Problem
Question 10, 8.2.11 HW Score: $97.24 \%, 28.2$ of 29 points Part 4 of 5 Points: 1.2 of 2 Save 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 $p=0.4$. Click here to view the standard normal distribution table (page 1). Click here to view the standard normal distribution table (page 2). (a) Describe the sampling distribution of $\hat{\mathrm{p}}$. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately 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 normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. D. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. Determine the mean of the sampling distribution of $\hat{p}$. $\mu_{\hat{p}}=0.4$ (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 $x=86$ or more individuals with the characteristic? That is, what is $P(\hat{p} \geq 0.43)$ ? $P(\hat{p} \geq 0.43)=\square($ Round to four decimal places as needed. $)$
Solution
Step 1 ：The sampling distribution of \(\hat{p}\) is approximately normal because \(n \leq 0.05 N\) and \(n p(1-p) \geq 10\).
Step 2 ：The mean of the sampling distribution of \(\hat{p}\) is \(\mu_{\hat{p}}=0.4\).
Step 3 ：The standard deviation of the sampling distribution of \(\hat{p}\) is \(\sigma_{\hat{p}}=0.034641\).
Step 4 ：The probability of obtaining \(x=86\) or more individuals with the characteristic is \(P(\hat{p} \geq 0.43)=\boxed{0.1932}\).
