Question

One year josh 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 2.89. ​Also,alice had the lowest ERA of any female pitcher at the school with an ERA of 3.31 . For the​ males, the mean ERA was 5.083 and the standard deviation was 0.672. For the​ females, the mean ERA was 4.032 and the standard deviation was 0.649. Find their respective​ z-scores. Which player had the better year relative to their​ peers, josh or alice ​? ​(Note: In​ general, the lower the​ ERA, the better the​ pitcher.)

Answer 1

Josh's ERA had a z-score of -3.26.

Alice's ERA had a z-score of -1.11.

Due to the lower z-score(ERA is a stat that the lower the better), Josh had a better year relative to his peers.

Explanation:

Z-score:

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Josh:

ERA of 2.89, mean of 5.083, standard deviation of 0.672. So

`X = 2.89, \mu = 5.083, \sigma = 0.672`, and the z-score is:

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

`Z = (2.89 - 5.083)/(0.672)`

`Z =  -3.26`

Josh's ERA had a z-score of -3.26.

Alice:

ERA of 3.31, mean of 4.032, standard deviation of 0.649. So

`X = 3.31, \mu = 4.032, \sigma = 0.649`, and the z-score is:

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

`Z = (3.31 - 4.032)/(0.649)`

`Z =  -1.11`

Alice's ERA had a z-score of -1.11.

Which player had the better year relative to their​ peers, josh or alice ?

Due to the lower z-score(ERA is a stat that the lower the better), Josh had a better year relative to his peers.