Question

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). For the provided data, the regression $t$-test is appropriate.

b). At the $5\%$level, the data give adequate evidence to infer that the predictor variable "high" temperature is beneficial for predicting the "low" temperature.

Step-by-step solution

Construction of residual plot using MINITAB (Part a)

Step 1: From the drop-down menu. Select Stat $>$Regression $>$Regression

Step 2: In the Response column, type Low.

Step 3: In Predictors, fill in the High and Low columns.

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

Step 5: Click OK.

MINITAB Output (Part a)

OUTPUT FROM MINITAB:

The normal probability plot of residuals (Part a)  

Procedure for MINITAB:

Step 1: Select Stat $>$Regression $>$Regression from the drop-down menu.

Step 2: In the Response column, type Low.

Step 3: In Predictors, fill in the High and Low columns.

Step 4: Select Normal probability plot of residuals from the Graphs menu.

Step 5: Click OK.

OUTPUT FROM MINITAB:

The following is the regression inferences assumptions:

Line of population regression:

For each value $x$of the predicator variable, the conditional mean of the response variable$y$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. $\sigma$is standard deviation

Typical populations include:

The response variable follows a normal distribution.

Observations made independently:

The response variable observations are unrelated to one another.

To examine whether the graph shows a violation of one or more of the regression inference assumptions.

• 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 they are linear pattern

For the provided data, the regression $t$-test is appropriate.

Appropriate Hypotheses (Part b)

The following are the suitable hypotheses:

Hypothesis of nullity:

$H_0:{\beta }_1=0$

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

Another possibility:

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

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

Rule of Rejection:

If $p$-value $\le \alpha (=0.05)$, reject the null hypothesis $H_0$.

 Procedure for MINITAB (Part b)

MINITAB can be used to find the test statistic and p-value.

Step 1: Select Stat $>$Regression $>$Regression from the drop-down menu.

Step 2: In the Response column, type Low.

Step 3: In Predictors, fill in the High and Low columns.

Step 4: Select Normal probability plot of residuals from the Graphs menu.

Step 5: Click OK.

MINITAB output:

Conclusion (Part b)

• Use the $\alpha =0.05$significance level.
• The $p$-value is lower than the level of significance in this case.
• That is, $p$-value $(=0.000)<\alpha (=0.05)$.
• According to the rejection criterion, there is sufficient evidence to reject the null hypothesis $H_0$at $\alpha =0.05$.