Discussion week9

Week 9, CLT & Gen Fnct

Fundamentals of Computational Mathematics

CUNY MSDS DATA 605, Fall 2018

Rose Koh

10/24/2018

Chapter 9,10

Section 9.1.5

5. 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 success. Could such a low average be considered just bad luck or should he be sent back to the minor leagues? Comment on the assumption of Bernoulli trials in this situation.

Let \(X\) be the number of times he hits - X has a Binomial distribution with n=300, p = 0.3

The probability to get a batting average of 0.267 or less is then the probability to have 80 hits or less in 300times at bats: \(P(X \leq 80) = B_{300,0.3}(80)\)

We use a normal approximation: \(B_{300,0.3} \approx N_{300*0.3, 300*0.3*0.7}\)

So,

\(P(X \leq 80) = B_{300,0.3}(80) \approx N_{90,63}(80)\)

\(= N_{0,1}(\frac{80-90}{\sqrt{63}}\)

\(= N_{0,1}(-1.26)\)

\(= 1- N_{0,1}(1.26)\)

\(= 1- 0.9131\)

\(=0.0869\)

The Rookie has a prob. of approx. 9% to show a batting averrage of 0.267, even if his true standard is 0.3. The low average is probably not due to bad luck. but rather poor performance.