The accompanying table shows results from regressions performed on data from a random sample of 21 cars. The response (y) variable is $\mathrm{CITY}$ (fuel consumption in milgal). The predictor $(x)$ variables are WT (weight in pounds), DISP (engine displacement in liters), and HWY (highway fuel consumption in milgal), Which regression equation is best for predicting city fuel consumption? Why?
Regression Table
\begin{tabular}{c|c|c|c|c}
\hline Predictor (x) Variables & P-Value & $\mathbf{R}^{2}$ & Adjusted $\mathrm{R}^{2}$ & Regression Equation \\
\hline WT/DISP/HWY & 0.000 & 0.944 & 0.934 & CITY $=6.91-0.00131 W T-0.258$ DISP $+0.651 \mathrm{HWY}$ \\
\hline WTIDISP & 0.000 & 0.748 & 0.720 & $\mathrm{CITY}=37.6-0.00159 W \mathrm{WT}-1.33 \mathrm{DISP}$ \\
\hline WT/HWY & 0.000 & 0.943 & 0.937 & $\mathrm{CITY}=6.71-0.00159 \mathrm{WT}+0.674 \mathrm{HWY}$ \\
\hline DISP/HWY & 0.000 & 0.935 & 0.928 & $\mathrm{CITY}=1.87-0.623 \mathrm{DISP}+0.709 \mathrm{HWY}$ \\
\hline WT & 0.000 & 0.712 & 0.697 & $\mathrm{CITY}=42.4-0.00605 \mathrm{WT}$ \\
\hline DISP & 0.000 & 0.658 & 0.640 & $\mathrm{CITY}=28.7-2.96 \mathrm{DISP}$ \\
\hline HWY & 0.000 & 0.925 & 0.921 & $\mathrm{CITY}=-3.14+0.822 \mathrm{HWY}$ \\
\hline
\end{tabular}
Choose the corred answer below.
A. The equation CTrY $=-3.14+0.822$ HWY is best because it has a low $P$-value and its $R^{2}$ and adfusted $R^{2}$ values are comparable to the $R^{2}$ and adjusted $R^{2}$ values of equations with more predictor variables.
B. The equation CTY $=6.91-0.0013 \mathrm{rWT}-0.2580 \mathrm{ISP}+0.65$ 1HWY is best because it has a bow $P$-value and the highest value of $R^{2}$
C. The equation CITY $=6.91-0.00131 W T-0.25801 S P+0.651 H W Y$ is best because it uses al of the avalabio prodictor variables.
D. The equation CITY $=6.71-0.00159$ WT +0.674 HiWY is best because it has a low P-value and the highest adfusted value of $R^{2}$.
Answer
Final Answer: \(\boxed{\text{D. The equation CITY } =6.71-0.00159 \text{ WT } +0.674 \text{ HWY is best because it has a low P-value and the highest adjusted value of } R^{2}}\).
Step 1 ：The question is asking us to determine which regression equation is best for predicting city fuel consumption. To determine this, we need to consider the P-value, R-squared value, and adjusted R-squared value. The P-value tells us the significance of the predictor variables, the R-squared value tells us the proportion of the variance in the dependent variable that is predictable from the independent variable(s), and the adjusted R-squared value takes into account the number of predictors in the model. The best model would have a low P-value and high R-squared and adjusted R-squared values.
Step 2 ：Looking at the table, we can see that all models have a P-value of 0.000, which means all predictor variables are significant. The model with the highest R-squared value is the one with WT/DISP/HWY as predictor variables, and the model with the highest adjusted R-squared value is the one with WT/HWY as predictor variables.
Step 3 ：Therefore, we need to choose between these two models. Since the adjusted R-squared value takes into account the number of predictors in the model, it is a better measure for model comparison. Therefore, the model with WT/HWY as predictor variables is the best for predicting city fuel consumption.
Step 4 ：Final Answer: \(\boxed{\text{D. The equation CITY } =6.71-0.00159 \text{ WT } +0.674 \text{ HWY is best because it has a low P-value and the highest adjusted value of } R^{2}}\).
