please do only e and show work!!

12. Mrs. Bracken asks a random sample of 500 adults whether they favor giving vouchers to parents of school- age children that can be exchanged for education at any public or private school of their choice. Each school would be paid by the government on the basis of how many vouchers it collected. Suppose that 45% of the population favor this idea (a) What is the mean of the sampling distribution of p , the proportion of adults in samples of size n = 500 who favor giving parents of school-age children these vouchers? (1) (b) What is the standard deviation of p ? Interpret this value. (R) (c) Check that you can use the Normal approximation for the distribution of p. (!) (d) What is the probability that more than half of the sample are in favor? Show your work. ( 2) (e) If a second random sample of 500 is selected, what is the probability that the two sample proportions are less than 3 percentage points apart? (3)