Question

In Exercises 14.98-14.108, use the technology of your choice to do the following tasks.
a. Decide whether your can reasonably apply the conditional mean and predicted value t-interval procedures to the data. If so, then also do parts (b) - (h).
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 $95\%$Te 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).

14.10 PCBs and Pelicans. The data from Exercise $14.40$for shell thickness and concentration of PCBs of $60$Anacapa pelican eggs are on the WeissStats site. Specified value of the predictor variable: $220\mathrm{ppm}$.

Short answer

(a) The variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken

(b) There is no way to calculate from part (b) to part (f).

(c) There is no way to calculate from part (b) to part (f).

(d) There is no way to calculate from part (b) to part (f).

(e) There is no way to calculate from part (b) to part (f).

(f) There is no way to calculate from part (b) to part (f).

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 $41.40$for shell thickness and concentration of PCBs of $60$ Anacapa pelican egg as follows:

$PCB$	THICKNESS		199	0.46
139	0.21		214	0.22
166	0.23		177	0.23
175	0.24		205	0.25
260	0.26		208	0.26
204	0.28		320	0.28
138	0.29		191	0.29
316	0.29		305	0.3
396	0.3		230	0.3
46	0.31		204	0.32
218	0.34		143	0.35
173	0.36		175	0.36
220	0.37		119	0.39
147	0.39		216	0.41
216	0.42		185	0.42
216	0.46		236	0.47
177	0.22		356	0.22
246	0.23		289	0.23
296	0.25		324	0.26
188	0.26		109	0.27
89	0.28		265	0.29
198	0.29		193	0.29
122	0.3		203	0.3
250	0.3		214	0.3
256	0.31		150	0.34
261	0.34		229	0.35
132	0.36		236	0.37
212	0.37		144	0.39
171	0.4		232	0.41
164	0.42		87	0.44
			237	0.49

Part (a) Step 3: Explanation

Using Exercise $14.40$, determine if it is reasonable to apply the conditional mean and predated value $t$-interval procedures to data.
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 linear trend.
As a result, for the variables thickness and PCB, assumption $1-3$ for regression inferences is not violated.
The data from Exercise $14.40$ does not support the conclusion that PCB is beneficial for predicting THICKNESS.
As a result, the variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).

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

The variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).

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 variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).

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 variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).

Part (e) Step 1: Given information

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

Part (e) Step 2: Explanation

The variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).

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

The variables cannot be predicted using a linear model.
Part (a) clearly shows that the regression inference assumptions have been broken.
As a result, there is no way to calculate from part (b) to part (f).