Q 4.114 - Elementary Statistics

Question

a. Decide whether finding a regression line for the data is reasonable. If so, then also do parts (b)-(d).

b. Obtain the coefficient of determination.

c. Determine the percentage of variation in the observed values of the response variable explained by the regression, and interpret your answer.

d. State how useful the regression equation appears to be for making predictions.

 Body Fat. In the paper "Total Body Composition by DualPhoton ( ${}^{153}\mathrm{Gd}$ ) Absorptiometry" (American Joumal of Climical 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 Wciss Stats site.

Step-by-Step Solution

Answer

a )Because the observations are spread around a line in the scatterplot above, it is fair to find a regression line for the data.

b) The coefficient of determination is 0.627. 

c)As a result, the regression explains $62.7\%$ of the variation in the observed values of the response variable. It may be deduced that the linear association between age $\left(x\right)$ and percent  fat $\left(y\right)$ accounts for$62.7\%$ of the variability in the percent  fat.

d)The regression equation is not useful in making predictions because the percentage of variation in the observed values of the response variable explained by the regression is$62.7\%$ which is greater than $50 \%.$.

Part a Step 1 Given Information

 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 Wciss Stats site. 

Part a Step 2 Explanation

The scatterplot for theprovided data can be drawn by using the MINITAB:

Step 1: Choose Graph > Scatterplot

Step 2: Choose With Connect Line, and then click OK.

Step 3: Under $\mathbf{Y}$ variables, enter a column of \%FAT

Step 4: Under X variables, enter a column of AGE

Step 5: Click OK

The scatterplot obtained will be

Because the observations are spread around a line in the scatterplot above, it is fair to find a regression line for the data.

Part b Step 1 Given Information

 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 Wciss Stats site. 

Part b Step 2 Explanation

The coefficient of determination can be calculatedby using MINITAB as follows:

Step 1: Choose Stat > Regression >Regression

Step 2: In Response, enter the column \%FAT

Step 3: In Predictors, enter the column AGE

Step 4: Click OK

The MINITAB output will be:

As a result, the coefficient of determination is 0.627.

Part c Step 1 Given Information

 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 Wciss Stats site. 

Part c Step 2 Explanation

The correlation coefficient will be$0.627.$

As a result, the regression explains $62.7\%$ of the variation in the observed values of the response variable. It may be deduced that the linear association between age $\left(x\right)$ and percent $ fat $\left(y\right)$ accounts for $62.7\%$ of the variability in the percent  fat.

Part d Step 1 Given Information

 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 Wciss Stats site. 

Part d Step 2 Explanation

The coefficient of determination will be $0.627$

Therefore, the percentage of variation in the observed values of the response variable explained by the regression is $62.7\%$. It can be interpreted as the$62.7\%$ of the variability in the $\%$fat is accounted for the linear relationship between the age $\left(x\right)$ and $\%$fat (y)

The regression equation is not useful in making predictions because the percentage of variation in the observed values of the response variable explained by the regression i$62.7\%$ which is greater than $50\%$.