In Exercises 14.70-14.80, use the technology of your choice to do the following tasks.
a. Decide whether you can reasonably apply the regression t-test. If so, then also do part (b).
b. Decide, at the 55 significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

14.75 High and Low Temperature. The data from Exercise 14.39 for average high and low temperatures in January for a random sample of 50 cities are on the Weiss Stats site.

The given data from Exercise 14.39 for average high and low temperatures in January for a random sample of 50 cities as follows:

The MINITAB is used to create a normal probability plot of residuals.
PROCEDURE FOR MINITAB:
Step 1: Select Stat > Regression > Regression from the drop-down menu.
Step 2: In Response, enter the column Low
Step 3: In Predictors, enter the columns High.
Step 4: Select Normal probability plot of residuals from the Graphs menu.
Step 5: Click the OK button.
The MINITAB output will be:

The following is the assumption for regression inferences:
Regression line for the population:
For any value of the predictor variable $\mathrm{X}$is the conditional mean of the response variable $Y$is ${\beta }_0+{\beta }_{1}X$.
The response variable's $Y$standard deviation is the same as the explanatory variable's $X$standard deviation. The standard deviation is represented by the symbol $\sigma$.
Normal populations:
The response variable's distribution is normally distributed.
Independent observations:
The response variable observations are unrelated to one another.

Examine the graph for evidence of a violation of one or more of the regression inference assumptions.

• 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 pattern.

As a result, for the variables average high January temperature and average low January temperature, the assumptions 1-3 for regression inferences are not broken.
As a result, the regression t-test is a reasonable choice for the provided data.

To decide, at the $55$ significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

The null hypothesis is indicated as follows:
$H_0:{\beta }_1=0$
That is to suggest, the predictor variable "High" temperature is useless for forecasting "Low" temperature.
The alternative hypothesis is indicated as follows:
$H_{\alpha }:{\beta }_1\ne 0$
In other words, the predictor variable "High" temperature can be used to forecast "Low" temperature.
Rejection Rule:
If $p$is less than a $(0.05)$, reject the null hypothesis $\left(H_0\right)$
MINITAB can be used to find the test statistic and p-value.
PROCEDURE FOR MINITAB:
Step 1: Select Stat > Regression > Regression
Step 2: In Response, enter the column Low.
Step 3: In Predictors, enter the columns High.

Step 4: Click the OK button.

The MINITAB output will be:

The test statistic value is $30.91$, and the p-value is $0.000$, according to the MINITAB result.
Use the $\alpha =0.05$ significance level.
The p-value is lower than the level of significance in this case.
In other words, the p-value $(=0.000)<\alpha (=0.05)$
As a result of the rejection rule, it may be argued that at $\alpha =0.05$there is evidence to reject the null hypothesis $\left(H_0\right)$.
So, the data support the conclusion that the predictor variable "high" temperature is beneficial in predicting "low" temperature at the $5\%$level.