A simple random sample of size $n=60$ is obtained from a population with $\mu=52$ and $\sigma=5$. Does the population need to be normally distributed for the sampling distribution of $\bar{x}$ to be approximately normally distributed? Why? What is the sampling distribution of $\bar{x}$ ? Does the population need to be normally distributed for the sampling distribution of $\bar{x}$ to be approximately normally distributed? Why? A. No because the Central Limit Theorem states that regardless of the shape of the underlying population, the sampling distribution of $\bar{x}$ becomes approximately normal as the sample size, $n$, increases. B. Yes because the Central Limit Theorem states that the sampling variability of nonnormal populations will increase as the sample size increases. C. No because the Central Limit Theorem states that only if the shape of the underlying population is normal or uniform does the sampling distribution of $\bar{x}$ become approximately normal as the sample size, $n$, increases. D. Yes because the Central Limit Theorem states that only for underlying populations that are normal is the shape of the sampling distribution of $\bar{X}$ normal, regardless of the sample size, $n$. What is the sampling distribution of $\bar{x}$ ? Select the correct choice below and fill in the answer boxes within your choice. (Type integers or decimals rounded to three decimal places as needed.) A. The sampling distribution of $\bar{x}$ is normal or approximately normal with $\mu_{\bar{x}}=\square$ and $\sigma_{\bar{x}}=\square$. B. The sampling distribution of $\bar{x}$ follows Student's t-distribution with $\mu_{\bar{x}}^{-}=\square$ and $\sigma_{\bar{x}}^{-}=\square$. C. The sampling distribution of $\bar{x}$ is skewed left with $\mu_{\bar{x}}=\square$ and $\sigma_{\bar{x}}=\square$. D. The sampling distribution of $\bar{x}$ is uniform with $\mu_{\bar{x}}=\square$ and $\sigma_{\bar{x}}=\square$.
Solution
Step 1 ：We are given a simple random sample of size \(n=60\) from a population with \(\mu=52\) and \(\sigma=5\).
Step 2 ：We are asked whether the population needs to be normally distributed for the sampling distribution of \(\bar{x}\) to be approximately normally distributed.
Step 3 ：The answer is no. According to the Central Limit Theorem, regardless of the shape of the underlying population, the sampling distribution of \(\bar{x}\) becomes approximately normal as the sample size, \(n\), increases.
Step 4 ：We are also asked to find the sampling distribution of \(\bar{x}\).
Step 5 ：To do this, we first calculate the standard deviation of the sampling distribution, \(\sigma_{\bar{x}}\), which is equal to \(\sigma / \sqrt{n}\).
Step 6 ：Substituting the given values, we get \(\sigma_{\bar{x}} = 5 / \sqrt{60} \approx 0.645\).
Step 7 ：The mean of the sampling distribution, \(\mu_{\bar{x}}\), is equal to the population mean, \(\mu\), which is 52.
Step 8 ：Therefore, the sampling distribution of \(\bar{x}\) is normal or approximately normal with \(\mu_{\bar{x}}=52\) and \(\sigma_{\bar{x}}=0.645\).
Step 9 ：\(\boxed{\text{Final Answer: }}\) The answer to the first question is no because the Central Limit Theorem states that regardless of the shape of the underlying population, the sampling distribution of \(\bar{x}\) becomes approximately normal as the sample size, \(n\), increases. The answer to the second question is that the sampling distribution of \(\bar{x}\) is normal or approximately normal with \(\mu_{\bar{x}}=52\) and \(\sigma_{\bar{x}}=0.645\).
