Link to Problem: HERE
A rookie is brought to a baseball club on the assumption that he will have a \(.300\) batting average. 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?
Assuming the rookie has a batting average of \(.300\) and he has had 300 at bats means that we can model the distribution with bernoulli trials methodology.
Expected Number of Hits: \(.3(300)=90\) Standard Deviation: \(\sqrt{300(.3)(.7)}=7.93\)
Actual Number of Hits: \(.267(300)=80.1\)
\(P(S_{80.1} < 90)=P(S^*_{80.1} < \frac{80.1-90}{7.93})=P(S^*_{80.1} <-1.25)\)
This can be calculated directly or using the table provided by the text book. According to Figure 9.4, an approximation of \(1-(0.5+ .3849)=0.1151\) is reasonable.
As to the question of whether or not this indicates that the rookie is not actually a \(.300\) batter, this is a bit more difficult. Being 1.25 standard deviations below the mean only has approiximately an 11% chance of happening given our priors. This is a bit unlikely, but not impossible so. It is within the 95% confidence interval.
Comment on the assumption of Bernoulli trials in this situation.
In order to model this situation, we must assume that every at bat is independent of each other. It is unclear whether or not this is a reasonable assumption. There are a number of game situations that may change how each at bat progresses and strategies signalled by the team may also result in a difference in the players at bats. Furthermore, there may be a psychological aspect at play when a player considers themselves on a hot streak or in a slump.