Question

In Exercises 14.98-14.108, use the technology of your choice to do the following tasks.
a. Decide whether you can reasonably apply the conditional mean and predicted value t-interval procedures to the data. If so, then also do parts $\left(b\right)-\left(f\right)$
b. Determine and interpret a point estimate for the conditional mean of the response variable corresponding to the specified value of the predictor variable.
c. Find and interpret a $95\%$ confidence interval for the conditional mean of the response variable corresponding to the specified value of the predictor variable.
d. Determine and interpret the predicted value of the response variable corresponding to the specified value of the predictor variable.
e. Find and interpret a $95\%$ prediction interval for the value of the response variable corresponding to the specified value of the predictor variable.
f. Compare and discuss the differences between the confidence interval that you obtained in part (c) and the prediction interval that you obtained in part (e).

14.98 Birdies and Score. The data from Exercise 14.34 for number of birdies during a tournament and final score of $63$ women golfers are on the WeissStats site. Specified value of the predictor variable:$12$birdies.

Short answer

(a) Apply the conditional mean and predated value $t$-interval technique to the data seems reasonable.

(b) The conditional mean of the response variable corresponding to the predictor variable $x=12$has a point estimate of $294.672$.
(c) The conditional mean of the response variable has a $95\%$confidence interval of $(293.076,296.267)$.

(d) The predicted value of the score corresponding to the predicted variable is $294.672$
(e) The $95\%$prediction interval for the conditional mean of the response variable is $(285.837,303.507)$.

(f) The prediction interval is greater than the confidence interval.

Step-by-step solution

Part (a) Step 1: Given information

To decide whether reasonably apply the conditional mean and predicted value$t$-interval procedures to the data.

Part (a) Step 2: Explanation

The data from Exercise $14.34$for number of birdies during a tournament as:

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 3: Explanation

It is evident from the residual map that the residuals are in the horizontal band.
The linear structure of residuals is evident from the normal probability plot of residuals.
Hence, the assumption for regression inferences for the variables birdies and scores is not broken.
As a result, implementing the conditional mean and predated value t-interval technique to the data seems reasonable.

Part (b) Step 1: Given information

To determine and interpret a point estimate for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

Part (b) Step 2: Explanation

Let, $MINITAB$procedure is as follows:

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 Options, enter $12$ under Prediction interval for new observations.
Step 5: In Confidence Level, enter $95.$
Step 6: In Storage, Choose Fits, Confidence limits, SEs of fits, and Prediction limits.
Step 7: Click OK.

And the $MINITAB$output is as follows:

The point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=12$is $294.672$based on the $MINITAB$output.
As a result, the conditional mean of the response variable corresponding to the predictor variable $x=12$has a point estimate of $294.672$.

Part (c) Step 1: Given information

To find and interpret a $95\%$confidence interval for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

Part (c) Step 2:Explanation

The$95\%$confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=12$calculated from $MINITAB$output is $(293.076,296.267)$.

The typical score for golfers with $12$birdies is between $293.076$and $296.267$, according to$95$ percent of the time.
As a result, the conditional mean of the response variable has a $95$percent confidence interval of $(293.076,296.267)$.

Part (d) Step 1: Given information

To determine and interpret the predicted value of the response variable corresponding to the specified value of the predictor variable.

Part (d) Step 2: Explanation

The predicted value of the score corresponding to the predicted variable $x=12$is $294.672$, according to $MINITAB\text{'}s$ output in part (a).

As a result, $294.672$is the predicted value of the score corresponding to the predicted variable.

Part (e) Step 1: Given information

To find and interpret a $95\%$prediction interval for the value of the response variable corresponding to the specified value of the predictor variable.

Part (e) Step 2: Explanation

The $95$percent prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=12$is $(285.837,303.507)$, according to the $MINITAB$output in section (a).

There is a $95\%$confident that a golfer's score with $12$birdies will fall between $285.837$and $303.507$.
As a result, the conditional mean of the response variable has a $95$percent prediction interval of $(285.837,303.507)$.

Part (f) Step 1: Given information

To compare and discuss the differences between the confidence interval that obtained in part (c) and the prediction interval that obtained in part (e).

Part (f) Step 2: Explanation

Parts (c) and (e) show that the confidence interval and prediction interval are both centered around the $12$ birdies' predicted value.
In addition, the prediction interval exceeds the confidence interval.
As a result, the prediction interval is greater than the confidence interval.