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(b)−(f).
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 a95%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 valte of the predictor variable.
e. Find and interpret a95%prediction inferval 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).

Estriol Level and Birth Weight. The data from Exercise $14.42$for estriol levels of pregnant women and birth weights of their children are on the WeissStats site. Specified value of the predictor variable: $18mg/24hr$

Short answer

Part a) it is reasonably applying the conditional mean and predated value $t$-interval procedure data.

Part b) From the MINITAB output, the point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=18$is $32.47$hectograms

Part c) There is $95\%$the confidence that the conditional mean weight for the babies' mother with an estriol level of $18$lies between $31.05$and $24.53$hectograms.

Part d) predictor variable $x=18$is $32.47$hectograms

Part e)There is$95\%$certain that the weight for babies' mother with an estriol level of 18 lies between $24.53$and $40.41$hectograms

Part f) From part (c) and part (e), it is clear that the confidence interval and the prediction interval are centered at the predicted$18$estriol level. Moreover, the prediction interval is greater than the confidence level.

Step-by-step solution

Part a) Step 1: Explanation

Using the data from the Exercise $14.42$determine whether it is reasonable to apply the conditional mean and predated value-interval procedure to data.

There is an increasing level of variability in the residual plot versus the estriol level. According to the normal probability

As a result, for the variables weight and Estriol level, assumption 2 for regression inferences is violated. However, the regression inferences' normality assumption is not violated.

Therefore, it is reasonably applying the conditional mean and predated value $t$-interval procedure data.

Part b) Step 1: Explanation

predictor variable$x=18$by using MINITAB.
MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.
Step 2: In Response, enter the column Weight.
Step 3: In Predictors, enter the columns Estriol. Step 4: In Options, enter 18 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

MINITAB output:
Prediction for WEIGHT
Regression Equation

WEIGHT $=21.52+0.608$ESTRIOL

From the MINITAB output, the point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=18$is $32.47$ hectograms

Part c) Step 1: Explanation

Compute the $95\%$confidence interval for the conditional mean of the response variable corresponding to the predictor variable$x=18$by using MINITAB.

From the MINITAB output in part (b), the$95\%$confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=18$
There is$95\%$ confidence that the conditional mean weight for babies' mothers with an estriol level of
18 lies between $31.05$and $24.53$hectograms.

Part d) Step 1: Explanation

Compute the predicted value of the response variable corresponding to the predictor variable $x=53$by using MINITAB.

From the MINITAB output in part (b), the predicted value of the score corresponding to the predictor variable $x=18$is $32.47$hectograms.

The predictor variable $x=18$is$32.47$ hectograms

Part e) Step 1: Explanation

Compute the$95\%$prediction interval for the conditional mean of the response variable corresponding to the predictor variable x=18 by using MINITAB.
From the MINITAB output in part (b), the $95\%$prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=18$is$(24.53,40.41)$
It is $95\%$certain that the weight for babies' mother with an estriol level of $18$ lies between $24.53$ and $40.41$ hectograms

Part f) Step 1: Explanation

rom part (c) and part (e), it is clear that the confidence interval and the prediction interval are centered at the predicted $18$estriol level. Moreover, the prediction interval is greater than the confidence level.