Question

One year Hank had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of 3.36 . Also, Jane had the lowest ERA of any female pitcher at the school with an ERA of 3.49 . For the males, the mean ERA was 4.449 and the standard deviation was 0.792 . For the females, the mean ERA was 5.091 and the standard deviation was 0.669 . Find their respective z-scores. Which player had the better year relative to their peers, Hank or Jane

Answer 1

Hank had a z-score of -1.375.

Jane had a z-score of -2.39.

Jane had the better year relative to their peers, due to her lower z-score.

Explanation:

Z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Hank:

ERA of 3.36. For the males, the mean ERA was 4.449 and the standard deviation was 0.792. This means that we have to find Z when
`X = 3.36, \mu = 4.449, \sigma = 0.792`. So

`Z = (X - \mu)/(\sigma)`

`Z = (3.36 - 4.449)/(0.792)`

`Z = -1.375`

Hank had a z-score of -1.375.

Jane

ERA of 3.49. For the females, the mean ERA was 5.091 and the standard deviation was 0.669. This means that we have to find Z when
`X = 3.49, \mu = 5.091, \sigma = 0.669`. So

`Z = (X - \mu)/(\sigma)`

`Z = (3.49 - 5.091)/(0.669)`

`Z = -2.39`

Jane had a z-score of -2.39.

Which player had the better year relative to their peers, Hank or Jane

Low ERA is good, high is bad. This means that whoever had the lower z-score had the better year, and in this case, it's Jane