Assignment 6.

[This assignment covers work in Statistical Inference, Chapter 9.]

1. Suppose we wish to test the hypothesis that the proportion of defectives in a large `lot' of articles, $\theta$, is against the alternative that . We decide to do so by taking a sample of items and noting the number of defectives, $X$. The test is to reject $H_0$ for large values of $X$.

   Evaluate the probabilities of , $3$, , assuming .

2. Let $X_1, \ldots, X_n$ denote a random sample from a distribution with probability function . Show that x: is a best critical region for testing $H_0:\theta=\frac{1}{2}$ against . Use the central limit theorem to find $n$ and so that approximately and .
3. Let denote a random sample of size from a normal distribution, . Find a uniformly most powerful critical region of size for testing against .

4. Let $X_1, \ldots, X_n$ be a random sample of size from a Poisson distribution with mean $\theta$. Show that the critical region defined by is a uniformly most powerful critcal region for testing against . What is $\alpha$, the significance level of the test?
5. Given a random sample $X_1, \ldots, X_n$ from a Poisson distribution with parameter $\theta$, derive the likelihood ratio test of the hypothesis $H_0:\theta=\theta_0$ against $H_1:\theta \neq \theta_0$.
   How is this altered if $H_1$ is $\theta > \theta_0$?
   

   Using the fact that, for $n$ large, has an approximate chi-square distribution, explain how the test for $H_0:\theta=1$ would be carried out.

6. Suppose that after $n$ tosses of a coin heads have appeared. That is, the random variable $X_i$ is the number of heads on the ith toss ($0$ or $1$). Derive the likelihood ratio test of the hypothesis where $\theta$ is the probability of a head.

   Show that the likelihood ratio is
   

   
   
   Draw roughly the graph of $\lambda$ as a function of $\bar{x}$, given that is symmetric about . Thus show that the likelihood ratio test is equivalent to rejecting $H_0$ if , for an appropriate constant $c$.

7. Suppose we have independent random samples of size $n$ taken from each of two Poisson distributions with parameters $\theta_1$, $\theta_2$ respectively. Derive the LR test of against . Assuming that $n$ is large, say how the test would be carried out.