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 $\left(b\right)-\left(h\right)$.
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 $95\%$ 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).
14.100 Movie Grosses. The data from Exercise 14.36 on domestic and overseas grosses for a random sample of 50 movies are on the WeissStats site. Specified value of the predictor variable: S 100 million.

Short answer

(a) For the variables abroad and domestic, the regression inference assumptions $(1\&3)$are violated. That is, the model does not work well with the data. Furthermore, the residuals do not follow a normal distribution.

(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 $14.36$ on domestic and overseas grosses is as follows:

DOMESTIC	OVERSEAS		44.3	10.5
49	28.1		2	1.7
70	11.3		4.2	2.3
16	13.6		1.5	6
125	71.1		36.9	20.8
64.6	62.1		51.8	80.8
56	51.1		79.7	165.8
9	25.5		41.2	43.2
25.9	8.7		12.5	21.6
162.8	262.2		408	283.2
73.1	211.1		1.8	7.7
2.1	17.4		12.8	56.1
31.1	92		216.4	530.5
182.2	93.1		6	3.1
83.7	221.6		1.6	3.7
54.3	73.4		18.6	27.6
17.7	15.6		448.1	636.3
93.8	68		13.4	22.2
31.6	7.9		45.5	22.8
148.8	293		19.2	126.4
218.8	330.6		262	490.2
18.5	3.6		65.4	237.6
1	3.1		35.3	46.2
5.7	4		25.3	7.4
126.5	276.9		27.1	103.4
			41.1	31

Part (a) Step 3: Explanation

Using Exercise $14.36,$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 fall into a funnel form.
It is obvious from the normal probability plot of residuals that the residuals follow a small linear pattern.
As a result, for the variables abroad and domestic, the regression inference assumptions ($1$and $3$) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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

For the variables abroad and domestic, the regression inference assumptions ($1$and $3$) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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

For the variables abroad and domestic, the regression inference assumptions ($1$ and $3$) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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

For the variables abroad and domestic, the regression inference assumptions ($1$ and 3) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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 $95\%$ prediction interval for the value of the response variable corresponding to the specified value of the predictor variable.

Part (e) Step 2: Explanation

For the variables abroad and domestic, the regression inference assumptions ($1$ and$3$) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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

For the variables abroad and domestic, the regression inference assumptions ($1$and$3$) are violated.
That is, the model does not work well with the data.
Furthermore, the residuals do not follow a normal distribution.
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).