Suppose a simple random sample of size $n=1000$ is obtained from a population whose size is $\mathrm{N}=1,000,000$ and whose population proportion with a specified characteristic is $p=0.26$. Complete parts (a) through (c) below. 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 $\hat{p}$. A. Approximately normal, $\mu_{\hat{p}}=0.26$ and $\sigma_{\hat{p}} \approx 0.0004$ B. Approximately normal, $\mu_{\hat{p}}=0.26$ and $\sigma_{\hat{p}} \approx 0.0139$ C. Approximately normal, $\mu_{\hat{p}}=0.26$ and $\sigma_{\hat{p}} \approx 0.0002$
Solution
Step 1 ：The sampling distribution of $\hat{p}$ is approximately normal due to the large sample size.
Step 2 ：The mean of the sampling distribution $\mu_{\hat{p}}$ is equal to the population proportion $p=0.26$.
Step 3 ：The standard deviation of the sampling distribution $\sigma_{\hat{p}}$ can be calculated using the formula $\sqrt{\frac{p(1-p)}{n}}$.
Step 4 ：Substitute $p = 0.26$ and $n = 1000$ into the formula, we get $\sigma_{\hat{p}} = \sqrt{\frac{0.26(1-0.26)}{1000}} \approx 0.0139$.
Step 5 ：Final Answer: The sampling distribution of $\hat{p}$ is approximately normal, with $\mu_{\hat{p}}=0.26$ and $\sigma_{\hat{p}} \approx 0.0139$. So, the correct option is B. Approximately normal, $\mu_{\hat{p}}=0.26$ and $\sigma_{\hat{p}} \approx 0.0139$.
