Answer:
Ron's ERA has a z-score of -2.03.
Karla's ERA has a z-score of -1.86.
Due to the lower z-score, Ron had a better yean than Karla relative to their peers.
Step-by-step explanation:
Z-score:
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
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 pvalue, we get the probability that the value of the measure is greater than X.
In this question:
Since the lower the ERA, the better the pitcher, whoever's ERA has the lower z-score had the better year relative to their peers.
Ron
ERA of 3.06, so ![X = 3.06](https://tex.z-dn.net/?f=X%20%3D%203.06)
For the males, the mean ERA was 5.086 and the standard deviation was 0.998. This means that ![\mu = 5.086, \sigma = 0.998](https://tex.z-dn.net/?f=%5Cmu%20%3D%205.086%2C%20%5Csigma%20%3D%200.998)
So
![Z = -2.03](https://tex.z-dn.net/?f=Z%20%3D%20-2.03)
Ron's ERA has a z-score of -2.03.
Karla
ERA of 3.28, so ![X = 3.28](https://tex.z-dn.net/?f=X%20%3D%203.28)
For the females, the mean ERA was 4.316 and the standard deviation was 0.558. This means that ![\mu = 4.316, \sigma = 0.558](https://tex.z-dn.net/?f=%5Cmu%20%3D%204.316%2C%20%5Csigma%20%3D%200.558)
So
![Z = -1.86](https://tex.z-dn.net/?f=Z%20%3D%20-1.86)
Karla's ERA has a z-score of -1.86.
Which player had the better year relative to their peers, Ron or Karla?
Due to the lower z-score, Ron had a better yean than Karla relative to their peers.