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?

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?

cdquestions Admin · Accepted Answer

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

Answer

$\beta < \alpha$

Answer

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

$\alpha < \beta$

Answer

The test $\psi_0$ is the uniformly most powerful test of level $\alpha$ for testing $H_0: \theta = \theta_0$ against $H_1: \theta < \theta_0$

The Correct Option is B, C

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