Chapter 13, Problem 81PS

A baseball analyst would like to study various team statistic for a recent season to determine which variables might be useful to predicting the number of winds achieved by teams during the season. He begins by using a team’s earned run average (ERA), a measure of pitching performance, to predict the number of wins. He collects the team ERA and team wins for each of the 30 Major League Baseball teams and stores these data in Baseball. (Hint: First determine which are the independent and dependent variables.)

a. Assuming a linear relationship, use the least-squares method to compute the regression coefficient $b_0$ and $b_1.$

b. Interpret the meaning of $Y$ intercept, $b_0,$ and the slope, $b_1,$ in this problem.

c. Use the prediction line developed in (a) to predict the mean number of wins for a team with an ERA of 4.50.

d. Compute the coefficient of determination, $r^2,$ and interpret its meaning.

e. Perform a residual analysis on your results and determine the adequacy of the fit of the model.

f. At the 0.05 level of significance, is there evidence of a linear relationship between the number of wins and the ERA?

g. Construct a $95\%$ confidence interval estimate of the mean number of wins expected for teams with an ERA of 4.50.

h. Construct a $95\%$ prediction interval of the number of wins for an individual team that has an ERA of 4.50.

i. Construct a $95\%$ confidence interval estimate of the population slope.

j. The 30 teams constitute a population. In order to use statistical inference, as in (f) through (i), the data must be assumed to represent a random sample. What “population� would this sample be drawing conclusions about?

k. What other independent variables might you consider for inclusion in the model?

l. What conclusions can you reach concerning the relationship between ERA And wins?