Question

Let \(X\) have the pdf \(f(x ; \theta)=\theta^{x}(1-\theta)^{1-x}, x=0,1\), zero elsewhere. We test \(H_{0}: \theta=\frac{1}{2}\) against \(H_{1}: \theta<\frac{1}{2}\) by taking a random sample \(X_{1}, X_{2}, \ldots, X_{5}\) of size \(n=5\) and rejecting \(H_{0}\) if \(Y=\sum_{1}^{h} X_{i}\) is observed to be less than or equal to a constant \(c\). (a) Show that this is a uniformly most powerful test. (b) Find the significance level when \(c=1\). (c) Find the significance level when \(c=0\). (d) By using a randomized test, as discussed in Example \(4.6 .4\), modify the tests given in parts (b) and (c) to find a test with significance level \(\alpha=\frac{2}{32}\).

Short answer

To provide a short answer: The critical test is a uniformly most powerful test as it meets the necessary conditions outlined by the Neyman-Pearson Lemma for such a test. The significance levels for \(c = 1\) and \(c = 0\) need to be determined using the binomial distribution under the null hypothesis \(H_0\). Modifying the test to achieve a given significance level involves creating a randomized test to adjust the acceptance and rejection regions.

Step-by-step solution

Understand the Test Settings

The first part of the task requires to prove that the given test is a uniformly most powerful (UMP) test. As per Neyman-Pearson Lemma, a UMP test has the largest power among all level alpha tests. It needs to be shown that the given test satisfies the criteria outlined in the Neyman-Pearson Lemma.

Show that the Test is UMP

According to the Neyman-Pearson Lemma, the likelihood ratio test is the most powerful for testing a simple hypothesis against another simple hypothesis. The likelihood ratio test rejects \(H_0\) if \( \frac{f(x;\theta)}{f(x;\theta_0)} > k \). Here, the null hypothesis is \(H_0: \theta = \frac{1}{2}\) and alternative hypothesis is \(H_1: \theta < \frac{1}{2}\). We get:\[\frac{ (\frac{1}{2})^{Y}(1-\frac{1}{2})^{n-Y}}{\theta^{Y}(1-\theta)^{n-Y}} \leq k\]Using the algebra of inequalities, this can be rewritten to show that \(Y \leq c\) is a UMP test.

Calculate the Significance Level for c=1

The significance level is calculated as the probability that the null hypothesis is incorrectly rejected, given that it is true. Here, this is \(P(Y \leq 1 | \: H_0)\). Use the binomial distribution and conditions to solve this.

Calculate the Significance Level for c=0

Similarly, for c=0, we calculate the significance level as \(P(Y \leq 0 | \: H_0)\). This probability can be calculated using similar methods.

Modify the Test to Match a Specific Significance Level

The test needs to be modified to attain a specific significance level. A randomized test is used here. The acceptance and rejection regions are adjusted in a probabilistic manner to attain the desired significance level \(\alpha = \frac{2}{32}\)