Chapter 8, Problem 10RE

Variability in Baseball Suppose, during the course of a typical season, a batter has 500 at-bats. This means the player has the opportunity to get a hit 500 times during the course of a season. Further, suppose a batter is a career 0.280 hitter (he averages 280 hits every 1000 at-bats or he has 280 successes in 1000 trials of the experiment), so the population proportion of hits is 0.280.

1. a. Assuming each at-bat is an independent event, describe the sampling distribution of $\hat{p}$, the proportion of hits in 500 at-bats over the course of a season.
2. b. Would it be unusual for a player who is a career 0.280 hitter to have a season in which he hits at least 0.310?
3. c. Would it be unusual for the player who hit 0.310 one season to hit below 0.255 the following season?
4. d. Explain why a career 0.280 hitter could easily have a batting average between 0.260 and 0.300.
5. e. Use the result of part (d) to explain that a player who hit 0.260 in a season may not be a worse player than one who hit 0.300.