One year 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 . Also, had the lowest ERA of any female pitcher at the school with an ERA of . For the males, the mean ERA was and the standard deviation was . For the females, the mean ERA was and the standard deviation was . Find their respective z-scores. Which player had the better year relative to their peers, or ? (Note: In general, the lower the ERA, the better the pitcher.) ### Comparing the ERA of Two Pitchers Using Z-ScoresIn this educational exercise, we are comparing the performance of two pitchers, Hank and Susan, from different groups (males and females) to determine who had a better year relative to their peers.#### Given Data:- **Hank's Performance:** - ERA (Earned Run Average): 2.59 - Male group's mean ERA: 4.536 - Male group's standard deviation: 0.639- **Susan's Performance:** - ERA (Earned Run Average): 2.68 - Female group's mean ERA: 4.771 - Female group's standard deviation: 0.915 #### Goal:Calculate the z-scores for both Hank and Susan to determine who had the better performance in their respective groups. A lower ERA indicates a better performance in pitching.#### Z-Score Calculation:The z-score formula used to compare each pitcher's ERA with their respective group is:\[ z = \frac{(X - \mu)}{\sigma} \]where:- \( X \) is the individual's ERA (Hank's or Susan's).- \( \mu \) is the mean ERA of the group (male or female).- \( \sigma \) is the standard deviation of the group's ERA (male or female).##### Hank's Z-Score:Using Hank's values:\[ z = \frac{(2.59 - 4.536)}{0.639} = \frac{(-1.946)}{0.639} \approx -3.05 \]##### Susan's Z-Score:Using Susan's values:\[ z = \frac{(2.68 - 4.771)}{0.915} = \frac{(-2.091)}{0.915} \approx -2.28 \]### Questions:- **Who had a better year relative to their peers?** - Since the lower ERA indicates a better performance, we also seek a more negative z-score (indicating how much better they were compared to the average, considering the standard deviation).#### Options:A. Susan had a better year because of a lower z-score.B. Hank had a better year because of a lower z-score.C. Hank had a better year because of a higher z-score.D. Susan had a better year because of a higher z-score.

Let X represents the mean ERA for a male pitcher.Y represents the mean ERA for a female…

Answered: Hank had an ERA with a z-score of…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

See similar textbooks

Related questions

Question

One year 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 . Also, had the lowest ERA of any female pitcher at the school with an ERA of . For the males, the mean ERA was and the standard deviation was . For the females, the mean ERA was and the standard deviation was . Find their respective z-scores. Which player had the better year relative to their peers, or ? (Note: In general, the lower the ERA, the better the pitcher.)

### Comparing the ERA of Two Pitchers Using Z-Scores

In this educational exercise, we are comparing the performance of two pitchers, Hank and Susan, from different groups (males and females) to determine who had a better year relative to their peers.

#### Given Data:
- **Hank's Performance:**
- ERA (Earned Run Average): 2.59
- Male group's mean ERA: 4.536
- Male group's standard deviation: 0.639

- **Susan's Performance:**
- ERA (Earned Run Average): 2.68
- Female group's mean ERA: 4.771
- Female group's standard deviation: 0.915

#### Goal:
Calculate the z-scores for both Hank and Susan to determine who had the better performance in their respective groups. A lower ERA indicates a better performance in pitching.

#### Z-Score Calculation:
The z-score formula used to compare each pitcher's ERA with their respective group is:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the individual's ERA (Hank's or Susan's).
- \( \mu \) is the mean ERA of the group (male or female).
- \( \sigma \) is the standard deviation of the group's ERA (male or female).

##### Hank's Z-Score:
Using Hank's values:
\[ z = \frac{(2.59 - 4.536)}{0.639} = \frac{(-1.946)}{0.639} \approx -3.05 \]

##### Susan's Z-Score:
Using Susan's values:
\[ z = \frac{(2.68 - 4.771)}{0.915} = \frac{(-2.091)}{0.915} \approx -2.28 \]

### Questions:
- **Who had a better year relative to their peers?**
- Since the lower ERA indicates a better performance, we also seek a more negative z-score (indicating how much better they were compared to the average, considering the standard deviation).

#### Options:
A. Susan had a better year because of a lower z-score.
B. Hank had a better year because of a lower z-score.
C. Hank had a better year because of a higher z-score.
D. Susan had a better year because of a higher z-score.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Expert Solution