standard error of estimate. b. Determine the multiple coefficient of determination. c. Determine the F-statistic.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is positive first-order autocorrelation, given that: the Durbin-Watson statistic d = 0.686, n = 16, k = 1 and $\alpha =$ 0.05.

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%.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation $H _ { 1 } :$ There is first-order autocorrelation, given that the Durbin-Watson statistic d = 1.89, n = 28, k = 3 and $\alpha =$ 0.05.

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

Excel and Minitab both provide the p-value for testing each coefficient in the multiple regression model. In the case of $b _ { 2 }$ , this represents the probability that:

Given the multiple linear regression equation, ŷ = b₀+ b₁x₁ + b₂x₂, the value of b₂ is the estimated average increase in y for a one unit increase in x₂, whilst holding x₁ constant.

For a multiple regression model with n = 35 and k = 4, the following statistics are given: SS_y = 500 and SSE = 100. The coefficient of determination is: