More Olympic athletes In Exercises 5 and 11, you described the relationship between height (in inches) and weight (in pounds) for Olympic track and field athletes. The scatterplot shows this relationship, along with two regression lines. The regression line for the shotput, hammer throw, and discus throw athletes (blue squares) is y^=−115+5.13x. The regression line for the remaining athletes (black dots) is y^=−297+6.41x

a. How do the regression lines compare?

b. How much more do you expect a 72-inch discus thrower to weigh than a 72-inch sprinter?

It is given that:

$$\begin{gathered} \stackrel{⏜}{y}=-115+5.13x \\ \stackrel{⏜}{y}=-297+6.41x \end{gathered}$$

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

As a result, we can see in the scatterplot that the two regression lines are virtually parallel. The equations of the regression lines confirm this because $5.13$ and $6.41$ are fairly close together. However, the intercepts are extremely different since the constants $115$ and $297$ in the regression lines' equations are very different. This is also seen in the scatterplot, where the shotput, hammer throw, and discus throw athletes' regression lines are above the other regression lines.

It is given that:

$$\begin{gathered} \stackrel{⏜}{y}=-115+5.13x \\ \stackrel{⏜}{y}=-297+6.41x \end{gathered}$$

As a result, the weight of a discus thrower with a diameter of $72$ inches is computed as follows:

$$\begin{gathered} \stackrel{⏜}{y}=-115+5.13x \\ =-115+5.13\left(72\right) \\ =254.36 \end{gathered}$$

And the weight of a $72-$inch sprinter is calculated as:

$$\begin{gathered} \stackrel{⏜}{y}=-297+6.41x \\ =-297+6.41\left(72\right) \\ =164.52 \end{gathered}$$

Thus, the difference between the two predicted values is:

$$\begin{gathered} =254.36-164.52 \\ =89.84 \end{gathered}$$

As a result, we may estimate that a $72-$inch discus thrower will weigh $89.84$ pounds more than a $72-$inch printer.