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 2.54. ​Also, Amber had the lowest ERA of any female pitcher at the school with an ERA of 3.45. For the​ males, the mean ERA was 3.841 and the standard deviation was 0.989. For the​ females, the mean ERA was 4.113 and the standard deviation was 0.757. Find their respective​ z-scores. Which player had the better year relative to their​ peers, Hank or Amber​? ​(Note: In​ general, the lower the​ ERA, the better the​ pitcher.

Answer 1

Solution :

Given that ,

mean = = 3.841 (Hank)

standard deviation =  = 0.989

x = 2.54

Using z-score formula,

z = x - /   

z = 2.54 - 3.841 / 0.989

z = -1.32 (Hank)

mean = = 4.113 (Amber)

standard deviation =  = 0.757

x = 3.45

Using z-score formula,

z = x - /   

z = 3.45 - 4.113 / 0.757

z = -0.88 (Amber)

Hank player had the better year relative to their​ peers, because Hank z-score -1.32 is smaller than Amber z-score -0.88.