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 inter. val that you obtained in part (c) and the prediction interval that you obtained in part (e).

14.99 U.S. Presidents. The data from Exercise 14.35 for the ages at inauguration and of death of the presidents of the United States are on the WeissStats site. Specified value of the predictor variable: 53 years.

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 $x=53$has a point estimate of $68.17$.

(c) The conditional mean of the response variable has a $95\%$confidence interval of $(64.87,71.47)$.

(d) The predicted value of the score corresponding to the predicted variable is $68.17$.

(e) The conditional mean of the response variable has a $95\%$prediction interval of $(48.23,88.11)$.

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

Step-by-step solution

Part (a) Step 1: Given information

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

Part (a) Step 2: Explanation

The data from Exercise 14.35 for the ages at inauguration and death as:

INAUGURATION	DEATH		49	49
57	67		51	57
61	90		47	71
57	83		55	67
57	85		55	71
58	73		54	58
57	80		42	60
61	78		51	72
54	79		56	67
68	68		55	57
51	71		51	60
49	53		54	90
64	65		51	63
50	74		60	88
48	64		62	78
65	77		43	46
52	56		55	64
56	66		56	81
46	63		61	93
54	70		69	93

Part (a) Step 3: Explanation

The residual plot clearly shows that the residuals lie within the horizontal band.
It is obvious from the normal probability plot of residuals that the residuals follow a fairly linear trend.
As a result, for the variables death and inauguration, the regression inference assumptions are not violated.
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, the $MINITAB$ procedure is as follows:
Step 1: Choose Stat >Regression>Regression.
Step 2: In Response, enter the column Death.
Step 3: In Predictors, enter the column lnauguration.
Step 4: In Options, enter $53$ 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 as follows:

Prediction for death:

The conditional mean of the response variable corresponding to the predictor variable$x=53$is $68.17$, according to the $MINITAB$output.
As a result, the conditional mean of the response variable corresponding to the predictor variable $x=53$has a point estimate of $68.17$.

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 $95$percent confidence interval for the conditional mean of the response variable corresponding to the predictor variable$x=53$is $(64.87,71.47)$, according to the $MINITAB$output in part (b).
The conditional mean death age for presidents inaugurated at age $53$lies between $64.87$and $71.47$, according to $95$ percent confidence.
As a result, the conditional mean of the response variable has a $95$percent confidence interval of $(64.87,71.47)$.

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 $x=53$is $68.17$, according to the $MINITAB$output in part (b).

As a result, $68.17$ 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 $x=53$is $(48.23,88.11)$, according to the $MINITAB$output in part (b).
Interpretation: The death age for presidents inaugurated at the age of $53$is estimated to be between $48.23$and $88.11$.
As a result, the conditional mean of the response variable has a $95$ percent prediction interval of $(48.23,88.11)$.

Part (f) Step 1: Given information

Compare and discuss the differences between the confidence inter. val that you obtained in part (c) and the prediction interval that you obtained in part (e).

Part (f) Step 2: Explanation

The confidence interval and the prediction interval are clearly centered on the expected value of the $53$ inauguration age, as shown in parts (c) and (e).
In addition, the prediction interval exceeds the confidence interval.
As a result, the prediction interval is greater than the confidence interval.