Q. 14.80 Body Fat. In the paper "Total Bo... [FREE SOLUTION]

Chapter 14: Q. 14.80 (page 571)

Body Fat. In the paper "Total Body Composition by Dual-Photon ( ${}^{153}G i d$ ) 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.
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

Expert verified

a). As a result, the variables birds and scores do not violate assumptions $1 - 3$ for regression conclusions. As a result, the regression $t$ -test is appropriate for the supplied data.

b). As a result, the results support the conclusion that the predictor variable "years of adult" is beneficial for predicting "body fat" at the $5 %$ level.

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, enter $%$ Fat.

Step 3: In Predictors, type age into the columns.

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

Step 5: Click the OK button.

MINITAB Output:

Construction of normal probability of residuals using MINITAB

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

Step 2: In the Response column, Enter $%$ Fat.

Step 3: In Predictors, Enter Age into the columns.

Step 4: In Graphs, Enter normal probability plot of residuals.

Step 5: Click the OK button.

MINITAB Output:

Regression inferences assumptions

The following is the regression inferences assumptions:

Line of population regression:

For each value $X$ of the predicator variable, the response variable conditional mean $Y$ is $尾_{0} + 尾_{1} X$ .

Standard deviations are equal:

The response variable's $(Y)$ standard deviation is the same as the explanatory variable's $(X)$ standard deviation. $蟽$ 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.

- 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 trend.

- As a result, the variables birds and scores do not violate assumptions $1 - 3$ for regression conclusions. As a result, the regression $t$ -test is appropriate for the supplied data.

Appropriate Hypotheses (Part b)

The following are the suitable hypotheses:

Hypothesis of nullity:

$H_{0} : 尾_{0} = 0$

In other words, the predictor variable "age" is ineffective in predicting " $%$ fat."

Alternative hypothesis:

$H_{a} : 尾_{1} 鈮� 0$

That example, the predictor variable "age" can be used to forecast " $%$ fat."

Rule of Rejection:

Reject the null hypothesis $H_{0}$ . If $p - v a l u e 鈮� 伪 (= 0.05)$ .

Finding test statistics and p-value (Part b)

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

Step 2: In the Response column, Enter $%$ Fat.

Step 3: In Predictors, Enter Age into the columns.

Step 4: Click the OK button.

MINITAB Output:

Conclusion (Part b)

Use the $伪 = 0.05$ significance level.
The significance level is less than the $p$ -value.
Specifically, $p - v a l u e (= 0.000) < 伪 (= 0.05)$ .
As a result of the rejection rule, it may be argued that at $伪 = 0.05$ , there is evidence to reject the null hypothesis $H_{0}$ .
As a result, the results support the conclusion that the predictor variable "years of adult" is beneficial for predicting "body fat" at the $5 %$ level.