The lengths of adult males' hands are normally distributed with mean 189 mm and standard deviation is 7.4 mm. Suppose that 43 individuals are randomly chosen. Round all answers to 4 where possible. a. What is the distribution of ? - N( b. For the group of 43, find the probability that the average hand length is less than 191. c. Find the first quartile for the average adult male hand length for this sample size. d. For part b), is the assumption that the distribution is normal necessary? O NoO Yes
The lengths of adult males' hands are normally distributed with mean 189 mm and standard deviation is 7.4 mm. Suppose that 43 individuals are randomly chosen. Round all answers to 4 where possible. a. What is the distribution of ? - N( b. For the group of 43, find the probability that the average hand length is less than 191. c. Find the first quartile for the average adult male hand length for this sample size. d. For part b), is the assumption that the distribution is normal necessary? O NoO Yes
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 Description:
The lengths of adult males' hands are normally distributed with a mean of 189 mm and a standard deviation of 7.4 mm. Suppose that 43 individuals are randomly chosen. Round all answers to 4 decimal places where possible.
### Questions:
1. **What is the distribution of \( \bar{X} \)?**
\[
\bar{X} \sim N\left( \boxed{189}, \boxed{\frac{7.4}{\sqrt{43}}} \right)
\]
2. **For the group of 43, find the probability that the average hand length is less than 191.**
\[
\boxed{0.9726}
\]
3. **Find the first quartile for the average adult male hand length for this sample size.**
\[
\boxed{187.0133}
\]
4. **For part b), is the assumption that the distribution is normal necessary?**
\[
\boxed{\text{No}} \quad \boxed{\text{Yes}}
\]
#### Explanation for Graphs/Diagrams:
There are no specific graphs or diagrams in this problem. The problem is solely focused on understanding and applying the concept of normal distribution and the Central Limit Theorem to calculate probabilities and critical values based on the given parameters.
#### Detailed Steps:
1. **Determine the distribution of \( \bar{X} \):**
- If \( X \sim N(\mu, \sigma^2) \), then \( \bar{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right) \), where \( n \) is the sample size.
- Given \( \mu = 189 \), \( \sigma = 7.4 \), and \( n = 43 \).
- Thus, \( \bar{X} \sim N\left( 189, \frac{7.4}{\sqrt{43}} \right) \).
2. **Probability calculation:**
- Convert the problem to a standard normal distribution and use Z-scores.
- The mean \( \mu_{\bar{X}} = 189 \) and standard error \( \sigma_{\bar{X}} = \frac{7.4}{\sqrt{43}} \approx 1.128](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46f664ac-b184-42bb-885a-212e0d7b7171%2Fa9a02557-070e-42df-96f8-de3713772827%2Fknir0u_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Description:
The lengths of adult males' hands are normally distributed with a mean of 189 mm and a standard deviation of 7.4 mm. Suppose that 43 individuals are randomly chosen. Round all answers to 4 decimal places where possible.
### Questions:
1. **What is the distribution of \( \bar{X} \)?**
\[
\bar{X} \sim N\left( \boxed{189}, \boxed{\frac{7.4}{\sqrt{43}}} \right)
\]
2. **For the group of 43, find the probability that the average hand length is less than 191.**
\[
\boxed{0.9726}
\]
3. **Find the first quartile for the average adult male hand length for this sample size.**
\[
\boxed{187.0133}
\]
4. **For part b), is the assumption that the distribution is normal necessary?**
\[
\boxed{\text{No}} \quad \boxed{\text{Yes}}
\]
#### Explanation for Graphs/Diagrams:
There are no specific graphs or diagrams in this problem. The problem is solely focused on understanding and applying the concept of normal distribution and the Central Limit Theorem to calculate probabilities and critical values based on the given parameters.
#### Detailed Steps:
1. **Determine the distribution of \( \bar{X} \):**
- If \( X \sim N(\mu, \sigma^2) \), then \( \bar{X} \sim N\left( \mu, \frac{\sigma^2}{n} \right) \), where \( n \) is the sample size.
- Given \( \mu = 189 \), \( \sigma = 7.4 \), and \( n = 43 \).
- Thus, \( \bar{X} \sim N\left( 189, \frac{7.4}{\sqrt{43}} \right) \).
2. **Probability calculation:**
- Convert the problem to a standard normal distribution and use Z-scores.
- The mean \( \mu_{\bar{X}} = 189 \) and standard error \( \sigma_{\bar{X}} = \frac{7.4}{\sqrt{43}} \approx 1.128
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