A rookie is brought to a baseball club on the assumption that he will have a 0.300 batting average based on his past performance. (Batting average is the ratio of the number of hits to the number of times at bat.) In the first year, he comes to bat 400 times during the season and his batting average is .348. Assume that his at-bats can be considered Bernoulli trials with probability 0.3 for success. Give both qualitative and quantitative arguments about what is likely to happen to the player’s batting performance next season. Be sure to discuss the merits of any assumptions that are made.
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Given: he will have a 0.300 batting average based on his past performance, In the first year, he comes to bat 400 times during the season and his batting average is 0.348
To find: a probability to show a batting average of 0.348
Solution:
- Now we have given that he will have a 0.300 batting average based on his past performance.
- So in order to get batting average of 0.348, rookie had 139 hits in the 400 times he was at bats. Let X be the number of times he hits and - X has a Binomial distribution with n = 400 and p = 0.3.
- The probability of getting a batting average of 0.348 or less is then the probability to have 139 hits or less in 400 times at bats:, that is
- P(X ≤ 139) = B(400,0.3)(139)
- Now, we won’t find this probability in a table, as n is so large.
- Instead, we are using a normal approximation.
B(400,0.3)(130) = N(400·0.3,400·0.3·0.7)
- Then:
P(X ≤ 139) = B(400,0.3)(80) = N(120,84)(80)
= N(0,1)( 139 − 120 / √ 84 )
= N(0,1)(2.07)
= 1 − N(0,1)(2.07)
Answer:
The rookie therefore has a probability of approx. 1 − N(0,1)(2.07) % to show a batting average of 0.348
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