(Solved): in details please 6.3.9. Let \( X_{1}, X_{2}, \ldots, X_{n} \) be a random sample from a Bernoulli \ ...
in details please
6.3.9. Let \( X_{1}, X_{2}, \ldots, X_{n} \) be a random sample from a Bernoulli \( b(1, \theta) \) distribution. where \( 0<\theta<1 \). (a) Show that the likelihood ratio test of \( H_{0}: \theta=\theta_{0} \) versus \( H_{1}: \theta \neq \theta_{0} \) is besed upon the statistic \( Y=\sum_{i=1}^{n} X_{i} \). Obtain the null distribution of \( Y \). (b) For \( n=100 \) and \( \theta_{0}=1 / 2 \), find \( c_{1} \) so that the test rejects \( H_{0} \) when \( Y \leq c_{1} \) or \( Y \geq c_{2}=100-c_{1} \) has the approximate significance level of \( \alpha=0.05 \). Hint: Use the Central Limit Theorem.