Suppose X1, ..., X, 2.2.d. " Bernoulli(p), i.e. P(X = x) = p"(1 -p)1- where p E (0, 1) is unknown. (a) Find a sufficient statistic for p, and state the theorem(s) you used to determine that it is sufficient. (b) Suppose n = 73, and the value of EXi is 71. Construct a two-sided 99% confidence interval for p. (c) Assume a reasonable guess for the value of p is p* = 0.967. If we want a two- sided 99% confidence interval for p to have a length of 0.01, find the sample size we would need. (d) Suppose we obtain p = 0.97. Use your sample size calculated in part (c) to construct a two-sided 99% confidence interval for p. (e) Give one way we can make the confidence interval in part (d) shorter