4) In baseball, people frequently calculate players\' batting averages. The batt

Question

4) In baseball, people frequently calculate players' batting averages. The batting average is defined as 1000 * nhits/nat bats. Assume that for every at bat a certain player has a 30% chance of getting a hit (expected batting average 300) and that the probability of getting a hit at every at bat is independent of the others. Also assume that the player gets exactly 20 at bats each week and that there are 25 weeks in the baseball season. For this player, what is the expected (mean) number of hits per week? Using the square root rule for a counting experiment, what is the expected range in the number of hits per week? Using the result above, what is the expected range in the batting average calculated weekly? The typical MLB player has a batting average of 250 and the best average is usually around 325. How many weeks would you have to observe a 250 hitter and a 325 hitter before the separation between their expected average (75) is greater than the sum of the expected errors due to counting? a. b. c. d.

Explanation / Answer

a. Expected number of hits per week = 20* 0.3 = 6 Hits

b. Expected range in the number of hits per week = +- *sqrt() = 6 +- * sqrt(6) = (3.55 , 8.45) so 3.55 to 8.45 hits are range.

c. so Expected range of batting average calculated weekly

Lower Value of range = 3.55 / 20 * 1000 = 178

Upper value of range = 8.45/20 * 1000 = 423

d. Batting average = 250 ; best average = 325

Weekly hit expected average for typical MLB player = (250/1000) * 20 = 5

Weekly hit expected average for typical MLB player = (325/1000) * 20 = 6.5

so we have to find the number of weeks for which that difference will be greater than 1.5

Pr( Diff. > 1.5 ; 6; 2.45) = Pr(MLB player > 6.5; 6; 2.45) - Pr(MLB Player < 5; 2.45)

= 0.424 - 0.285 = 0.139

so expected number of weeks = 0.139 * 20 = 2.78 or say 3 weeks

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