Body Fat. The data from Exercise 14.80 for age and body fat of 18 randomly selected adults are on the WeissStats site.

a. Do the data provide sufficient evidence to conclude that, for adults, age and percentage of body fat are positively linearly correlated?

b.Remove the potential outlier and repeat part (a).

c. Compare your results with and without the removal of the potential outlier and state your conclusions.

Body Fat. The data from Exercise 14.80 for age and body fat of 18 randomly selected adults are on the WeissStats site.

Check whether or not it is reasonably apply the correlation t-test procedure by using the data from Exercise 14.80.

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

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

Hence, the assumptions $1.3$ for the regression inferences are not violated for the variables birdies and scores. Hence, the regression $t$-test is reasonable to apply for the given data.

Therefore, it is reasonably applying the correlation $t$-test procedure for given data.

Check whether the data provide sufficient evidence to conclude that, for adults, age and percentage of body fat are positively Inearly correlated or not.

The test hypotheses are as follows:

Null hypothesis:

$$

H_{e}: \rho=0

$$

That is, age and percentage of body fat are not positively linearly correlated.

Alternative hypothesis:

$$

H_{\alpha}: \rho>0

$$

That is, age and percentage of body fat are positively linearly correlated.

Obtain the correlation and $p$-value between age and \%tat by using MINITAB.

MINITAB procedure:

Step 1: Select Stat > Basic Statistics > Correlation.

Step 2: In Variables, select age and \%fat from the box on the left.

Step 3: Click OK.

MINITAB output:

Correlation: AGE, \%FAT

From the MINITAB output, the value of correlation between age and $\%$ fat is $0.792$ and the p-value for right-tail test is 0 .

Conclusion:

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

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

That is, $p$-value $(=0)<\alpha(=0.05)$.

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 sufficlent evidence to conclude that, for adults, age and percentage of body fat are positively linearly correlated at $5 \%$ significance level.

Find the correlation and $p$-value between age and $\%$ fat when removing the potential outlier.

MINITAB procedure:

Step 1: Select Stat >Basic Statistics > Correlation.

Step 2: In Variables, select age and \%fat from the box on the left.

Step 3: Click OK.

MINITAB output:

Correlation: AGE, \%FAT

From the MINITAB cutput, the value of correlation between age and $\%$ fat is $0.873$ and the $p$-value for right-tail test is 0 .

Conclusion:

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

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

That is, $p$-value $(=0)<\alpha(=0.05)$.

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 adults, age and percentage of body fat are positively linearly correlated at $5 \%$ significance level.

Compare the results with and without the removal of the potential outlier.

Correlation with outlier:

The value of correlation between age and $\%$ fat is $0.792$ and the $p$-value for right-tail test is 0 .

Thus, the data provide sufficient evidence to conclude that, for adults, age and percentage of body fat are positively linearly correlated because the null hypothesis is rejected at $5 \%$ significance level.

Correlation without outlier:

The value of correlation between age and $\%$ fat is $0.873$ and the p-value for right-tail test is 0 . Thus, the data provide sufficient evidence to conclude that, for adults, age and percentage of body fat are positively linearly correlated because the null hypothesis is rejected at $5 \%$ significance level.

From the results, it is clear that both conclusions are same but the correlation value for without outlier is larger when compared to with outlier.