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 $95\%$ 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).

14.103 High and Low Temperature. The data from Exercise $14.31$for average high and low temperatures in January of a random sample, of $50$cities are on the WeissStats site. Specified value of the predictor variable: $55°\mathrm{F}$.

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

The data from Exercise $14.31$ for average high and low temperatures in January of a random sample as follows:

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.
Hence, the regression inference assumptions for the variables average high January temperature and average low January temperature are not violated.
As a result, implementing the conditional mean and predated value $t$-interval technique to the data seems reasonable.

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

The $MINITAB$ procedure as follows:
Step 1: Choose Stat >Regression>Regression.
Step 2: In Response, enter the column Low.
Step 3: In Predictors, enter the column High.
Step 4: In Options, enter 55 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:
The Prediction for LOW:

The conditional mean of the response variable corresponding to the predictor variable $x=55$is $42.85$.

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.

The $95$ percent confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=55$is $(41.67,44.04)$, according to the$MINITAB$ output in part (b).
The conditional mean low temperature with a mean high temperature of $55$degrees lies between $41.67$and $44.04$degrees, according to $95$percent confidence.
As a result, the conditional mean of the response variable has a $95\%$confidence interval of $(41.67,44.04)$.

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

The predicted value of the score corresponding to the predicted variable $x=55$ is $42.85$, according to the $MINITAB$output in part (b).
As a result, the predicted value of the score associated with the predicted variable is $42.85$.

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

The $95$percent prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=55$is $(34.42,51.28)$, according to the $MINITAB$output in part (b).
As a result, there is a $95\%$confidence interval between $34.42$and $51.28$degrees for the low temperature with a mean high temperature of $55$ degrees.

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

Parts (c) and (e) show that the confidence interval and prediction interval are centered on the predicted value of $55$ degrees for the mean high temperature.
In addition, the prediction interval exceeds the confidence interval.

As a result, the prediction interval is greater than the confidence interval.