MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![**Problem Statement:**
Letitia swam the girls 100m freestyle in 56.5 seconds. Ben swam the boys 100m freestyle in a time of 50.3 seconds. The mean time for the girls 100m at the swim meet was 58.6 seconds with a standard deviation of 2.3 seconds. The mean time for the boys 100m at the swim meet was 52.1 seconds with a standard deviation of 1.8. Who swam the faster relative time?
**Explanation:**
In this question, students are asked to determine who performed better in relation to their respective group averages by comparing the individual times of Letitia and Ben with the mean times of their respective groups.
To solve this, students can calculate the z-score for each swimmer:
- **Z-Score Formula:**
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where \(X\) is the individual's time, \(\mu\) is the mean time, and \(\sigma\) is the standard deviation.
- **Letitia's Z-Score Calculation:**
\[
Z_{Letitia} = \frac{(56.5 - 58.6)}{2.3}
\]
- **Ben's Z-Score Calculation:**
\[
Z_{Ben} = \frac{(50.3 - 52.1)}{1.8}
\]
Each z-score indicates how many standard deviations a swimmer's time is from the mean time of their group. The swimmer with the higher (less negative) z-score is considered to have swum the faster relative time.](https://content.bartleby.com/qna-images/question/4f4dfc46-ba00-462b-b0f5-c6a6c0eb61ba/b3e9b491-5ff8-453f-8396-825e02d38f52/w7do0io_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Letitia swam the girls 100m freestyle in 56.5 seconds. Ben swam the boys 100m freestyle in a time of 50.3 seconds. The mean time for the girls 100m at the swim meet was 58.6 seconds with a standard deviation of 2.3 seconds. The mean time for the boys 100m at the swim meet was 52.1 seconds with a standard deviation of 1.8. Who swam the faster relative time?
**Explanation:**
In this question, students are asked to determine who performed better in relation to their respective group averages by comparing the individual times of Letitia and Ben with the mean times of their respective groups.
To solve this, students can calculate the z-score for each swimmer:
- **Z-Score Formula:**
\[
Z = \frac{(X - \mu)}{\sigma}
\]
where \(X\) is the individual's time, \(\mu\) is the mean time, and \(\sigma\) is the standard deviation.
- **Letitia's Z-Score Calculation:**
\[
Z_{Letitia} = \frac{(56.5 - 58.6)}{2.3}
\]
- **Ben's Z-Score Calculation:**
\[
Z_{Ben} = \frac{(50.3 - 52.1)}{1.8}
\]
Each z-score indicates how many standard deviations a swimmer's time is from the mean time of their group. The swimmer with the higher (less negative) z-score is considered to have swum the faster relative time.
Expert Solution
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Given info:
Letitia swam the girls 100 m freestyle in 56.5 seconds. Ben swam the boys 100 m freestyle in a time of 50.3 seconds.
The mean and standard deviation for the girls 100m at the swim meet is =58.6 seconds and =2.3 seconds respectively.
The mean and standard deviation for the boys 100m at the swim meet is =52.1 seconds and =1.8 seconds respectively.
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