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.)

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

[tex]X = 2.89, \mu = 5.083, \sigma = 0.672[/tex], and the z-score is:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{2.89 - 5.083}{0.672}[/tex]

[tex]Z = -3.26[/tex]

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

[tex]X = 3.31, \mu = 4.032, \sigma = 0.649[/tex], and the z-score is:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{3.31 - 4.032}{0.649}[/tex]

[tex]Z = -1.11[/tex]

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.