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
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%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source olf Variation } & \text { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ Interpret the coefficient $b _ { 2 }$ .

Accepted Answer

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

Question 2

In multiple regression, 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.

Accepted Answer

Multicollinearity can lead to inflated standard errors and t-values, which may result in incorrectly concluding that some variables are not significant predictors when they actually are.

Question 3

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

Multicollinearity primarily affects the precision and interpretation of the individual regression coefficients (making their t-tests unreliable) but does not directly impact the F-test for the overall significance of the regression model. The F-test assesses whether at least one predictor variable has a non-zero coefficient, and it can still be significant even in the presence of multicollinearity.

Question 4

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%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ Is there enough evidence at the 5% significance level to infer that the cholesterol level and the age at death are negatively linearly related?

Accepted Answer

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

Question 5

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation H_A : There is negative first-order autocorrelation, given that the Durbin-Watson statistic d = 2.50, n = 40, k = 3 and $\alpha =$ 0.05.

Accepted Answer

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

Question 6

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%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ Is there enough evidence at the 5% significance level to infer that the number of points by which the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

Accepted Answer

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

Question 7

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). $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & t \
\hline \text { Constant } & 2.794 & a & 6.404 \
x _ { 1 } & b & 0.007 & - 0.025 \
x _ { 2 } & 0.383 & 0.072 & c \
\hline
\end{array}$$ se = d R2 = e. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \begin{array} { l } 
\text { Source of } \
\text { Variation }
\end{array} & \text { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & f & i & j & l \
\text { Error } & g & 11.884 & k & \
\hline \text { Total } & h & 26.887 & & \
\hline
\end{array}
\end{array}$$

Accepted Answer

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

Question 8

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 $y ^ { \prime } = \log y ( \text { provided } y \geq 0 )$ , $$y ^ { \prime } = y ^ { 2 }$$ , $y ^ { \prime } = \sqrt { y } ( \text { provided } y \geq 0 )$ , and $y ^ { \prime } = 1 / y$ .

Accepted Answer

Transforming the dependent variable is a common approach to address non-normality or heteroscedasticity in regression analysis. The transformations mentioned, such as logarithmic, square, square root, and reciprocal, are indeed among the most commonly used techniques to stabilize variance and make the data distribution more normal.

Question 9

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation
HA : 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

The computer output for the multiple regression model $$y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \varepsilon$$ 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). $$\begin{array} { | c | c c c | } 
\hline \text { Predictor } & \text { Coef } & \text { StDev } & \text { T } \
\hline \text { Constant } & \boldsymbol { a } & 0.120 & 3.18 \
x _ { 1 } & 0.068 & \boldsymbol { b } & 3.38 \
x _ { 2 } & 0.024 & 0.010 & \boldsymbol { c } \
\hline
\end{array}$$ se = d R2 = e. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \mathrm { SS } & \mathrm { MS } & \mathrm { F } \
\hline \text { Regression } & 2 & 7.382 & \boldsymbol { g } & \boldsymbol { i } \
\text { Error } & 22 & f & h & \
\hline \text { Total } & 24 & 7.530 & & \
\hline
\end{array}
\end{array}$$

Accepted Answer

The answer of The computer output for the multiple regression...

Question 11

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

Accepted Answer

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

Question 12

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%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ 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

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

Question 13

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%. $$\begin{array}{l}
\text { ANALYSIS OF VARIANCE }\
\begin{array} { | l | c c c c | } 
\hline \text { Source of Variation } & \mathrm { df } & \text { SS } & \text { MS } & \text { F } \
\hline \text { Regression } & 3 & 936 & 312 & 3.477 \
\text { Error } & 36 & 3230 & 89.722 & \
\hline \text { Total } & 39 & 4166 & & \
\hline
\end{array}
\end{array}$$ Is there enough evidence at the 10% significance level to infer that the model is useful in predicting length of life?

Accepted Answer

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

Question 14

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

Accepted Answer

The Durbin-Watson test is commonly used to check for autocorrelation in the residuals of a multiple regression model. A value of 2 for the Durbin-Watson test statistic indicates no autocorrelation, while values less than 2 or greater than 2 indicate positive or negative autocorrelation, respectively.

Question 15

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, $\hat { \beta } _ { 1 }$ = -5.30 $S _ {\hat { \beta } _ { 1 }}$ = 1.5

Accepted Answer

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

Question 16

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation
HA : 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 17

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

When the response surface perfectly fits every data point, there are no errors between the observed and predicted values, leading to SSE = 0, $S _ { \varepsilon }$ = 0, and $R ^ { 2 }$ = 1, indicating a perfect fit.

Question 18

In multiple regression with k independent variables, the t-tests of the individual coefficients allow us to determine whether $\beta _ { i } \neq 0$ (for i = 1, 2, …, k), which tells us whether a linear relationship exists between $x _ { i }$ and y.

Accepted Answer

The t-tests for individual coefficients in a multiple regression model test the null hypothesis that each coefficient is equal to zero ($\beta_i = 0$). Rejecting this null hypothesis ($\beta_i \neq 0$) for any coefficient suggests that there is a statistically significant linear relationship between the corresponding independent variable $x_i$ and the dependent variable $y$, controlling for the other variables in the model.

Question 19

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 20

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

This statement is true. The F-test in multiple regression is used to test the overall significance of the regression model. A large F statistic indicates that the regression model is significant and useful for predicting the outcome variable y, while a small F statistic indicates that the model is not significant and provides little information about y.

Quiz 16: Multiple Regression

In multiple regression, 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.

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.

Test the hypotheses: H0:H _ { 0 } :H0​: There is no first-order autocorrelation HA : There is negative first-order autocorrelation, given that the Durbin-Watson statistic d = 2.50, n = 40, k = 3 and α=\alpha =α= 0.05.

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

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

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

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, β^1\hat { \beta } _ { 1 }β^​1​ = -5.30 Sβ^1S _ {\hat { \beta } _ { 1 }}Sβ^​1​​ = 1.5

Test the hypotheses: H0:H _ { 0 } :H0​: There is no first-order autocorrelation HA : 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.

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 with k independent variables, the t-tests of the individual coefficients allow us to determine whether βi≠0\beta _ { i } \neq 0βi​=0 (for i = 1, 2, …, k), which tells us whether a linear relationship exists between xix _ { i }xi​ and y.

Test the hypotheses: $H _ { 0 } :$ There is no first-order autocorrelation H_A : There is negative first-order autocorrelation, given that the Durbin-Watson statistic d = 2.50, n = 40, k = 3 and $\alpha =$ 0.05.

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

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, $\hat { \beta } _ { 1 }$ = -5.30 $S _ {\hat { \beta } _ { 1 }}$ = 1.5

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

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 with k independent variables, the t-tests of the individual coefficients allow us to determine whether $\beta _ { i } \neq 0$ (for i = 1, 2, …, k), which tells us whether a linear relationship exists between $x _ { i }$ and y.