Chapter 8, Problem 48E

(a)

To determine

To explain why you cannot use that model to estimate the fat content of a $400$ calorie sandwich.

Explanation of Solution

From the previous question it is given that the association between the fat and calories of the sandwiches is given. The scatterplot and the mean and standard deviation of the fat and calories is also given. So, we have given that:

  $\begin{array}{l}r=0.947\\ {\mu }_1=472.7\\ {\mu }_2=20.6\\ {\sigma }_1=144.2\\ {\sigma }_2=9.8\end{array}$

And the regression line is:

  $\text{<mover accent="true"><mo>C</mo><mo>^</mo></mover> alories}=185.651+13.934(\text{Fat)}$

Thus, we can say that we cannot use that model to estimate the fat content of a sandwich with $400$ 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 a dependent variable on calories.

(b)

To determine

To make that estimate using the appropriate model.

Answer to Problem 48E

The fat content of a $400$ calorie sandwich is $16.09$ .

Explanation of Solution

From the previous question it is given that the association between the fat and calories of the sandwiches is given. The scatterplot and the mean and standard deviation of the fat and calories is also given. So, now we have,

  $\begin{array}{l}r=0.947\\ {\mu }_1=20.6\\ {\mu }_2=472.7\\ {\sigma }_1=9.8\\ {\sigma }_2=144.2\end{array}$

Thus, the regression model can be calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.947\times \frac{9.8}{144.2}\\ =0.083\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =20.6-0.064\times 472.7\\ =-9.65\\ \text{<mover accent="true"><mo>F</mo><mo>^</mo></mover> at}=\alpha +\beta (\text{Calories)}\\ =-9.65+0.083(\text{Calories)}\end{array}$

Thus, the fat content of a $400$ calorie sandwich is:

  $\begin{array}{l}\text{<mover accent="true"><mo>F</mo><mo>^</mo></mover> at}=\alpha +\beta (\text{Calories)}\\ =-9.65+0.083(\text{Calories)}\\ =-9.65+0.083\times 400\\ =16.09\end{array}$

The fat content of a $400$ calorie sandwich is $16.09$ .