Final answer:
For hypothesis testing with a small sample size (n<30) and unknown population variance, the t-distribution is used. If the sample size is large (n>30), or the population variance is known, the normal distribution is used. The Central Limit Theorem assures that the sampling distribution of the sample mean is approximately normal for large sample sizes.
Step-by-step explanation:
When you are given information about the population mean, sample mean, sample standard deviation, and sample size, and you are asked to determine what distribution to use for a hypothesis test, the typical distribution that you will use is the t-distribution or the normal distribution, depending on specific conditions. If the population variance is known or the sample size is large (typically n > 30), the normal distribution can be used. However, if the population variance is unknown and the sample size is small (n < 30), you would use the t-distribution. In the case mentioned, where the sample size is 20 and the population distribution is assumed to be normal, the proper distribution to use for the hypothesis testing is the t-distribution, because even though the population is normal, the population variance is unknown and the sample size is relatively small.
Furthermore, when working with exercises related to sampling, such as drawing multiple samples from a population, it's important to recognize that the calculated sample means will form their own distribution known as a sampling distribution. According to the Central Limit Theorem, the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, provided the sample size is sufficiently large (usually n > 30).
For example, if you take repeated samples of size n=100 from a population with a mean of 75 and standard deviation of 4.5, the sampling distribution of the sample mean is expected to be normal due to the large sample size. When the U.S. Census Bureau conducts a survey with 200 people, and the sample mean is 8.2 minutes with a known standard deviation of 2.2 minutes, again, due to the large sample size, the sampling distribution is assumed to be normal.