Question

Transcribed image text:

One year Ted 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 2.52. For the males, the mean ERA was 3.998 and the standard deviation was 0.779 . For the females, the mean ERA was 3.899 and the standard deviation was 0.992 . Find their respective Z-scores. Which player had the better year relative to their peers, Ted or Amber? (Note: In general, the lower the ERA, the better the pitcher.)
Ted had an ERA with a z-score of
Amber had an ERA with a z-score of
(Round to two decimal places as needed.)

Answer 1

Ted's Z-score is -1.87 and Amber's Z-score is -1.39, indicating that Ted had a better year relative to his peers because his Z-score is lower, and in the context of ERA, a lower score is better.

Step-by-step explanation:

To calculate the Z-scores for both Ted and Amber, we use the Z-score formula:

Z = (X - μ) / σ

where X is the value for which we wish to calculate the Z-score, μ is the mean of the dataset, and σ is the standard deviation.

For Ted, with his ERA of 2.54, the mean ERA of 3.998 and standard deviation of 0.779 for males:

Z = (2.54 - 3.998) / 0.779

Z = -1.87 approximately (after rounding to two decimal places).

For Amber, with her ERA of 2.52, the mean ERA of 3.899 and standard deviation of 0.992 for females:

Z = (2.52 - 3.899) / 0.992

Z = -1.39 approximately (after rounding to two decimal places).

Since we are looking at ERAs where lower scores represent better performance, we must consider that a lower Z-score indicates a better performance relative to their peers. Ted's Z-score of -1.87 is lower than Amber's Z-score of -1.39, which means Ted had a better year relative to his peers.