Final answer:
The questions are related to conducting hypothesis tests in statistics, specifically to comparing sample means to known values or between groups. These tests involve the use of t-distribution or z-distribution depending on whether the population standard deviation is known, and they use the Central Limit Theorem when the sample size is large.
Step-by-step explanation:
The subject of this question pertains to conducting a hypothesis test in the field of statistics, which is a branch of mathematics. When looking at the provided scenarios, such as determining if the mean credit hours completed, the mean study time, or the mean enrollment differs from a known average or between different groups, we are dealing with the realm of inferential statistics, specifically hypothesis testing and confidence intervals.
For example, in the case where a sample mean and standard deviation are given, such as a sample mean study time of 18.5 hours with a standard deviation of 1.5 hours for 25 students, you would typically use the t-distribution to test against the national average if the population standard deviation is unknown. If the population standard deviation is known, as in the example where it is 1.5 hours, a z-distribution would be used for hypothesis testing.
The scenarios provided require the use of the Central Limit Theorem, which implies that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the distribution of the population. Each of these examples uses this concept to form the basis of conducting hypothesis tests to compare means.