Chapter 8, Problem 46E

(a)

To determine

To explain do you think a linear model is appropriate in this situation.

Answer to Problem 46E

Yes, the linear model is appropriate.

Explanation of Solution

In the question 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 can say that the linear model for this situation is appropriate because as we see in the scatterplot of the fat and calories that it shows they have a positive and fairly strong linear relationship as the points in the scatterplot shows the upward direction and is less scattered.

(b)

To determine

To describe the strength of this association.

Answer to Problem 46E

The strength of this association is strong.

Explanation of Solution

In the question 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. Thus, we can say about the strength of the association between the fat and the calories of the sandwich that the relationship between the two is strong because the value of the correlation coefficient is nearly to one. As it is given that $r=0.947$ .

(c)

To determine

To write the equation of the regression line to estimate calories from the fat content.

Answer to Problem 46E

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

Explanation of Solution

In the question 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:

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

Thus, the equation of the regression line to estimate calories from the fat content is calculated as:

  $\begin{array}{l}\beta =r\times \frac{{\sigma }_1}{{\sigma }_2}\\ =0.947\times \frac{144.2}{9.8}\\ =13.934\\ \alpha ={\mu }_1-\beta \times {\mu }_2\\ =472.7-13.934\times 20.6\\ =185.651\\ \text{<mover accent="true"><mo>C</mo><mo>^</mo></mover> alories}=\alpha +\beta (\text{Fat)}\\ =185.651+13.934(\text{Fat)}\end{array}$

(d)

To determine

To explain the meaning of the slope.

Explanation of Solution

In the question 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, the slope of the regression line interprets that every additional gram of fat adds, on average, $13.934$ calories in sandwiches.

(e)

To determine

To explain the meaning of the y -intercept.

Explanation of Solution

In the question 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 interpret from the y -intercept of the regression line that a fat-free sandwich should have $185.651$ calories which is not practically possible as there is nothing that value zero.

(f)

To determine

To explain what does it mean if a certain sandwich has a negative residual.

Explanation of Solution

In the question 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:

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

As we know that the residual is calculated as by taking the difference between the actual value minus the predicted value. Thus, if a certain sandwich has negative residual then the sandwich has fewer calories than is predicted by the linear model. And it could have a wrong effect.