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

Answer: (B), (C)

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 2

The correct option is (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): $\alpha < \beta$