MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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).
**(a)** Find the mean of \( \hat{P} \), where \( \hat{P} \) is the proportion of minority member applications in a random sample of 900 that is drawn from all applications.
\[ \boxed{} \]
**(b)** Find the standard deviation of \( \hat{P} \).
\[ \boxed{} \]
**(c)** Compute an approximation for \( P(\hat{p} \leq 0.08) \), which is the probability that there will be 8% or fewer minority member applications in a random sample of 900 drawn from all applications. Round your answer to four decimal places.
\[ \boxed{} \]
[Continue]
**Explanation of Formulas and Results:**
1. **Mean of \( \hat{P} \)**:
- The mean \( \mu_{\hat{P}} \) of the sample proportion \( \hat{P} \) can be found using the formula:
\[ \mu_{\hat{P}} = p \]
where \( p \) is the true proportion of all minority member applications.
2. **Standard Deviation of \( \hat{P} \)**:
- The standard deviation \( \sigma_{\hat{P}} \) of the sample proportion \( \hat{P} \) is calculated using:
\[ \sigma_{\hat{P}} = \sqrt{\frac{p(1 - p)}{n}} \]
where \( p \) is the true proportion of minority applications and \( n \) is the sample size.
3. **Approximation for Probability \( P(\hat{p} \leq 0.08) \)**:
- To approximate the probability that the sample proportion \( \hat{P} \) is less than or equal to 0.08, we can use the normal distribution as an approximation to the binomial distribution:
\[ Z = \frac{\hat{p} - \mu_{\hat{P}}](https://content.bartleby.com/qna-images/question/43ea54d5-bb78-413a-8bda-c29f678bdd37/9e23ad10-9a50-43d5-91e6-6d09ea738425/4aacq4_processed.jpeg)
Transcribed Image Text:**Topic: Statistical Analysis of Minority Representation in Hiring**
A minority representation group accuses a major bank of racial discrimination in its recent hires for financial analysts. Exactly 10% of all applications were from minority members, and exactly 8% of the 900 open positions were filled by members of the minority.
**Answer the following:**
(If necessary, consult a [list of formulas](#)).
**(a)** Find the mean of \( \hat{P} \), where \( \hat{P} \) is the proportion of minority member applications in a random sample of 900 that is drawn from all applications.
\[ \boxed{} \]
**(b)** Find the standard deviation of \( \hat{P} \).
\[ \boxed{} \]
**(c)** Compute an approximation for \( P(\hat{p} \leq 0.08) \), which is the probability that there will be 8% or fewer minority member applications in a random sample of 900 drawn from all applications. Round your answer to four decimal places.
\[ \boxed{} \]
[Continue]
**Explanation of Formulas and Results:**
1. **Mean of \( \hat{P} \)**:
- The mean \( \mu_{\hat{P}} \) of the sample proportion \( \hat{P} \) can be found using the formula:
\[ \mu_{\hat{P}} = p \]
where \( p \) is the true proportion of all minority member applications.
2. **Standard Deviation of \( \hat{P} \)**:
- The standard deviation \( \sigma_{\hat{P}} \) of the sample proportion \( \hat{P} \) is calculated using:
\[ \sigma_{\hat{P}} = \sqrt{\frac{p(1 - p)}{n}} \]
where \( p \) is the true proportion of minority applications and \( n \) is the sample size.
3. **Approximation for Probability \( P(\hat{p} \leq 0.08) \)**:
- To approximate the probability that the sample proportion \( \hat{P} \) is less than or equal to 0.08, we can use the normal distribution as an approximation to the binomial distribution:
\[ Z = \frac{\hat{p} - \mu_{\hat{P}}
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