Final answer:
For a population with known parameters and small sample sizes, a t-distribution is typically used for hypothesis testing, whereas for larger samples, the z-distribution can be applied due to the Central Limit Theorem. The characteristics of the distribution for sampling depend on the population parameters and sample size.
Step-by-step explanation:
The subject of this question is Mathematics, specifically statistics. We need to determine which distribution to use for hypothesis testing given the characteristics of the population and sample.
24. Supporting Answer
For a sample size of 20, with a population mean of 13, a sample mean of 12.8, and sample standard deviation of two, you would use the t-distribution for hypothesis testing since the sample size is relatively small and the population standard deviation is not known. The normality of the underlying population supports the use of the t-distribution.
69. Supporting Answer
When drawing 100 samples of size 40 from a normally-distributed population with a mean of 50 and standard deviation of four, the sampling distribution of the sample mean would be expected to be normally distributed with a mean of 50 and a smaller standard deviation, calculated using the standard error of the mean formula, which in this case is the population standard deviation divided by the square root of the sample size (n).
6. Supporting Answer
Identifying factors from the U.S. Census Bureau study:
- X = Sample mean = 8.2 minutes
- σ = Known standard deviation = 2.2 minutes
- n = Sample size = 200
25. Supporting Answer
For a population mean of 25 and standard deviation of five, a sample mean of 24, and a sample size of 108, you should use the z-distribution for hypothesis testing. The substantial sample size justifies the use of the Central Limit Theorem which allows for the normal approximation.