SCENARIO 17-2 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 4 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit

\left( X _ { 1 } \right)

, the amount of insulation in inches

\left( X _ { 2 } \right)

, the number of windows in the house

\left( X _ { 3 } \right)

, and the age of the furnace in years

\left( X _ { 4 } \right)

. Given below are the EXCEL outputs of two regression models.

Model 1

\begin{array}{lr}\hline{\text { Regression Statistics }} \\\hline \text { R Square } & 0.8080 \\\text { Adjusted R Square } & 0.7568 \\\text { Observations } & 20 \\\hline\end{array}

\text { ANOVA }

\begin{array}{lrrrrrr}\hline & d f & & {\text { SS }} & M S & F & \text { Significance } F \\\hline \text { Regression } && 4 & 169503.4241 & 42375.86 & 15.7874 & 0.0000 \\\text { Residual } && 15 & 40262.3259 & 2684.155 & & \\\text { Total } && 19 & 209765.75 & & & \\\hline\end{array}

\begin{array}{lrrrrrrr}\hline & \text { Coefficients } & \text { Standard Error } &{\text { t Stat }} & \text { P-value } & \text { Lower 90.0\% } & \text { Upper 90.0\% } \\\hline \text { Intereept } & 421.4277 & 77.8614 & 5.4125 & 0.0000 & 284.9327 & 557.9227 \\\mathrm{X}_{1} \text { (Temperature) } & -4.5098 & 0.8129 & -5.5476 & 0.0000 & -5.9349 & -3.0847 \\\mathrm{X}_{2} \text { (Insulation) } & -14.9029 & 5.0508 & -2.9505 & 0.0099 & -23.7573 & -6.0485 \\\mathrm{X}_{3} \text { (Windows) } & 0.2151 & 4.8675 & 0.0442 & 0.9653 & -8.3181 & 8.7484 \\\mathrm{X}_{4} \text { (Furnace Age) } & 6.3780 & 4.1026 & 1.5546 & 0.1408 & -0.8140 & 13.5702\end{array}

\text { Model } 2

\begin{array}{lr}\hline {\text { Regression Statistics }} \\\hline \text { R Square } & 0.7768 \\\text { Adjusted R Square } & 0.7506 \\\text { Observations } & 20 \\\hline\end{array}

\text { ANOVA }

$SCENARIO 17-2 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 4 variables to predict heating costs: the daily minimum outside temperature in degrees of Fahrenheit \left( X _ { 1 } \right) , the amount of insulation in inches \left( X _ { 2 } \right) , the number of windows in the house \left( X _ { 3 } \right) , and the age of the furnace in years \left( X _ { 4 } \right) . Given below are the EXCEL outputs of two regression models. Model 1 \begin{array}{lr} \hline{\text { Regression Statistics }} \\ \hline \text { R Square } & 0.8080 \\ \text { Adjusted R Square } & 0.7568 \\ \text { Observations } & 20 \\ \hline \end{array} \text { ANOVA } \begin{array}{lrrrrrr} \hline & d f & & {\text { SS }} & M S & F & \text { Significance } F \\ \hline \text { Regression } && 4 & 169503.4241 & 42375.86 & 15.7874 & 0.0000 \\ \text { Residual } && 15 & 40262.3259 & 2684.155 & & \\ \text { Total } && 19 & 209765.75 & & & \\ \hline \end{array} \begin{array}{lrrrrrrr} \hline & \text { Coefficients } & \text { Standard Error } &{\text { t Stat }} & \text { P-value } & \text { Lower 90.0\% } & \text { Upper 90.0\% } \\ \hline \text { Intereept } & 421.4277 & 77.8614 & 5.4125 & 0.0000 & 284.9327 & 557.9227 \\ \mathrm{X}_{1} \text { (Temperature) } & -4.5098 & 0.8129 & -5.5476 & 0.0000 & -5.9349 & -3.0847 \\ \mathrm{X}_{2} \text { (Insulation) } & -14.9029 & 5.0508 & -2.9505 & 0.0099 & -23.7573 & -6.0485 \\ \mathrm{X}_{3} \text { (Windows) } & 0.2151 & 4.8675 & 0.0442 & 0.9653 & -8.3181 & 8.7484 \\ \mathrm{X}_{4} \text { (Furnace Age) } & 6.3780 & 4.1026 & 1.5546 & 0.1408 & -0.8140 & 13.5702 \end{array} \text { Model } 2 \begin{array}{lr} \hline {\text { Regression Statistics }} \\ \hline \text { R Square } & 0.7768 \\ \text { Adjusted R Square } & 0.7506 \\ \text { Observations } & 20 \\ \hline \end{array} \text { ANOVA } \begin{array}{lrrllrr} \hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\ \hline \text { Intercept } & 489.3227 & 43.9826 & 11.1253 & 0.0000 & 396.5273 & 582.1180 \\ \mathrm{X}_{1} \text { (Temperature) } & -5.1103 & 0.6951 & -7.3515 & 0.0000 & -6.5769 & -3.6437 \\ \mathrm{X}_{2} \text { (Insulation) } & -14.7195 & 4.8864 & -3.0123 & 0.0078 & -25.0290 & -4.4099 \end{array} -Referring to Scenario 17-2 and allowing for a 1% probability of committing a type I error, what is the decision and conclusion for the test H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0 \text { vs. } H _ { 1 } : \text { At least one } \beta _ { j } \neq 0 , j = 1,2 , \cdots , 4 using Model 1? A) Do not reject H0 and conclude that the 4 independent variables have significant individual linear effects on heating costs. B) Reject H0 and conclude that the 4 independent variables taken as a group have significant linear effects on heating costs. C) Do not reject H0 and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs. D) Reject H0 and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs.$

\begin{array}{lrrllrr}\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\\hline \text { Intercept } & 489.3227 & 43.9826 & 11.1253 & 0.0000 & 396.5273 & 582.1180 \\\mathrm{X}_{1} \text { (Temperature) } & -5.1103 & 0.6951 & -7.3515 & 0.0000 & -6.5769 & -3.6437 \\\mathrm{X}_{2} \text { (Insulation) } & -14.7195 & 4.8864 & -3.0123 & 0.0078 & -25.0290 & -4.4099\end{array}

-Referring to Scenario 17-2 and allowing for a 1% probability of committing a type I error, what is the decision and conclusion for the test

H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0 \text { vs. } H _ { 1 } : \text { At least one } \beta _ { j } \neq 0 , j = 1,2 , \cdots , 4

using Model 1?
A) Do not reject H0 and conclude that the 4 independent variables have significant individual linear effects on heating costs.
B) Reject H0 and conclude that the 4 independent variables taken as a group have significant linear effects on heating costs.
C) Do not reject H0 and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs.
D) Reject H0 and conclude that the 4 independent variables taken as a group do not have significant linear effects on heating costs.

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