Question

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 the rest of the season. Using 66 Major League Baseball players from the 2010 season, 23 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. 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.

Answer 1

If each baseball player had the same batting average the entire season as in the first month, the least-squares regression line would be y = x. This represents a perfect one-to-one relationship, indicating no change in batting average over time.

Step-by-step explanation:

The question involves understanding the concept of a least-squares regression line in statistics, which is a tool used for predicting the relationship between two variables.

In the context of the question, if each baseball player had the same batting average the rest of the season as he did in the first month, the equation of the least-squares regression line would simply be y = x, which indicates a direct one-to-one relationship, meaning no change in performance. This is because the slope would be 1 (indicating that for every unit increase in x, y increases by the same amount), and the y-intercept would be 0 (since there is no fixed amount added to or subtracted from the batting average over time).

However, it's important to note that this scenario is hypothetical and does not reflect the actual complexities of predicting player performance based on early-season statistics, which would typically involve residual analysis and more data to establish trends.

Answer 2

The least-squares regression line, if each baseball player's batting average remained the same throughout the season as the first month's, would be y = x, indicating a one-to-one relationship with no changes over time.

Step-by-step explanation:

The question pertains to the application of least-squares regression in baseball statistics, specifically in predicting a player's batting average for the rest of the season based on the first month's performance. The student asks for the equation of the least-squares regression line if each baseball player's batting average would remain the same throughout the season as it was in the first month. In this hypothetical scenario, since the rest-of-season batting average would be exactly the same as the first-month batting average for each player, the least-squares regression line would have a slope of 1 (since there is a one-to-one relationship) and a y-intercept of 0 (as there's no difference to be accounted for). The resulting equation of the regression line would simply be y = x, indicating a direct correlation where no change is expected.