Chapter 15.2, Problem 10E

a.

To determine

Find the estimated regression equation that could be used to predict the average number of runs given up per inning given the average number of strikeouts per innings pitched.

Answer to Problem 10E

The estimated regression equation that could be used to predict the average number of runs given up per inning given the average number of strikeouts per innings pitched is $\widehat{\text{R/IP}}=0.6758-0.2838\text{SO/IP}$.

Explanation of Solution

Calculation:

The Major League Baseball (MLB) data for the season 2011consists of the average number of earned runs given up by the pitcher per nine innings (ERA), the average number of strikeouts per innings pitched (SO/IP), the average number of home runs per innings pitched (HR/IP) and the number of runs given up per innings pitched (R/IP).

Multiple linear regression model:

A multiple linear regression model is given as $\widehat{y}=b_0+b_1{x}_1+\mathrm{...}+b_p{x}_p$ where $\widehat{y}$ is the predicted value of response or dependent variable, and $x_1,x_2,\mathrm{...},x_p$ are the p predictor variables. The quantities $b_1,b_2,\mathrm{...},b_p$ are the estimated slopes corresponding to $x_1,x_2,\mathrm{...},x_p$ respectively and $b_0$ is the estimated intercept of the line, from the sample data.

Regression:

Software procedure:

Step by step procedure to get regression equation using MINITAB software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of R/IP.
• Under Continuous predictors, enter the columns ofSO/IP.
• Click OK.

The output using MINITAB software is given as,

Thus, the estimated regression equation that could be used to predict the average number of runs given up per inning given the average number of strikeouts per innings pitched is $\widehat{\text{R/IP}}=0.6758-0.2838\text{SO/IP}$.

b.

To determine

Find the estimated regression equation that could be used to predict the average number of runs given the average number of home runs per innings pitched.

Answer to Problem 10E

The estimated regression equation that could be used to predict the average number of runs given the average number of home runs per innings pitched is $\widehat{\text{R/IP}}=0.3081+1.347\text{HR/IP}$.

Explanation of Solution

Calculation:

Regression:

Software procedure:

Step by step procedure to get regression equation using MINITAB software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of R/IP.
• Under Continuous predictors, enter the column ofHR/IP.
• Click OK.

The output using MINITAB software is given as,

Thus, the estimated regression equation that could be used to predict the average number of runs given the average number of home runs per innings pitched is $\widehat{\text{R/IP}}=0.3081+1.347\text{HR/IP}$.

c.

To determine

Find the estimated regression equation that could be used to predict the average number of runs given the average number of strikeouts per innings pitched and the average number of home runs per innings pitched.

Answer to Problem 10E

The estimated regression equation that could be used to predict the average number of runs given the average number of strikeouts per innings pitched and the average number of home runs per innings pitched is $\widehat{\text{R/IP}}=0.5365-0.2483\text{SO/IP}+1.032\text{HR/IP}$.

Explanation of Solution

Calculation:

Regression:

Software procedure:

Step by step procedure to get regression equation using MINITAB software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• Under Responses, enter the column of R/IP.
• Under Continuous predictors, enter the columns ofSO/IP andHR/IP.
• Click OK.

The output using MINITAB software is given as,

Thus, the estimated regression equation that could be used to predict the average number of runs given the average number of strikeouts per innings pitched and the average number of home runs per innings pitched is $\widehat{\text{R/IP}}=0.5365-0.2483\text{SO/IP}+1.032\text{HR/IP}$.

d.

To determine

Predict the average number of runs given up per inning for A.J. Burnett using the regression equation of part c.

Answer to Problem 10E

The predicted average number of runs given up per inning for A.J. Burnett is 0.4757.

Explanation of Solution

Calculation:

The average number of strikeouts and the average number of home runs per innings pitched for A.J. Burnet are 0.91 and 0.16, respectively.

From part (c), it is found that estimated regression equation that could be used to predict the average number of runs given the average number of strikeouts per innings pitched and the average number of home runs per innings pitched is $\widehat{\text{R/IP}}=0.5365-0.2483\text{SO/IP}+1.032\text{HR/IP}$

Thus, using the given information the predicted average number of runs given up per inning for A.J. Burnett is,

$\begin{array}{c}\widehat{\text{R/IP}}=0.5365-0.2483\left(\text{0}\text{.91}\right)+1.032\left(0.16\right)\\ =0.5365-0.225953+0.16512\\ \approx 0.4757\end{array}$

Thus, the predicted average number of runs given up per inning for A.J. Burnett is 0.4757.

It is to be noted that the actual average number of runs given up per inning for A.J. Burnett was 0.6.

Thus, the predicted value is slightly less than the actual value. Hence, the residual is positive, which underestimate the model slightly.

e.

To determine

Explain about the suggestion to use the earned run average as another independent variable.

Explanation of Solution

Calculation:

Correlation Coefficient:

Software procedure:

Step by step procedure to get correlation coefficient using MINITAB software is given as,

• Choose Stat > Basic Statistics > Correlation.
• Under Variables, enter the column of R/IP and ERA.
• Click OK.

The MINITAB output is given as,

Thus, the correlation coefficient between R/IO and ERA is 0.964.

Due to the high correlation coefficient, it can be said that if a pitcher gives up more runs per innings pitched then pitcher’s run average also increase. Thus, automatically there will be high value of coefficient of determination in the regression analysis using ERA as predictor variable.

Thus, the suggestion does not make any sense.

In addition, the ERA can be used as the predictor dependent variable.