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Answer:
(a) The <em>z</em>-score of Roger is -1.56 and the <em>z</em>-score of Amber is -1.13.
(b) Roger had a better year relative to his peers.
Step-by-step explanation:
If X follows N (<em>µ, σ</em>₂), then , is a standard normal variate with mean, E (Z) = 0 and Var (Z) = 1. That is, Z follows N (0, 1).
Let <em>X</em> = ERA of male pitchers and <em>Y</em> = ERA of ERA of female pitchers.
It is provided that the mean and standard deviation of <em>X</em> are:
Also, the mean and standard deviation of <em>Y</em> are:
ER of Roger is 3.14 and ERA of Amber is 3.38.
(1)
Compute the <em>z</em>-score of Roger as follows:
Thus, the <em>z</em>-score of Roger is -1.56.
Compute the <em>z</em>-score of Amber as follows:
Thus, the <em>z</em>-score of Amber is -1.13.
(2)
Compute the probability of ERA's that are greater than Roger's ERA as follows:
This implies that 94% of the other male pitchers had an ERA higher than 3.14.
Compute the probability of ERA's that are greater than Amber's ERA as follows:
This implies that 87% of the other female pitchers had an ERA higher than 3.38.
As it is provided that the lower the ERA the better the pitcher, then it can be concluded that Roger had a better year relative to his peers.