Question

High and Low Temperature. The data from Exercise $14.34$for average high and low temperatures in January of a random sample of $50$cities are on the WeissStats site. The specified value of the predictor variable: $55\circ F$.

Short answer

(a) It is applying the conditional mean and predated value $t$-interval procedure to the given data in a reasonable manner.

(b) The conditional mean of the response variable corresponding to the predictor variable $x=55$has a point estimate of $42.85$.

(c) The conditional mean of the response variable has a 95% confidence interval of $(41.67,44.04)$.

(d) $42.85$is the predicted value of the score corresponding to the predicted variable.

(e)$(34.42,51.28)$is the $95\%$prediction interval for the conditional mean of the response variable.

(f) The prediction interval exceeds the confidence interval.

Step-by-step solution

Part (a) Step 1: Given information

To determine whether the conditional mean and predated value t-interval procedures are appropriate.

Given that,

Explanation

Calculation:

• It is obvious from the residual plot that the residuals fall into the horizontal band.
• It is obvious from the normal probability plot of residuals that the residuals follow a fairly linear trend.

As a result, the regression inference assumptions for the variables average high January temperature and average low January temperature are not violated.

Conclusion:
Therefore, it is reasonably applying the conditional mean and predated value t-interval procedure for given data.

Part (b) Step 1: Given information

Given that, the conditional mean of the response variable is estimated as a point estimate.

Explanation

Calculation:

MINITAB Procedure:

Steps

1 . ChooseStat >Regression>Regression.
2. In Response, enter the columnLow.
3. In Predictors, enter the columnHigh.
4. In Options, enter $55$ underPrediction interval for new observations.
5. In Confidence Level, enter$95$.
6. In Storage, ChooseFits, Confidence limits, SEs of fits, andPrediction limits.
7. ClickOK.
MINITAB output:
LOW Prediction
Regression Equation
$LOW=-7.57+0.9168HIGH$
Variable Setting
HIGH 55

: Conclusion

The point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=55$ from the MINITAB output is $42.85$.

Part (c) Step 1: Given information

Given that, the conditional mean of the response variable has a$95\%$ confidence interval.

Explanation

Calculation:

The $95$percent confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=55$is calculated using the MINITAB output in part (b) $(41.67,44.04)$.

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.

Conclusion:
As a result, the conditional mean of the response variable has a $95\%$ confidence interval of $(41.67,44.04)$

Part (d) Step 1: Given information

Given that, the score corresponding to the predicted variable's predicted value.

Explanation

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

Part (e) Step 1: Given information

Given that, the conditional mean of the answer variable's 95 percent prediction interval.

Step 2: Explanation

Calculation:
The $95$ percent prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=55$ is calculated using the MINITAB output in part (b) $(34.42,51.28)$.
Interpretation:
There is a$95$percent chance that the low temperature will be between $34.42$and $51.28$degrees, with a mean high temperature of$55$ degrees.
Conclusion:
As a result, the conditional mean of the response variable's $95$ percent prediction interval is $(34.42,51.28)$.

Part (f) Step 1: Given information

To find, the difference between a confidence interval and a prediction interval.

Explanation

Calculation:

Comparison:

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.

Conclusion:

The prediction interval exceeds the confidence interval.