Question

It’s still early We expect that a baseball player who has a high batting average in the first month of the season will also have a high batting average for the rest of the season. Using 66 Major League Baseball players from a recent season,33 a least-squares regression line was calculated to predict rest-of-season batting average y from first-month batting average x. Note: A player’s batting average is the proportion of times at-bat that he gets a hit. A batting average over 0.300 is considered very good in Major League Baseball.

a. State the equation of the least-squares regression line if each player had the same batting average the rest of the season as he did in the first month of the season.

b. The actual equation of the least-squares regression line is y^=0.245+0.109x

Predict the rest-of-season batting average for a player who had a 0.200 batting average the first month of the season and for a player who had a 0.400 batting average the first month of the season.

c. Explain how your answers to part (b) illustrate regression to the mean.

Short answer

Part (a)$\stackrel{⏜}{y}=x$

Part (b) For a player with a $0.200$batting average in the first month of the season, the expected rest of the season batting average is $0.2668$For a player that had a $0.400$batting average in the first month of the season, the expected rest of the season batting average is $0.2886$

Part (c) The batting average prediction appears to be closer to the mean.

Step-by-step solution

Part (a) Step 1: Given information

The batting average of a player is the percentage of at-bats in which he obtains a hit. In Major League Baseball, a batting average of over $0.300$ is considered excellent.

Part (a) Step 2: Explanation

Let $x$represent the batting average for the first month of the season and y represent the batting average for the rest of the season.

The data for the explanatory variable $x$will be the same as the data for the response variable $y$ if each player has the same batting average in the first month as the rest of the season. If the data for the two variables is the same, the linear model will forecast that they are equal, and so the linear model will presume that the expected $y-$variable is equal to the predicted $x-$variable. As a result, the least square regression line looks like this:

$\overline{y}=x$

Part (b) Step 1: Explanation

The least-squares regression line is stated in the question as:

$\stackrel{⏜}{y}=0.245+0.109x$

As a result, for a player who has a$0.200$ batting average in the first month of the season, we will predict the remainder of the season's batting average by:

$$\begin{gathered} \stackrel{⏜}{y}=0.245+0.109x \\ =0.245+0.109(0.200) \\ =0.2668 \end{gathered}$$

Thus, for a player with a $0.200$batting average in the first month of the season, the expected remainder of the season batting average is $0.2668$

And for a player who has a batting average in the first month of the season, we'll predict his batting average for the rest of the season by:

$$\begin{gathered} \stackrel{⏜}{y}=0.245+0.109x \\ =0.245+0.109(0.400) \\ =0.2886 \end{gathered}$$

Thus, for a player who had a $0.400$ batting average in the first month of the season, the expected remainder of the season batting average is $0.2886$

Part (c) Step 1: Explanation

From part (b), we have,

For a player that had a $0.200$ batting average in the first month of the season, the expected rest of the season batting average is $0.2668$

For a player that had a $0.400$batting average in the first month of the season, the expected rest of the season batting average is $0.2886$

As a result, we expect the mean batting average to be between $0.200$and $0.300$as batting averages above $0.300$are regarded extremely good, and we expect the majority of the players to be good while only a few to be exceptional. The player who has a batting average of $0.200$in the first month has a batting average that is lower than the mean in the first month. And the player who has a batting average of $0.400$ in the first month has a batting average that is higher than the mean. Finally, we note that a player with a $0.200$ first-month batting average has a higher predicted batting average for the rest of the season, whereas a player with a $0.400$ first-month batting average has a lower predicted batting average for the rest of the season, indicating that the batting average prediction appears to be closer to the mean.