Question

One year Thomas had the lowest ERA (earned-run average, mean number of runs yielded per nine inning pitched) of any male pitcher at his school, with an ERA of 3.31. Also, Karla had the lowest ERA of any female pitcher at the school with an ERA of 3.02. For the males, the mean ERA was 4.837 and the standard deviation was 0.541. For Females, the mean ERA was 4.533 and the standard deviation was 0.539. Find their respective z-scores. Which player had the better year relative to their peers, Thomas or Karla? (Note: In general, the lower the ERA, the better the pitcher.)


Thomas had an ERA with a z-score of?_______


Karla had an ERA with a z-score of? ________


Which player had a better year in comparison with their peers?

Answer 1

Answer:

Thomas had an ERA with a z-score of -2.82.

Karla had an ERA with a z-score of -2.81.

Thomas had the better year compared with his peers, since his ERA had the lower Zscore.

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

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 ERA is the Earned Runs Average per 9 innings. This mean that the lower the ERA is, the better it is. So, between Thomas and Karla, whoever has the lower Zscore had the better season.

Thomas

For the males, the mean ERA was 4.837 and the standard deviation was 0.541. This means that $\mu = 4.837, \sigma = 0.541$.

Thomas ERA was 3.31. This means that $X = 3.31$

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

$Z = \frac{3.31 - 4.837}{0.541}$

$Z = -2.82$

Thomas had an ERA with a z-score of -2.82.

Karla

For Females, the mean ERA was 4.533 and the standard deviation was 0.539. This means that $\mu = 4.533, \sigma = 0.539$.

Karla ERA was 3.02. So $X = 3.02$.

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

$Z = \frac{3.02 - 4.533}{0.539}$

$Z = -2.81$

Karla had an ERA with a z-score of -2.81

Which player had a better year in comparison with their peers?

Thomas had the better year compared with his peers, since his ERA had the lower Zscore.

Answer 2

The Zscores and performance of Karla and Thomas are :

• Thomas Zscore = - 2.823
• Karla Zscore = - 2.087
• Thomas performed better compared to Karla.

Given that :

Male data (Thomas) :

Mean ERA, μ = 4.837

Standard deviation, σ = 0.541

Lowest Score, X = 3.31

Female data (Karla) :

Mean ERA, μ = 4.533

Standard deviation, σ = 0.539

Lowest Score, X = 3.02

The standardized score, Zscore which gives the number of standard deviations a certain value is from the mean is calculated using the relation :

Zscore = (X - μ) ÷ σ

Thomas's Zscore :

Zscore = (3.31 - 4.837) ÷ 0.541

Zscore = - 1.527 ÷ 0.541

Zscore = - 2.823

Hence, THOMAS Zscore is 2.823 standard deviations below the mean.

Karla's Zscore :

Zscore = (3.02 - 4.533) ÷ 0.539

Zscore = - 1.513 ÷ 0.539

Zscore = - 2.087

Hence, Karla's Zscore is 2.087 standard deviations below the mean.

- 2.823 < - 2.087 ; Hence Thomas has a lower ERA compared to Karla, hence, he performed better than Karla

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