I am not quite sure what the questions are asking, do they want numbers or an explanation?

Suppose you are interested in finding out how happy the students are at UNR. You design a survey that probes aspects of an individual’s emotional life, and using this survey are able to compute an accuracy index score ranging from 0 (completely miserable) to 100 (nauseatingly happy). You know that the standard deviation of the population (sigma) equals 10; however you do not know, and want to know, what the mean of the population (mu) is. Rather than give the survey to every student who attend UNR and calculate mu, you give the survey to a random sampling of 100 students. You collect the accuracy indices and compute the mean of your 100 samples and find the mean (x-bar) = 80.

1. Based on the central limit theorem, what is the distribution that your
sample mean is drawn from?

I got N(80,1)

2. What would the distribution be if instead of 100 individuals, you sampled
25?

I got N(80,2)

3. f you sampled 25 instead of 100 students, would your sample mean (x-
bar) be more or less likely to be close to mu?

I said greater but I am not sure how to figure it out

The "standard error" of the mean is the s.d. of all possible sample means. This is what you have just calculated. The "margin of error" of an estimate is often considered to be about 2x the standard error. So for your sample of 100 the margin of error is about 2x1 = +/- 2. For a sample of 25 the margin of error in this case would be about 2x2 = +/- 4. Which sample would most likely to be closest to the true mean?