Elijah and Tyler, two high school juniors, conducted a survey on 13 students at their school, asking the students whether they would like the school to offer an after-school art program, counted the number of "yes" answers, and recorded the sample proportion. 12 out of the 13 students responded "yes." They repeated this 100 times and built a distribution of sample means.

(a) What is this distribution called?
(b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed?
Explain your reasoning.
(c) Calculate the variability of this distribution and state the appropriate term used to refer to
this value.
(d) Suppose that the students were able to recruit a few more friends to help them with sampling,
and are now able to collect data from random samples of 25 students. Once again, they
record the number of \yes" answers, and record the sample proportion, and repeat this 100
times to build a new distribution of sample proportions. How will the variability of this new
distribution compare to the variability of the original distribution?

b. The given sample proportion is 12 out of 13. If this is a typical sample proportion, then most sample proportions will lie about 12, but there could be a few unusually low sample proportions and thus the distribution could be skewed left.

c. The variability of the sampling distribution of the sample proportion is measured by the standard error of the proportion which is the square root of the product of the proportion p and 1 - p divided by the sample size n.

S.E = √((p(1 - p))/n)

= √((12/13(1 - 12/13))/13)

= √((12/13 * 1/13)/13)

= 0.073905344867

= 0.0739

d. Given n = 25, as the sample size increases, the standard error of the proportion will decrease and the variability in the sample proportion will decrease too.