Chapter 8, Problem 47E

(a)

To determine

To explain why you cannot use that model to estimate the fat content of a burger with $600$ calories.

Explanation of Solution

From the previous exercise we have that we are examining the association between the fat and calories in fast-food hamburgers. Thus, the data is as below:

Fat	Calories
19	410
31	580
34	590
35	570
39	640
39	680
43	660

Also, it is given that,

  $\begin{array}{l}r=0.9606\\ {\mu }_1=590\\ {\mu }_2=34\\ {\sigma }_1=89.81\\ {\sigma }_2=7.80\end{array}$

And the regression line is:

  $\text{<mover accent="true"><mo>C</mo><mo>^</mo></mover> alories}=211+11.056(\text{Fat)}$

Thus, we can say that we cannot use that model to estimate the fat content of a burger with $600$ calories because the regression was for predicting calories from fat not the other way round. So, it will not be appropriate as the regression line we are using has fat as an independent variable and we have to find out the value from the equation which has the fat as an dependent variable on calories.

(b)

To determine

To estimate the fat content of a burger with $600$ calories using an appropriate model.

Answer to Problem 47E

The fat content of a burger with $600$ calories is $34.8$ grams.

Explanation of Solution

From the previous exercise we have that we are examining the association between the fat and calories in fast-food hamburgers. Thus, the data is as below:

Fat	Calories
19	410
31	580
34	590
35	570
39	640
39	680
43	660

Now, we have that,

  $\begin{array}{l}r=0.9606\\ {\mu }_1=34\\ {\mu }_2=590\\ {\sigma }_1=7.80\\ {\sigma }_2=89.81\end{array}$

And the new regression line will be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.9606\times \frac{7.80}{89.81}\\ =0.083\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =34-0.083\times 590\\ =-14.96\\ =-15\end{array}$

  $\begin{array}{l}\text{<mover accent="true"><mo>F</mo><mo>^</mo></mover> at}=\alpha +\beta (\text{Calories)}\\ =-15+0.083(\text{Calories)}\end{array}$

Thus, the fat content of a burger with $600$ calories is as:

  $\begin{array}{l}\text{<mover accent="true"><mo>F</mo><mo>^</mo></mover> at}=\alpha +\beta (\text{Calories)}\\ =-15+0.083(\text{Calories)}\\ =-15+0.083\times 600\\ =34.8\end{array}$

The fat content of a burger with $600$ calories is $34.8$ grams.