Chapter 3.2, Problem 46E

(a)

To determine

To explain how do you the regression lines compare.

Explanation of Solution

It is given in the question that:

  $\begin{array}{l}\widehat{y}=170+11.8x\\ \widehat{y}=88+24.5x\end{array}$

In the given scatterplot, we note that the regression line of the drink products is much steeper than the regression line of the food products. This is confirmed by the equations of the regression lines because $24.5$ is much higher than $11.8$ . We also note that the intercept of the food products is much higher than the intercept of the drink products because $170$ is much higher than $88$ in the equations of the regression lines. This is also noticeable in the scatterplot because the regression line for the food products lies higher than the regression line for the drink product on the left side of the graph.

(b)

To determine

To explain how much more calories do you expect to find in a food item with $5$ grams of fat compared to the drink item with $5$ grams of fat.

Answer to Problem 46E

We expect to find $18.5$ calories more in a food item with $5$ grams of fat compared to the drink item with $5$ grams of fat.

Explanation of Solution

It is given in the question that:

  $\begin{array}{l}\widehat{y}=170+11.8x\\ \widehat{y}=88+24.5x\end{array}$

Thus, the amount of calories we expect to find in a food item with $5$ grams of fat is calculated as:

  $\begin{array}{l}\widehat{y}=170+11.8x\\ =170+11.8(5)\\ =229\end{array}$

And the amount of calories we expect to find in a food item with $5$ grams of fat is calculated as:

  $\begin{array}{l}\widehat{y}=88+24.5x\\ =88+24.5(5)\\ =210.5\end{array}$

Now, the difference between the two predicted values is:

  $\begin{array}{l}=229-210.5\\ =18.5\end{array}$

This then implies that we expect to find $18.5$ calories more in a food item with $5$ grams of fat compared to the drink item with $5$ grams of fat.