Suppose a simple random sample of size $n=75$ is obtained from a population whose size is $N=30,000$ 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. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. 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 $n p(1-p)<10$. Determine the mean of the sampling distribution of $\hat{p}$. $\mu_{p}=\square$ (Round to one decimal place as needed.) Determine the standard deviation of the sampling distribution of $\hat{p}$. $\sigma_{\hat{p}}=\square$ (Round to six decimal places as needed.)
Solution
Step 1 ：Given values are: sample size \(n = 75\), population size \(N = 30000\), and population proportion \(p = 0.6\).
Step 2 ：Check if \(n \leq 0.05N\) and \(np(1-p) \geq 10\).
Step 3 ：Calculate the mean of the sampling distribution of \(\hat{p}\) which is equal to \(p\). So, \(\mu_{\hat{p}} = p = 0.6\).
Step 4 ：Calculate the standard deviation of the sampling distribution of \(\hat{p}\) using the formula \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\).
Step 5 ：The shape of the sampling distribution of \(\hat{p}\) is approximately normal because the sample size is less than or equal to 5% of the population size and \(np(1-p)\) is greater than or equal to 10.
Step 6 ：The mean of the sampling distribution of \(\hat{p}\) is \(\boxed{0.6}\).
Step 7 ：The standard deviation of the sampling distribution of \(\hat{p}\) is approximately \(\boxed{0.056568}\).
