Question

Let \( X_1, X_2, X_3, X_4 \) be a random sample from an \( N(\theta, 1) \) distribution, where \( \theta \in (-\infty, \infty) \). Suppose the null hypothesis \( H_0: \theta = 1 \) is to be tested against the hypothesis \( H_1: \theta<1 \) at \( \alpha = 0.05 \) level of significance. For what observed values of \( \sum_{i=1}^4 X_i \), the uniformly most powerful test would reject \( H_0 \)?

• A. -1
• B. 0
• C. 0.5
• D. 0.8

Hint:

In hypothesis testing, the test statistic is often based on the sample mean, and rejection regions are determined using the critical values for the normal distribution.

Answer 1

The Correct Option is A, B, C

Step 1: Setting up the hypothesis test.
We are testing \( H_0: \theta = 1 \) against \( H_1: \theta<1 \). The test statistic is based on the sample mean, as it is a normal distribution.
Step 2: Finding the critical region.
For a normal distribution with mean \( \theta \) and variance 1, the rejection region for the test is determined by the critical value corresponding to \( \alpha = 0.05 \). For \( \theta = 1 \), the critical value of \( \sum_{i=1}^4 X_i \) is found using the z-score formula.
Step 3: Conclusion.
The uniformly most powerful test will reject \( H_0 \) when the sum \( \sum_{i=1}^4 X_i \) is less than or equal to 0, as this falls in the lower tail of the distribution under \( H_1 \).