Question

More Starbucks In Exercises 6 and 12, you described the relationship between fat (on Page Number: 204 grams) and the number of calories in products sold at Starbucks. The scatterplot shows this relationship, along with two regression lines. The regression line for the food products (blue squares) is y^=170+11.8x. The regression line for the drink products

(black dots) is y^=88+24.5x

a. How do the regression lines compare?

b. How many more calories do you expect to find in a food item with 5 grams of fat compared to a drink item with 5 grams of fat?

Short answer

Part (a) 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.

Part (b) 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.

Step-by-step solution

Part (a) Step 1: Given information

It is given in the question that:

$$\begin{gathered} \stackrel{⏜}{y}=170+11.8x \\ \stackrel{⏜}{y}=88+24.5x \end{gathered}$$

Part (a) Step 2: Concept

The least-squares regression line reduces the sum of squares of vertical distances between the observed points and the line to zero.

Part (a) Step 3: Explanation

The regression line for drink products is substantially steeper than the regression line for food goods, as shown in the scatterplot. The equations of the regression lines confirm this because $24.5$ is substantially greater than $11.8$ In the equations of the regression lines, $170$ is substantially higher than $88$ hence the intercept of the food goods is much greater than the intercept of the drink products. This is also seen in the scatterplot, where the regression line for food goods is higher on the left side of the graph than the regression line for drink products.

Part (b) Step 1: Calculation

It is given in the question:

$$\begin{gathered} \stackrel{⏜}{y}=170+11.8x \\ \stackrel{⏜}{y}=88+24.5x \end{gathered}$$

As a result, the number of calories we should expect to discover in a food item containing $5$grams of fat is estimated as follows:

$$\begin{gathered} \stackrel{⏜}{y}=170+11.8x \\ =170+11.8\left(5\right) \\ =229 \end{gathered}$$

And the number of calories in a food item containing $5$grams of fat is calculated as follows:

$$\begin{gathered} \stackrel{⏜}{y}=88+24.5x \\ =88+24.5\left(5\right) \\ =210.5 \end{gathered}$$

Now, the difference between the two predicted values is:

$$\begin{gathered} =229-210.5 \\ =18.5 \end{gathered}$$

This means that a food item containing $5$grams of fat will contain $18.5$calories more than a drink containing $5$ grams of fat.