1. Let Yi, .... Yn " Bernoulli(p). For this problem, you will assess the quality of the CLT approximation to the binomial distribution. You may find the following applets helpful: . https://homepage.divms.uiowa.edu/ mbognar/applets/bin.html . https://homepage.divms.uiowa.edu/ mbognar/appletsormal.html (a) Consider the standardized sample mean: Vn(Y - P) Vp(1 - p Write an expression approximationg the CDF: F(c) = p vn(Y -p) VP(1 - p) sc ) Denote the approximation by F(c). What is F(2)? Calculate a specific number. (b) Find c* in terms of c, n, p such that: F(c) = P Vn(Y - P) VP (1 - P) Sc = P (nysc * ) How can you calculate F(c) exactly? (Hint: what is the sampling distribution of nY?)(c) For each of the values of n and p in the table below, calculate F(2) exactly and fill in the first empty column. Calculate the absolute difference between the exact probability and the approximation you calculated earlier and fill in the second empty column. Round to three decimal places. p F(2) |F(2) - F(2)1 16 0.01 49 0.01 144 0.01 16 0.20 49 0.20 144 0.20 (d) Often it is said that the approximation is good when n is "large enough". What do you notice about the approximation error as n increases? (e) However, "large enough" varies in different contexts. What do you notice about the approximation error for larger sample sizes when p is small? Do you think that n = 50 is sufficient to achieve a good approximation if p = 0.01