Question

U.S. Presidents. The data from Exercise 14.35 for the ages at inauguration and of death of the presidents of the United States are on the WeissStats site. Do the data provide sufficient evidence to conclude that, for U.S. presidents, age at inauguration and age at death are positively linearly correlated?

Short answer

For U.S. presidents, age at inauguration and age at death are positively linearly correlated at $5\%$ significance level.

Step-by-step solution

Step 1:Given information

The data from Exercise 14.35 for the ages at inauguration and of death of the presidents of the United States are on the WeissStats site.

Step 2:Explaination

Check whether or not it is reasonably apply the correlation $t$-test procedure by using the data from Exercise 14.35.

- From the residual plot, it is clear that the residuals are fall in the horizontal band.

- From the normal probability plot of residuals, it is clear that the residuals are in the linear pattern.

Hence, the assumptions for the regression inferences are not violated for the variables inauguration and death.

Check whether the data provide sufficient evidence to conclude that, for U.S. presidents, age at inauguration and age at death are positively linearly correlated.

The test hypotheses are as follows:

Null hypothesis:

$H_0:\rho =0$

That is, the inauguration ages and the death ages of U.S. presidents are not positively linearly correlated.

Alternative hypothesis:

$H_a:\rho >0$

That is, the inauguration ages and the death ages of U.S. presidents are positively linearly correlated.

Obtain the correlation and P-value between inauguration and death by using MINITAB.

MINITAB procedure:

Step 1: Select Stat >Basic Statistics > Correlation.

Step 2: In Variables, select inauguration and death from the box on the left.

Step 3: Click OK.

MINITAB output:

Correlation: INAUGURATION, DEATH

Pearson correlation of INAUGURATION and DEATH $=0.606$

$\mathrm{P}-\mathrm{Value}=0.000$

From the above MINITAB output, the value of correlation between inauguration and death is

$0.606$and the P-value for right-tail test is $0.000\left(=\frac{0.000}{2}\right)$.

Conclusion:

Use the significance level, $\alpha =0.05$.

Here, p-value is less than the level of significance.

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

Therefore, by the rejection rule, it can be concluded that there is evidence to reject the null hypothesis $\left(H_0\right)$at $\alpha =0.05$

Thus, the data provide sufficient evidence to conclude that, for U.S. presidents, age at inauguration and age at death are positively linearly correlated at $5\%$significance level.