Chapter 10.3, Problem 58E

a.

To determine

Identify the variable that can be eliminated first to possibly improve the given model.

Answer to Problem 58E

The variable that can be eliminated first to possibly improve the given model is $X_2$.

Explanation of Solution

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In order to improve a regression model, the most insignificant independent variable in the model should be eliminated first. The high p-value indicates that the variable is insignificant.

From the multiple regression model, it is clear that the variable $X_2$ has the highest p-value. Hence, one should try to eliminate the variable $X_2$ to possibly improve the given model.

b.

To determine

Describe about the $R^2$ of the model.

Explain whether the change in $R^2$ represents that removing the variable chosen in Part (a) is a good or bad idea.

Answer to Problem 58E

The coefficient of determination $R^2$ is 15.0%.

Explanation of Solution

The larger value of $R^2$ indicates that the model is better. From the multiple regression output, it is clear that the coefficient of determination $R^2$ is 15.0%.

Generally, even eliminating an insignificant variable tends to decrease the value of $R^2$. However, if $R^2$ is not decreased by any substantial amount, it indicates that removing $X_2$ is a good idea. If there is a substantial decrease in $R^2$, it is an indication that removing $X_2$ is a bad idea.

c.

To determine

Obtain the p-value of ANOVA for the original 3-predictor model.

Explain whether the type of change in the p-value for ANOVA represents that removing the variable chosen in Part (a) is a good or bad idea.

Answer to Problem 58E

The p-value of ANOVA for the original 3-predictor model is 0.322.

Explanation of Solution

From the multiple regression output, it is clear that the p-value of ANOVA for the original 3-predictor model is 0.322.

In common, removing an insignificant variable tends to improve the model. Thus, it causes the p-value of ANOVA to decrease.

Thus, if the p-value decreases after removing $X_2$, it is an indication that removing $X_2$ is a good idea. If the p-value increases after removing $X_2$, it is an indication that removing $X_2$ is a bad idea.

d.

To determine

Obtain the F-statistic from ANOVA for the given model.

Explain whether there is a change in F-statistic for ANOVA when removing an insignificant variable.

Answer to Problem 58E

The F-statistic of ANOVA for the given model is 1.23.

Explanation of Solution

From the multiple regression output, it is clear that the F-statistic of ANOVA for the given model is 1.23.

In the F-statistic, the numerator is divided by fewer degrees of freedom and the denominator is divided by more degrees of freedom.

The variable that explains little variation than other variables is dropped out as it is an insignificant variable. Hence, if an insignificant variable is dropped out, it causes the F-statistic to increase.

If the F-statistics is increased after removing the insignificant variable, then removing that variable is good idea. Otherwise, it is a bad idea.