Chapter 15.2, Problem 9E

The Professional Golfers Association (PGA) maintains data on performance and earnings for members of the PGA Tour. For the 2012 season Bubba Watson led all players in total driving distance, with an average of 309.2 yards per drive. Some of the factors thought to influence driving distance are club head speed, ball speed, and launch angle. For the 2012 season Bubba Watson had an average club head speed of 124.69 miles per hour, an average ball speed of 184.98 miles per hour, and an average launch angle of 8.79 degrees. The DATAfile named PGADrivingDist contains data on total driving distance and the factors related to driving distance for 190 members of the PGA Tour (PGA Tour website, November 1, 2012). Descriptions for the variables in the data set follow.

Club Head Speed: Speed at which the club impacts the ball (mph).

Ball Speed: Peak speed of the golf ball at launch (mph).

Launch Angle: Vertical launch angle of the ball immediately after leaving the club (degrees).

Total Distance: The average number of yards per drive.

1. a. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the club head speed.
2. b. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the ball speed.
3. c. A recommendation has been made to develop an estimated regression equation that uses both club head speed and ball speed to predict the average number of yards per drive. Do you agree with this? Explain.
4. d. Develop an estimated regression equation that can be used to predict the average number of yards per drive given the ball speed and the launch angle.
5. e. Suppose a new member of the PGA Tour for 2013 has a ball speed of 170 miles per hour and a launch angle of 11 degrees. Use the estimated regression equation in part (d) to predict the average number of yards per drive for this player.

To determine

Find the estimated regression equation that could be used to predict the average number of yards per drive given the club head speed.

Answer to Problem 9E

The estimated regression equation that could be used to predict the average number of yards per drive given the club head speed is $\widehat{y}=124.7160+1.3943\left(\text{Club Head Speed}\right)$.

Explanation of Solution

Calculation:

The Professional Golfers Association (PGA) data consist of information regarding to the speed at which the club impacts the ball (club head speed), peak speed of the golf ball at launch (ball speed), vertical launch angle of the ball immediately after leaving the club (launch angle), and the average number of yards per drive (total distance).

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 k 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.

Denote y as the dependent variable total distance.

Regression:

Software procedure:

Step-by-step procedure to get regression equation using EXCEL software:

• Open the file PGADrivingDist.
• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range enter $E$1:$E$191.
• Under Input X Range enter $B$1:$B$191.
• Click the box of Labels.
• Under Output Range enter $H$1.
• Click OK.

Output obtained using EXCEL software is given as follows:

Thus, the estimated regression equation that could be used to predict the average number of yards per drive given the club head speed is $\widehat{y}=124.7160+1.3943\left(\text{Club Head Speed}\right)$.

To determine

Find the estimated regression equation that could be used to predict the average number of yards per drive given the ball speed.

Answer to Problem 9E

The estimated regression equation that could be used to predict the average number of yards per drive given the ball speed is $\widehat{y}=117.14+0.9876\left(\text{Ball}\text{Speed}\right)$.

Explanation of Solution

Calculation:

The regression equation can be obtained using excel software.

Regression:

Software procedure:

Step-by-step procedure to get the regression equation using EXCEL software:

• Open the file PGADrivingDist.
• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range enter $E$1:$E$191.
• Under Input X Range enter $C$1:$C$191.
• Click the box of Labels.
• Under Output Range enter $H$24.
• Click OK.

Output obtained using EXCEL software is given as follows:

Thus, the estimated regression equation that could be used to predict the average number of yards per drive given the ball speed is $\widehat{y}=117.14+0.9876\left(\text{Ball}\text{Speed}\right)$.

To determine

Explain whether developing an estimated regression equation that uses both club head speed and ball speed to predict the average number of yards per drive is useful or not.

Explanation of Solution

Calculation:

Regression:

Software procedure:

Step-by-step procedure to get the regression equation using EXCEL software:

• Open the file PGADrivingDist.
• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range enter $E$1:$E$191.
• Under Input X Range enter $B$1:$C$191.
• Click the box of Labels.
• Under Output Range enter $H$24.
• Click OK.

Output obtained using EXCEL software is given as follows:

Thus, the estimated regression equation that could be used to predict the average number of yards per drive given both club head speed and ball speed is as follows:

$\widehat{y}=113.85-2.231\left(\text{Club Head}\right)+2.515\left(\text{Ball}\text{Speed}\right)$.

In the given output, $R^2=77.14\%$.

Thus, only 77.14% variability in total distance can be explained by variability in club head and ball speed.

Scatter Diagram:

Software procedure:

Step-by-step procedure to obtain scatter diagram using EXCEL software:

• Open the file PGADrivingDist.
• Select the cell Range $B$1:$C$191.
• Go to insert and select scatter plot.
• Click OK

Output obtained using EXCEL software is given as follows:

The scatter diagram exhibits a strong positive linear relationship between ball speed and club head speed. Both the variables in the same model are not recommended as the linear effect of one variable describes the linear effect of the other variable. As a result, the other variable will be of little additional value. Thus, there is a chance of multicollinearity.

Thus, developing an estimated regression equation that uses both club head speed and ball speed to predict the average number of yards per drive is not useful.

To determine

Find the estimated regression equation that could be used to predict the average number of yards per drive given the ball speed and the launch angle.

Answer to Problem 9E

The estimated regression equation that could be used to predict the average number of yards per drive given both ball speed and launch angle is as follows:

$\widehat{y}=81.60+1.0927\left(\text{Ball Speed}\right)+1.646\left(\text{Launch Angle}\right)$.

Explanation of Solution

Calculation:

The regression equation can be obtained using Excel software.

Regression:

Software procedure:

Step-by-step procedure to get the regression equation using EXCEL software:

• Open the file PGADrivingDist.
• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range enter $E$1:$E$191.
• Under Input X Range enter $C$1:$D$191.
• Click the box of Labels.
• Under Output Range enter $H$24.
• Click OK.

Output obtained using EXCEL software is given as follows:

Thus, the estimated regression equation that could be used to predict the average number of yards per drive given both ball speed and launch angle is as follows:

$\widehat{y}=81.60+1.0927\left(\text{Ball Speed}\right)+1.646\left(\text{Launch Angle}\right)$.

To determine

Predict the average number of yards per drive for the new player using the obtained equation in part (d).

Answer to Problem 9E

The predicted average number of yards per drive for the new player is 285.465 yards.

Explanation of Solution

Calculation:

The ball speed of a new PGA player is 170 mph with launch angle of 11 degrees.

From Part (b), it is found that the estimated regression equation that could be used to predict the average number of yards per drive given both club head speed and launch angle is $\widehat{y}=81.60+1.0927\left(\text{Ball Speed}\right)+1.646\left(\text{Launch Angle}\right)$.

Thus, the predicted average number of yards per drive for the new player is as follows:

$\begin{array}{c}\widehat{y}=81.60+1.0927\left(170\right)+1.646\left(11\right)\\ =81.60+185.759+18.106\\ =285.465\end{array}$

Thus, the predicted average number of yards per drive for the new player is 285.465 yards.