Suppose X1, ..., Xn are i.i.d. bernoulli trials with P(Xi = 1} = = 1 - P {Xi = 0}.

Suppose n is large. Consider testing H0 : > 0:5 versus H1 : > 0:5. Let ^{^}_n =Xi/n be the MLE of . Consider the large sample test which rejects when 2n^0.5(^{^}_n - 0.5) > z_1- where z_1- is the 1- quantile of the standard normal distribution.

(i) What is the limiting probability of a Type 1 error?

(ii) Show that the power tends to one against any fixed > 0:5 as n approaches infinity;

that is, if is true, the test rejects H0 with probability tending to one.