Art after school. Elijah and Tyler, two high school juniors, conducted a survey on 15 students at their school, asking the students whether they would like the school to offer an afterschool art program, counted the number of "yes" answers, and recorded the sample proportion. 14 out of the 15 students responded "yes". They repeated this 100 times and built a distribution of sample means. (Note that this question requires having reviewed Section 3.4.2 on the normal approximation to the binomial distribution.)

(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?