MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![Suppose a simple random sample of size \( n = 200 \) is obtained from a population whose size is \( N = 20,000 \) and whose population proportion with a specified characteristic is \( p = 0.4 \). Complete parts (a) through (c) below.
**Options:**
- A. Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) \geq 10 \).
- B. Not normal because \( n > 0.05N \) and \( np(1 - p) \geq 10 \).
- C. Not normal because \( n \leq 0.05N \) and \( np(1 - p) < 10 \).
- D. Approximately normal because \( n > 0.05N \) and \( np(1 - p) \geq 10 \).
**Determine the mean of the sampling distribution of \( \hat{p} \):**
\[
\mu_{\hat{p}} = p = 0.4 \quad \text{(Round to one decimal place as needed.)}
\]
**Determine the standard deviation of the sampling distribution of \( \hat{p} \):**
\[
\sigma_{\hat{p}} = 0.034641 \quad \text{(Round to six decimal places as needed.)}
\]
**(b) What is the probability of obtaining \( x = 90 \) or more individuals with the characteristic? That is, what is \( P(\hat{p} \geq 0.45) \)?**
\[
P(\hat{p} \geq 0.45) = 0.0745 \quad \text{(Round to four decimal places as needed.)}
\]
**(c) What is the probability of obtaining \( x = 66 \) or fewer individuals with the characteristic? That is, what is \( P(\hat{p} \leq 0.33) \)?**
\[
P(\hat{p} \leq 0.33) = 0.0918 \quad \text{(Round to four decimal places as needed.)}
\]](https://content.bartleby.com/qna-images/question/ca98936e-33c7-4402-95ae-aeafe8e3f8b8/65a746c7-73e9-48fe-9ae2-1035c655313f/2dtilj_processed.jpeg)
Transcribed Image Text:Suppose a simple random sample of size \( n = 200 \) is obtained from a population whose size is \( N = 20,000 \) and whose population proportion with a specified characteristic is \( p = 0.4 \). Complete parts (a) through (c) below.
**Options:**
- A. Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) \geq 10 \).
- B. Not normal because \( n > 0.05N \) and \( np(1 - p) \geq 10 \).
- C. Not normal because \( n \leq 0.05N \) and \( np(1 - p) < 10 \).
- D. Approximately normal because \( n > 0.05N \) and \( np(1 - p) \geq 10 \).
**Determine the mean of the sampling distribution of \( \hat{p} \):**
\[
\mu_{\hat{p}} = p = 0.4 \quad \text{(Round to one decimal place as needed.)}
\]
**Determine the standard deviation of the sampling distribution of \( \hat{p} \):**
\[
\sigma_{\hat{p}} = 0.034641 \quad \text{(Round to six decimal places as needed.)}
\]
**(b) What is the probability of obtaining \( x = 90 \) or more individuals with the characteristic? That is, what is \( P(\hat{p} \geq 0.45) \)?**
\[
P(\hat{p} \geq 0.45) = 0.0745 \quad \text{(Round to four decimal places as needed.)}
\]
**(c) What is the probability of obtaining \( x = 66 \) or fewer individuals with the characteristic? That is, what is \( P(\hat{p} \leq 0.33) \)?**
\[
P(\hat{p} \leq 0.33) = 0.0918 \quad \text{(Round to four decimal places as needed.)}
\]
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