Question

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.

Christian’s Response:

To find whether the rookie’s .267 a sign of bad luck or bad performance, we can check the p-value to monitor it’s significance. In this problem I will be using a one-sample binomial test to check for significance.

x <- 80 # successful hits
n <- 300 # times the rookie bat
p0 <- 0.3 # batting avg


binom.test(x,n,p=p0, alternative = 'less')
## 
##  Exact binomial test
## 
## data:  x and n
## number of successes = 80, number of trials = 300, p-value = 0.1149
## alternative hypothesis: true probability of success is less than 0.3
## 95 percent confidence interval:
##  0.0000000 0.3119456
## sample estimates:
## probability of success 
##              0.2666667

We received a p-value of 0.1149, which is not statistically significant. This means that the batting average of .267 can be attributed to chance. In this case, we can likely attribute this to bad luck. Though our p-value is insignificant, it is still important to look at variables that may have lead to this poor performance, such as fatigue and load management.