Final answer:
To conduct hypothesis testing with a normally distributed population, one would use a t-distribution when the population standard deviation is unknown and the sample size is small, and a normal distribution for large sample sizes. For the U.S. Census Bureau case, the sample mean is 8.2 minutes, the population standard deviation is 2.2 minutes, and the sample size is 200.
Step-by-step explanation:
To address a student's question regarding point estimates and hypothesis testing with normally distributed populations, let's clarify the concepts with examples from the information provided.
If a sample mean is 12.8, with a known population mean of 13, a sample standard deviation of 2, and a sample size of 20, the appropriate distribution for a hypothesis test would be the t-distribution, since the sample size is small and the population standard deviation is not known.
In another example, if a population has a mean of 25 and a standard deviation of 5, with a sample mean of 24 and a sample size of 108, you would use the normal distribution (Z-distribution) for hypothesis testing as the sample size is large enough to invoke the central limit theorem, which states that the sampling distribution of the sample mean will be approximately normal regardless of the population distribution.
For the U.S. Census Bureau example, we can identify X as the sample mean of 8.2 minutes, σ (sigma) as the population standard deviation of 2.2 minutes, and n as the sample size of 200.