Final answer:
The appropriate distribution for hypothesis testing varies on the sample size and the known information about the population's standard deviation. For small samples (n < 30), use the t-distribution, and for larger samples (n > 30) and known population standard deviation, use the normal distribution, as per the Central Limit Theorem.
Step-by-step explanation:
The student's question addresses the concept of the sampling distribution of the sample mean and touches upon statistical hypothesis testing. When a population has a known mean (such as 13 or 25 from the provided examples) and the sample size is sufficiently large, the distribution of the sample means can be expected to be normal according to the Central Limit Theorem. This is particularly true when the sample size is greater than 30, making it suitable to use the normal distribution for hypothesis tests.
For the student's scenario with a population mean of 13, a sample mean of 12.8, a sample standard deviation of two, and a sample size of 20, the appropriate distribution for a hypothesis test, assuming the underlying population is normal, would be the t-distribution due to the sample size being below 30 and the population standard deviation being unknown. Similarly, when a population mean is 25 with a standard deviation of five and the sample mean is 24 with a sample size of 108, the distribution to use for a hypothesis test would again be the normal distribution or Z-distribution, since the sample size is above 30, and the standard error of the mean can be determined for large samples.