I really need your help. Here's the problem:
A rookie is brought to a baseball club on the assumption that he will have a .300 batting average. (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 300 times and his batting average is .267. Assume that his at bats can be considered Bernoulli trials with probability .3 for successs. Could such a low average be considered just bad luck or should he be sent back to the minor leagues?
I know that, as his batting average was .267 in 300 times, he only hit 80 times.
I know I have to use the Central Limit Theorem, but how? First I thought I had to calculate, for example, the probability that he had hit between 75 to 85 times. Obviously knowing that this can be modeled by a Binomial(300,0.3) and aproximating this by a Normal (because n=300). So I could use the CLT that way.
But I think that what I really need to do is to give an interval, where I can assure, with high probability (90% for example), the real expectation is in that interval.
So, how I bound the real expectation using the CLT?
(Sorry if I wrote something wrong, English is not my first language)