MATLAB: An Introduction with Applications
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P(p<0.33)
![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.
- Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) \geq 10 \).
- Not normal because \( n > 0.05N \) and \( np(1 - p) < 10 \).
- Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) < 10 \).
Determine the mean of the sampling distribution of \( \hat{p} \).
\[ \mu_{\hat{p}} = p = 0.4 \] (Round to one decimal place as needed.)
Determine the standard deviation of the sampling distribution of \( \hat{p} \).
\[ \sigma_{\hat{p}} = 0.034641 \] (Round to six decimal places as needed.)
(b) What is the probability of obtaining \( x \geq 90 \) or more individuals with the characteristic? That is, what is \( P(\hat{p} \geq 0.45) \)?
\[ P(\hat{p} \geq 0.45) = 0.0745 \] (Round to four decimal places as needed.)
(c) What is the probability of obtaining \( x \leq 66 \) or fewer individuals with the characteristic? That is, what is \( P(\hat{p} \leq 0.33) \)?
\[ P(\hat{p} \leq 0.33) = 0.0339 \] (Round to four decimal places as needed.)](https://content.bartleby.com/qna-images/question/ca98936e-33c7-4402-95ae-aeafe8e3f8b8/f919a487-3d84-4872-bd39-f1cb218072eb/0ydpoqe_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.
- Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) \geq 10 \).
- Not normal because \( n > 0.05N \) and \( np(1 - p) < 10 \).
- Approximately normal because \( n \leq 0.05N \) and \( np(1 - p) < 10 \).
Determine the mean of the sampling distribution of \( \hat{p} \).
\[ \mu_{\hat{p}} = p = 0.4 \] (Round to one decimal place as needed.)
Determine the standard deviation of the sampling distribution of \( \hat{p} \).
\[ \sigma_{\hat{p}} = 0.034641 \] (Round to six decimal places as needed.)
(b) What is the probability of obtaining \( x \geq 90 \) or more individuals with the characteristic? That is, what is \( P(\hat{p} \geq 0.45) \)?
\[ P(\hat{p} \geq 0.45) = 0.0745 \] (Round to four decimal places as needed.)
(c) What is the probability of obtaining \( x \leq 66 \) or fewer individuals with the characteristic? That is, what is \( P(\hat{p} \leq 0.33) \)?
\[ P(\hat{p} \leq 0.33) = 0.0339 \] (Round to four decimal places as needed.)
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