(c) (5 points) We took n = 32 samples of X, and we get Yo = 2; Y1 = 8; Y2 = 8; Y3 = 8; YA = 6. Suppose that the hypotheses are . Null hypothesis Ho: X is a binomial random variable b(4, p) for some unknown p. . Alternative hypothesis H1: the null hypothesis is false. Can we reject the null hypothesis at a = 0.05 significance level?4. Let X be an unknown random variable. We ran 11 experiments for X , we denote by Y;- (z 2 0) the number of samples that are equal to i. (a) (5 points) Suppose that X is a binomial random variable ((4, p) with p unknown. Show that the maximum likelihood estimate 1? is equal 13 0 A Y1+2Y2+3Y3+4Y4 P = 4 :0. (Give a rigorous proof.) (b) (5 points) Show that the formula for p in part (a) is a sufficient statistic for p. (Give a rigorous proof. )