IGNOU MTE 11 2018 - Use the Neyman-Pearson Lemma to obtain the best critical region for te

Question

Use the Neyman-Pearson Lemma to obtain the best critical region for testing $\theta$ = $\theta _{0}$ against $\theta$ = $\theta _{1}$ > $\theta _{0}$ in the case of a normal population

N( $\theta$ , $\sigma^{2}$ ), where $\sigma^{2}$ is known. Hence find the power of the test.

14 Feb 2018

Answer :

Word Count : 606

The Neyman-Pearson Lemma provides a method for constructing the best test for simple hypotheses, i.e., a hypothesis testing problem where the null hypothesis $ H_0 $ and the alternative hypothesis $ H_1 $ specify exact values for the parameters. Let's break the problem into its components and solve it step by step: ### Problem Setup: We are given that the random variable $ X $ follows a normal distribution: \[ X \sim N(\theta, \sigma^2), \] where $ \theta $ is the mean of the population and $ \sigma^2 $ is the known variance. We are tasked with testing: \[ H_0: \theta = \theta_0 \quad \text{against} \quad H_1: \theta = \theta_1, \quad \theta_1 > \theta_0. \] ### Step 1: Likelihood Ratio Test According to the Neyman-Pearson Lemma, the best critical region (i.e., the most powerful test) is given by the likelihood ratio test, which uses the likelihood ratio statistic: \[ \Lambda(x) = \frac{L(\theta_0 | x)}{L(\theta_1 | x)}, \] where $ L(\theta | x) $ is the likelihood function for the parameter $ \theta $ given the observed data $ x $. For a normal distribution, the likelihood function for $ \theta $ given a sample

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