Question

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 WeissStats site.

a. Decide whether you can reasonably apply the regression $t$-test. If so, then also do part $b$.

b. Decide, at the $5\%$significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

Short answer

a). As a result, the regression $t$-test is appropriate for the supplied data.

b). As a result of the data, we may conclude that the predictor variable "high" temperature is beneficial for forecasting "low" temperature at the $5\%$level.

Step-by-step solution

Construction of the residual plot using MINITAB (Part a)

Procedure for MINITAB:

Step 1: From the drop-down menu, select Stat $>$Regression $>$Regression

Step 2: In the Response, select the column as Low.

Step 3: In Predictors, select the column as high.

Step 4: In Graphs, under Residuals vs the variables, enter the columns High.

Step 5: Click the OK button.

MINITAB Output (Part a)

Construction of the normal probability plot of residuals using MINITAB (Part a)

Procedure for MINITAB:

Step 1: From the drop-down menu, select Stat$>$Regression$>$Regression

Step 2: In the Response, select the column as Low.

Step 3: In Predictors, select the column as high.

Step 4: In Graphs, Click the normal probability of residuals.

Step 5: Click the OK button.

MINITAB Output

Regression inferences assumption (Part a)

Line of population regression:

• For each value $X$ of the predicator variable, the response variable conditional mean $\left(Y\right)$ is ${\beta }_0+{\beta }_{1}X$.

Standard deviations are equal:

• The response variable's $\left(Y\right)$ standard deviation is the same as the explanatory variable's $\left(X\right)$ standard deviation. Here, $\sigma$is used to represent the standard deviation.

Typical populations include:

• The response variable follows a normal distribution.

Independent observations:

• The responses variable's observations are unrelated to one another.

Graph suggestion for the regression inferences (Part a)

To examine whether the graph shows 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 trend.

• As a result, the regression inference assumptions $1.3$for the variables average high January temperature and average low January temperature are not violated.
• As a result, the regression $t$-test is appropriate for the supplied data.

Appropriate hypothesis data (Part b)

The following are the suitable hypotheses:

Null Hypothesis is:

$H_0:{\beta }_1=0$

That is, the high temperature predictor variable cannot be used to forecast "Low" temperature.

Alternative hypothesis is:

$H_a:{\beta }_1\ne 0$

In other words, the predictor variable "High" temperature can be used to forecast "Low" temperature.

Rule of Rejection:

Reject the null hypothesis $H_0$if $p-value\le \alpha (=0.05)$.

Finding test statistics and p value using MINITAB (Part b)

Procedure for MINITAB:

Step 1: From the drop-down menu, select Stat$>$Regression$>$Regression

Step 2: In the Response, select the column as Low.

Step 3: In Predictors, select the column as high.

Step 4: Click the OK button.

MINITAB Output (Part b)

 Conclusion (Part b)

• Use the $\alpha =0.05$ significance level.
• The significance level is less than the $p$-value.
• Specifically, $p-value(=0.000)<\alpha (=0.05)$.
• As a result of the rejection rule, it may be argued that at$\alpha =0.05$, there is evidence to reject the null hypothesis$H_0$.
• As a result of the data, we may conclude that the predictor variable "high" temperature is beneficial for forecasting "low" temperature at the $5\%$level.