Question

Let $\theta_0$ and $\theta_1$ be real constants such that $\theta_1 > \theta_0$. Suppose that a random sample is taken from a $N(\theta, 1)$ distribution, $\theta \in \mathbb{R}$. For testing $H_0: \theta = \theta_0$ against $H_1: \theta = \theta_1$ at level 0.05, let $\alpha$ and $\beta$ denote the size and the power, respectively, of the most powerful test, $\psi_0$. Then which of the following statements is/are correct?

• A. $\beta < \alpha$
• B. The test $\psi_0$ is the uniformly most powerful test of level $\alpha$ for testing $H_0: \theta = \theta_0$ against $H_1: \theta > \theta_0$
• C. $\alpha < \beta$
• D. The test $\psi_0$ is the uniformly most powerful test of level $\alpha$ for testing $H_0: \theta = \theta_0$ against $H_1: \theta < \theta_0$

Answer 1

The Correct Option is B, C

1. Hypothesis Testing Framework: - The most powerful test $\psi_0$ is derived from the Neyman-Pearson Lemma for testing $H_0: \theta = \theta_0$ against $H_1: \theta = \theta_1$ when $\theta_1 > \theta_0$. - The rejection region of $\psi_0$ is determined by the likelihood ratio. 

2. Uniformly Most Powerful (UMP) Test: - When $\theta_1 > \theta_0$, the test $\psi_0$ is the UMP test for testing $H_0: \theta = \theta_0$ against $H_1: \theta > \theta_0$ at level $\alpha$. 

3. Relationship Between $\alpha$ and $\beta$: - The size $\alpha$ of the test is the probability of rejecting $H_0$ when $H_0$ is true. - The power $\beta$ is the probability of rejecting $H_0$ when $H_1$ is true. - Since the test is designed to maximize power, $\beta > \alpha$ when $\theta_1 > \theta_0$. 

4. Correctness of Statements: - (A): $\beta < \alpha$ is incorrect because $\beta > \alpha$. - (B): $\psi_0$ is the UMP test for $H_0: \theta = \theta_0$ against $H_1: \theta > \theta_0$, which is correct. - (C): $\alpha < \beta$ is incorrect because it assumes $H_1$ as a one-sided alternative. - (D): $\psi_0$ is not UMP for $H_0: \theta = \theta_0$ against $H_1: \theta < \theta_0$, so it is incorrect