Question

U.S. Presidents. The data from Exercise $14.35$ for the ages at inauguration and of death for the presidents of the United States are on the Weiss Stats site.

Short answer

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

(b) The findings support the conclusion that the predictor variable "inauguration" is useful in predicting "death" at the $5\%$level.

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that, U.S. Presidents. The data from Exercise $14.35$for the ages at inauguration and of death for the presidents of the United States are on the Weiss Stats site. We need to decide that whether we can reasonably apply the regression t-lest. If so, then also do part (b).

Part (a) Step 2: Explanation

Given,

INAGURATION	DEATH
$57$	$67$
$61$	$90$
$57$	$83$
$57$	$85$
$58$	$73$
$57$	$80$
$61$	$78$
$54$	$79$
$68$	$68$
$51$	role="math" localid="1652287714545" $71$
$49$	$53$
$64$	$65$
$50$	$74$
$48$	$64$
$65$	$77$
$52$	$56$
$56$	$66$
$46$	$63$
$54$	$70$
$49$	$49$
$51$	$57$
$47$	$71$
$55$	$67$
$55$	$71$
$54$	$58$
$42$	$60$
$51$	$72$
$56$	$67$
$55$	57
$51$	$60$
$54$	$90$
$51$	$63$
$60$	$88$
$62$	$78$
$43$	$46$
$55$	$64$
$56$	$81$
$61$	$93$
$69$	$93$

Part(a) Step 3: Construct the residual plot

MINITAB is used to create the residual plot.

Procedure for MINITAB:

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

Step 2: In the Response column, type Death.

Step 3: In Predictors, fill in the Inauguration columns.

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

Step 5: Click the OK button.

OUTPUT FROM MINITAB:

Part(a) Step 4: Construct the normal probability plot

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 the Response column, type Death.

Step 3: Fill in the columns in Predictors. Inauguration

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

MINITAB output: Step 5: Click OK

The following is the assumption for regression inferences:

Line of population regression:

For each value $X$of the predicator variable, the conditional mean of the response variable $\left(Y\right)$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. The standard deviation is represented by the symbol $\sigma$.

Typical populations include:

The response variable follows a normal distribution.

Observations made independently:

The response variable observations are unrelated to one another.

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

Part (b) Step 1: Given information

Given in the question that, U.S. Presidents. The data from Exercise $14.35$for the ages a inauguration and of death for the presidents of the United States are on the Weiss Stats site. we need to 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.

Part (b) Step 2: Explanation

The following are the suitable hypotheses:

Hypothesis of nullity:

$H_0:{\beta }_1=0$

In other words, the predictor variable "inauguration" is ineffective in predicting "death."

Another possibility:

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

In other words, the predictor variable "inauguration" can be used to predict death.

Rule of Rejection:

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

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

Part (b) Step 2:  Procedure for MINITAB 

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

Procedure for MINITAB:

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

Step 2: In the Response column, type Death.

Step 3: In Predictors, fill in the Inauguration columns.

Step 4: Click the OK button.

OUTPUT FROM MINITAB:

DEATH vs. INAUGURATION: A Regression Analysis

Model Overview

The test statistic value is $4.63$, and the p-value is $0.000$, according to the MINITAB report.

Conclusion: Use the $\alpha =0.05$significance threshold.

The p-value is lower than the level of significance in this case.

That is, $(=0.000)<a(=0.05)$p-value.

As a result of the rejection rule, it may be concluded that at $\alpha =0.05$, there is evidence to reject the null hypothesis ($H_0$).