Step 1 :The Central Limit Theorem states that if the sample size is large enough (usually n > 30 is considered large enough), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. So, the population distribution does not need to be normal, but the sample size should be large. Therefore, the answer to the first part is B. Since the sample size is large enough, the population distribution does not need to be normal.
Step 2 :The mean of the sampling distribution of the sample mean (also known as the expected value of the sample mean) is equal to the population mean, which is 62 in this case. The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is equal to the population standard deviation divided by the square root of the sample size. So, it should be 18/sqrt(38). Therefore, the answer to the second part is A. Approximately normal, with \(\mu_{\bar{x}}=62\) and \(\sigma_{\bar{x}}=\frac{18}{\sqrt{38}}\).
Step 3 :Using python to simplify \(\sigma_{\bar{x}}=\frac{18}{\sqrt{38}}\)
Step 4 :\(\sigma_{\bar{x}}\) is approximately 2.92
Step 5 :Final Answer: (a) \(\boxed{\text{B. Since the sample size is large enough, the population distribution does not need to be normal.}}\) (b) \(\boxed{\text{A. Approximately normal, with } \mu_{\bar{x}}=62 \text{ and } \sigma_{\bar{x}} \approx 2.92}\)