A test of filtering software examined a sample of n = 100 messages. If the filtering software reduces the level of spam to 15%, this test only has a 33% chance of correctly rejecting H0: p ≥ 0.20. Suppose instead of using 100 messages, the test were to use n = 400.
(a) In order to obtain a p-value of 0.05, what must be the percentage of spam that gets through the filtering software?
(b) With the larger sample size, does p̂ need to be as far below p0 = 0.20 as when n = 100? Explain what threshold moves closer to p0.
(c) If in fact p = 0.15, what is the probability that the test with n = 400 correctly rejects H0?