Let \(B_{1}, \ldots, B_{n}\) be a sequence of independent and identically distributed random variables from a \(\operatorname{BERNOULLI}(\theta)\) distribution where the parameter space of \(\theta\) is \(\Omega=(0,1)\). Consider testing the null hypothesis \(H_{0}: \theta \leq \theta_{0}\) against the alternative hypothesis \(H_{1}: \theta>\theta_{0}\).

a. Describe an exact test of \(H_{0}\) against \(H_{1}\) whose rejection region is based on the BINOMIAL distribution.

b. Find an approximate test of \(H_{0}\) against \(H_{1}\) using Theorem 4.20 (Lindeberg and Lévy). Prove that this test is consistent and find an expression for the asymptotic power of this test for the sequence of alternatives given by \(\theta_{1, n}=\theta_{0}+n^{-1 / 2} \delta\) where \(\delta>0\).