Final answer:
The mean of the sampling distribution of the sample mean for a sample size of n=3 would still be 15, which is the population mean, according to the Central Limit Theorem. None of the multiple choice options provided are correct. For a sample size of 20 with known sample mean and standard deviation, the t-distribution would be used for hypothesis testing.
Step-by-step explanation:
The question is focused on the concept of sampling distributions and how they relate to the population mean. In the scenario provided, the sampling distribution of the sample mean for a random sample of size n = 3 policyholders, given a population mean (μ) of 15, would have the same mean of 15. This is because the mean of the sampling distribution of the sample mean is always equal to the population mean regardless of the sample size, according to the Central Limit Theorem.
For part (a) of the question, none of the options provided, A) 18, B) 5, C) 2, or D) 3, are correct since the mean of the sampling distribution should be 15, not any of these values. Additionally, the Central Limit Theorem plays a crucial role here as it indicates that as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution even if the original population distribution is not normal.
For the given exercise information with a population mean of 13, a sample mean of 12.8, and a sample size of 20, assuming a normal population distribution, the appropriate distribution for a hypothesis test would be a t-distribution since we do not have the population standard deviation and are instead using the sample standard deviation.