Problem
Suppose a simple random sample of size $n=200$ is obtained from a population whose size is $N=10,000$ and whose population proportion with a specified characteristic is $p=0$. 6 Complete parts (a) through (c) below. (a) Describe the sampling distribution of $\hat{p}$. Choose the phrase that best describes the shape of the sampling distribution below. A. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. B. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. C. Approximately nomal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$ D. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$ Determine the mean of the sampling distribution of $\hat{p}$ $\mu_{p}=0.6$ (Round to one decimal place as needed) Determine the standard deviation of the sampling distribution of $\hat{p}$ $\sigma_{\hat{p}}=0034641$ (Round to six decimal places as needed) (b) What is the probability of obtaining $x=122$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.61)$ ? $P(\hat{p} \geq 0.61)=0.3864$ (Round to four decimal places as needed) (c) What is the probability of obtaining $x=104$ or fewer individuals with the characteristic? That is, what is $P(\hat{p} \leq 0.52)$ ?
Solution
Step 1 ：The sampling distribution of \(\hat{p}\) is approximately normal because \(n \leq 0.05 \mathrm{~N}\) and \(n p(1-p) \geq 10\).
Step 2 ：The mean of the sampling distribution of \(\hat{p}\) is \(\boxed{0.6}\).
Step 3 ：The standard deviation of the sampling distribution of \(\hat{p}\) is \(\boxed{0.034641}\).
Step 4 ：The probability of obtaining \(x=122\) or more individuals with the characteristic is \(\boxed{0.3864}\).
Step 5 ：The probability of obtaining \(x=104\) or fewer individuals with the characteristic is \(\boxed{0.0105}\).
