In Exercises 14.70-14.80, use the technology of your choice to do the following tasks.
a. Decide whether you can reasonably apply the regression t-test. If so, then also do part (b).
b. Decide, at the 55 significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

14.80 Body Fat. In the paper "Total Body Composition by Dual-Photon (Gd) Absorptiometry" (American Journal of Clinical Nutrition, Vol. 40, pp. 834-839 ), R. Mazess et al, studied methods for quantifying body composition. Eighteen randomly selected adults were measured for percentage of body fat, using dual-photon absorptiometry. Each adult's age and percentage of body fat are shown on the WeissStats site.

To decide whether can reasonably apply the regression t-test. If so, then determine part (b).

The provided data as follows:

The MINITAB is used to create a plot of residuals.
PROCEDURE FOR MINITAB:
Step 1: Select Stat > Regression > Regression from the drop-down menu.
Step 2: In Response, enter the column Fat%
Step 3: In Predictors, enter the columns Age.
Step 4: Select columns Age under residuals from the Graphs menu.
Step 5: Click the OK button.
The MINITAB output will be:

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 Response, enter the column Fat%
Step 3: In Predictors, enter the columns Age.
Step 4: Select, normal probability plot from graph.
Step 5: Click the OK button.
The MINITAB output will be:

The following is the assumption for regression inferences:
Regression line for the population:
For any value of the predictor variable $X$, the conditional mean of the response variable $Y$is ${\beta }_0+{\beta }_{1}X$.
Equal standard deviation:
The response variable's $Y$ standard deviation is the same as the explanatory variable's $X$ standard deviation.
The standard deviation is represented by the symbol $\sigma$.
Normal populations:
The response variable's distribution is normally distributed.
Independent observations:
The response variable observations are independent to one another.
Examine the graph for any indications of 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 pattern.

The regression inferences' normality assumption is not violated here.

As a result, the regression inferences assumption $1-3$ are not violated.

As a result, the regression $t$-test is a reasonable choice for the provided data.

To decide, at the $55$significance level, whether the data provide sufficient evidence to conclude that the predictor variable is useful for predicting the response variable.

The null hypothesis is indicated as follows:

$H_0:{\beta }_1=0$
To put it another way, the predictor variable "age" is useless for predicting "percent fat."
The alternative hypothesis is indicated as follows:
$H_{\alpha }:{\beta }_1\ne 0$
In other words, the predictor variable "age" can be used to predict "percent fat."
Rejection Rule:

If $p$-value $\le \alpha (=0.05)$, reject the null hypothesis $\left(H_0\right)$.
MINITAB can be used to find the test statistic and $p$-value.
PROCEDURE FOR MINITAB:
Step 1: Select Stat > Regression > Regression.
Step 2: In Response, enter the column \%Fat.
Step 3: In Predictors, enter the columns Age.
Step 4: Click the OK button.

The MINITAB output will be:
Regression Analysis: FAT% versus AGE
Model Summary as follows:

The value of the test statistic is $5.19$, and the $p$-value is $0.000$, according to the MINITAB output.
Use the $\alpha =0.05$ significance threshold.
The $p$-value is lower than the level of significance in this case.
In other words, the $p$-value is $(=0.000)<\alpha (=0.05)$.
As a result of the rejection rule, it may be argued that at $\alpha =0.05$, there is evidence to reject the null hypothesis $\left(H_0\right)$.
Hence, the data support the conclusion that the predictor variable "years of adult" is useful for predicting "body fat" at the $5\%$ level.