During the course of a typical season, suppose 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.

(a) Assuming each at-bat is an independent event, describe the sampling distribution of , the proportion of hits in 500 at-bats over the course of a season.

(b) Would it be unusual for a player who is a career 0.280 hitter to have a season in which he hits 0.310?
(c) Would it be unusual for the player who hit 0.310 one season to hit below 0.255 the following season?
(d) Explain why a career 0.280 hitter could easily have a batting average between 0.260 and 0.300.
(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.