Question

How important are birdies (a score of one under par on a given hole) in determining the final total score of a woman golfer? From the U.S Women's Open website, we obtained data on number of birdies during a tournament ab=nd final score for \(63\) women golfers. The data are presented on the WeissStats site.

a. obtain and interpret the standard error of the estimate.

b. obtain a residual plot and a normal probability plot of the residuals.

c. decide whether you can reasonably consider Assumptions \(1-3\) for regression inferences met by the two variables under considerations.

Short answer

Part a. From the MINITAB output, the standard error of the estimate is \(4.34577\).

Part b. The residual plot and normal probability plot are obtained.

Part c. The assumptions for the regression inferences are not violated for the variables birdies and scores.

Step-by-step solution

Part a. Step 1. Given information

Given,

BIRDIES	SCORE		9	304
15	292		9	302
14	297		9	290
14	293		9	299
13	291		9	296
13	289		9	304
13	296		8	301
13	296		8	296
12	289		8	291
12	290		8	299
12	294		8	299
12	294		7	307
12	292		7	303
12	291		7	297
11	293		7	300
11	299		7	296
11	297		7	294
11	296		7	302
11	287		7	297
11	299		6	297
11	293		6	309
11	303		6	303
10	299		6	299
10	303		6	294
10	296		6	298
10	303		6	293
10	304		5	298
10	299		5	304
10	292		5	301
10	293		4	296
9	291		3	301
9	293		3	303

Part a. Step 2. Calculation

Find the standard error of the estimate by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column SCORE.

Step 3: In Predictors, enter the column BIRDIES.

Step 4: Click OK.

MINITAB output:

From the MINITAB output, the standard error of the estimate is \(4.34577\).

Interpretation:

The predicted scores in the sample differ on average from the observed scores by.

Conclusion:From the MINITAB output, the standard error of the estimate is \(4.34577\).

Part b. Step 1. Calculation

Construct the residual plot by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column SCORE.

Step 3: In Predictors, enter the column BIRDIES.

Step 4: In Graphs, enter the columns BIRDIES under Residuals versus the variables.

Step 5: Click OK.

MINITAB output:

Construct the normal probability plot by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column SCORE.

Step 3: In Predictors, enter the column BIRDIES.

Step 4: In Graphs, select Normal probability plot of residuals.

Step 5: Click OK.

MINITAB output:

Thus, the residual plot and normal probability plot are obtained.

Part c. Step 1. Calculation

The conditional mean of the response variable \((Y)\) is \(\beta _{0}+\beta _{1}X\), for each value \(X\) of predictor variable.

Equal standard deviation:

The standard deviation for the response variable \((Y)\) is same for the standard deviation for the explanatory variable \((X)\). The standard deviation is denoted as \(\sigma\).

Normal populations:

The distribution of the response variable follows normal.

Independent observations:

The observations of the response variable are independent of each other.

Check whether the graph suggests violation of one or more of the assumptions for the regression inferences.

• From the residual plot, it is clear that the residuals are fall in the horizontal band.
• From the normal probability plot of residuals, it is clear that the residuals are in the linear pattern.

Hence, the assumptions for the regression inferences are not violated for the variables birdies and scores.