Question

One year Ron had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched)

Answer 1

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 $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $X = 3.06$

For the males, the mean ERA was 5.086 and the standard deviation was 0.998. This means that $\mu = 5.086, \sigma = 0.998$

So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.06 - 5.086}{0.998}$

$Z = -2.03$

Ron's ERA has a z-score of -2.03.

Karla

ERA of 3.28, so $X = 3.28$

For  the females, the mean ERA was 4.316 and the standard deviation was 0.558. This means that $\mu = 4.316, \sigma = 0.558$

So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{3.28 - 4.316}{0.558}$

$Z = -1.86$

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.