suppose a simple random sample of size $n=75$ is obtained from a population whose size is $N=20.000$ and whose population proportion with a specified characteristic is $\mathrm{p}=0$ ? 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 $\hat{p}$. Choose the phrase that best describes the shape of the sampling distribution below. A. Not normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p)<10$. B. Approximately normal because $n \leq 0.05 \mathrm{~N}$ and $n p(1-p) \geq 10$. C. Approximately normal because $\mathrm{n} \leq 0.05 \mathrm{~N}$ and $\mathrm{np}(1-\mathrm{p})<10$ 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 $\hat{p}$. $\mu_{n}=\square$ (Round to one decimal place as needed.) Next
Solution
Step 1 ：Given that the sample size \(n=75\), the population size \(N=20,000\), and the population proportion \(p=0\).
Step 2 ：Check the conditions for normality of the sampling distribution of \(\hat{p}\).
Step 3 ：The first condition is \(n \leq 0.05N\). Substitute the given values to get \(75 \leq 0.05*20,000 = 1,000\), which is true.
Step 4 ：The second condition is \(np(1-p) \geq 10\). Substitute the given values to get \(75*0*(1-0) = 0\), which is not greater than or equal to 10.
Step 5 ：Therefore, the shape of the sampling distribution is not normal because both conditions are not satisfied.
Step 6 ：The mean of the sampling distribution of \(\hat{p}\) is equal to the population proportion \(p\). In this case, \(p=0\), so \(\mu_{\hat{p}}=0\).
Step 7 ：\(\boxed{\text{The shape of the sampling distribution is not normal and the mean is 0.}}\)
