Question

Body Fat. The data from Exercise 14.80 for age and body fat of 18 randomly selected adults are on the WeissStats site. Specified value of the predictor variable: 30 years.

a. Decide whether you can reasonably apply the conditional mean and predicfed value t-interval procedures to the data. If so, then also do parts (b) - (f).

b. Determine and interpret a point estimate for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

c. Find and interpret a 95% confidence interval for the conditional mean of the response variable corresponding to the specified value of the predictor variable.

d. Determine and interpret the predicted value of the response variable corresponding to the specified value of the predictor variable.

e. Find and interpret a 95 % prediction interval for the value of the response variable corresponding to the specified value of the prediclor variable.

f. Compare and discuss the differences between the confidence interval that you obrained in part (c) and the prediction interval that you obtained in part (e).

Short answer

Part a)it is reasonably applying the concitional mean and predated value $t$-interval procedure for given data.

Part b)the point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=30$is $19.66\%$.

Part c)There is $90\%$confident that the conditional mean body $\%$fat with 30 years old adults lies between $15.01\%$and $24.31\%$

Part d)the predicted value of the score corresponding to the predictor variable $x=30$is 19.66 %.

Part e)There is 95 \% certain that body $\%$fat with 30 years old adults lies between 6.61 \% and 32.71 %

Part f) the prediction interval is greater than the confidence interval

Step-by-step solution

Step 1:Given information

The data from Exercise 14.80 for age and body fat of 18 randomly selected adults are on the WeissStats site. Specified value of the predictor variable: 30 years.

Step 2:Explaination Part a)

Check whether or not it is reasonably apply the conditional mean and predated value t-interval procedure to data 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 resicuals, it is clear that the residuals are roughly linear patterri.

Hence, the assumptions $1-3$for the regression inferences are not violated for the variables birdles and scores. Hence, the regression t-test is reasonable to apply tor the given data.

Theretore, it is reasonably applying the concitional mean and predated value $t$-interval procedure for given data.

Step 2:Explaination Part b)

Finding the point estimate for the conditional mean of the response varlable corresponding to the predictor variable$x=30$by using MINITAB.

MINITAB procedure:

Step 1; Choose Stat > Regression > Regresslon.

Step 2; In Response, enter the column$\%$Fat.

Step 3: In Predictors, enter the columns Age.

Step 4: In Options,enter $30$under Preciction interval for new obserwations.

Step 5: In Confidence Level, enter $95$

Step f: In Storage, Choose Fits, Confidence limits, SEs of fils, and Prediction limits.

Step 7; ClickOK.

MINITAB output:

Prediction for $\%$FAT

From the MINITAB output, the point estimate for the conditional mean of the response variable corresponding to the predictor variable $x=30$ is$19.66\%$

Step 2:Explaination Part c)

Compute the $95\%$confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=30$by using MINITAB.

From the MINITAB output in part (b), the $95\%$confidence interval for the conditional mean of the response variable corresponding to the predictor variable $x=30$is $(15.01,24.31)$.

Interpretation:

There is $95\%$confident that the conditional mean body \%fat with 30 years old adults lies between $15.01\%$and$24.31\%$

Step 2:Explaination Part d)

Compute the predicted value of the response variable corresponding to the predictor variable $x=30$by using MINITAB.

From the MINITAB output in part (b), the predicted value of the score corresponding to the predictor variable $x=30$is $19.66\%$

Step 2:Explaination Part e)

Compute the $95\%$prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=30$by using MINITAB.

From the MINITAB output in part (b), the 95 % prediction interval for the conditional mean of the response variable corresponding to the predictor variable $x=30$is $(6.61,32.71)$.

Interpretation:

There is 95 % certain that body $\%$fat with 30 years old adults lies between 6.61 % and 32.71 %

Step 2:Explaination Part f)

Comparison:

From part (c) and part (e), it is clear that the confidence interval and the prediction interval are centered at the predicted value of 30 years old adults. Moreover, the prediction interval is greater than the confidence interval.