4. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height.of students at your college or university to within one inch with 93% confidence. How many male students must you measure?
4. The average height of young adult males has a normal distribution with standard deviation of 2.5 inches. You want to estimate the mean height.of students at your college or university to within one inch with 93% confidence. How many male students must you measure?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Problem Statement:**
4. The average height of young adult males has a normal distribution with a standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 99% confidence. How many male students must you measure?
**Explanation and Calculation:**
- **Normal Distribution:** The problem states that the height follows a normal distribution with a given standard deviation of 2.5 inches.
- **Margin of Error:** You wish to estimate the mean within one inch.
- **Confidence Level:** The desired confidence level is 99%.
**Calculation of Sample Size:**
To calculate the necessary sample size (n) for the given confidence level and margin of error, we use the formula for the sample size of a population mean:
\[ n = \left( \frac{Z \times \sigma}{E} \right)^2 \]
Where:
- \( Z \) is the Z-value corresponding to the desired confidence level (for 99% confidence, \( Z \approx 2.576 \)).
- \( \sigma = 2.5 \) inches (standard deviation).
- \( E = 1 \) inch (margin of error).
Plug in the values:
\[ n = \left( \frac{2.576 \times 2.5}{1} \right)^2 \]
\[ n = \left( \frac{6.44}{1} \right)^2 \]
\[ n = 6.44^2 \]
\[ n \approx 41.47 \]
Therefore, you would need to measure approximately 42 male students to achieve the desired confidence level.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F481b823d-d8c1-44b0-8540-541a45983b6e%2Fc5b4a35c-1871-4e16-b04d-cb07d4ab8c79%2F95mzey_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
4. The average height of young adult males has a normal distribution with a standard deviation of 2.5 inches. You want to estimate the mean height of students at your college or university to within one inch with 99% confidence. How many male students must you measure?
**Explanation and Calculation:**
- **Normal Distribution:** The problem states that the height follows a normal distribution with a given standard deviation of 2.5 inches.
- **Margin of Error:** You wish to estimate the mean within one inch.
- **Confidence Level:** The desired confidence level is 99%.
**Calculation of Sample Size:**
To calculate the necessary sample size (n) for the given confidence level and margin of error, we use the formula for the sample size of a population mean:
\[ n = \left( \frac{Z \times \sigma}{E} \right)^2 \]
Where:
- \( Z \) is the Z-value corresponding to the desired confidence level (for 99% confidence, \( Z \approx 2.576 \)).
- \( \sigma = 2.5 \) inches (standard deviation).
- \( E = 1 \) inch (margin of error).
Plug in the values:
\[ n = \left( \frac{2.576 \times 2.5}{1} \right)^2 \]
\[ n = \left( \frac{6.44}{1} \right)^2 \]
\[ n = 6.44^2 \]
\[ n \approx 41.47 \]
Therefore, you would need to measure approximately 42 male students to achieve the desired confidence level.
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