Question

One year Ron 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 3.06. Also, Karla had the lowest ERA of any female pitcher at the

school with an ERA of 3.28. For the males, the mean ERA was 5.086 and the standard deviation was 0.998. For

the females, the mean ERA was 4.316 and the standard deviation was 0.558. Find their respective Z-scores.

Which player had the better year relative to their peers, Ron or Karla? (Note: In general, the lower the ERA, the

better the pitcher.)

Answer 1

Ron's ERA has a z-score of -2.03.

Karla's ERA has a z-score of -1.86.

Due to the lower z-score, Ron had a better yean than Karla relative to their peers.

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.

In this question:

Since the lower the ERA, the better the pitcher, whoever's ERA has the lower z-score had the better year relative to their peers.

Ron

ERA of 3.06, so
`X = 3.06`

For the males, the mean ERA was 5.086 and the standard deviation was 0.998. This means that
`\mu = 5.086, \sigma = 0.998`

So

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

`Z = (3.06 - 5.086)/(0.998)`

`Z = -2.03`

Ron's ERA has a z-score of -2.03.

Karla

ERA of 3.28, so
`X = 3.28`

For the females, the mean ERA was 4.316 and the standard deviation was 0.558. This means that
`\mu = 4.316, \sigma = 0.558`

So

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

`Z = (3.28 - 4.316)/(0.558)`

`Z = -1.86`

Karla's ERA has a z-score of -1.86.

Which player had the better year relative to their peers, Ron or Karla?

Due to the lower z-score, Ron had a better yean than Karla relative to their peers.