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The uploaded document is a problem set on hypothesis testing from the "Testing of Hypothesis" course (STAT 432) at Egerton University's Mathematics Department. Here is a breakdown of the problems provided:

1. **Problem 1**: Develop a uniformly most powerful (UMP) test of size \( \alpha \) for a Bernoulli distribution parameter \( \theta \) with hypotheses \( H_0: \theta = \frac{1}{2} \) vs \( H_1: \theta < \frac{1}{2} \).

2. **Problem 2**: For a normal distribution \( N(\theta, 100) \), with sample size 25, find a UMP test of size 0.05 for \( H_0: \theta = 75 \) vs \( H_1: \theta > 75 \).

3. **Problem 3**: For \( N(\theta, 16) \), determine sample size and UMP test for \( H_0: \theta = 25 \) vs \( H_1: \theta < 25 \), given power values \( \pi(25) = 0.10 \) and \( \pi(23) = 0.90 \).

4. **Problem 4**: For \( N(\theta, 1) \), find the critical region for testing \( H_0: \theta = 2 \) vs \( H_1: \theta < 2 \) with \( n = 10 \) and \( \alpha = 0.05 \).

5. **Problem 5**: For \( N(0, \sigma^2) \) with unknown variance, show Monotone Likelihood Ratio (MLR) and derive UMP test for \( H_0: \theta = 4 \) vs \( H_1: \theta > 4 \).

6. **Problem 6**: For a Geometric distribution parameter \( \theta \), derive a UMP test of size \( \alpha \) for \( H_0: \theta \geq \theta_0 \) vs \( H_1: \theta < \theta_0 \).

7. **Problem 7**: For normal distribution with known mean and unknown variance, use MLR to derive UMP test for \( H_0: \sigma^2 \leq \sigma_0^2 \) vs \( H_1: \sigma^2 > \sigma_0^2 \).

8. **Problem 8**: For an exponential distribution with unknown parameter \( \beta \), derive a UMP test of size 0.05 for \( H_0: \beta = \frac{1}{2} \) vs \( H_1: \beta < \frac{1}{2} \).

9. **Problem 9**: For a Poisson distribution parameter \( \lambda \), derive a UMP test of size \( \alpha \) for \( H_0: \lambda \leq 4.0 \) vs \( H_1: \lambda > 4.0 \).

10. **Problem 10**: For a distribution with density \( f(x; \theta) = \theta x^{\theta - 1} e^{-x / \theta} \), derive a UMP test for \( H_0: \theta = 6 \) vs \( H_1: \theta > 6 \).

11. **Problem 11**: Test for significance in a Poisson model where the observed accident rate decreased from 3.0 to 2.5 over a four-week interval, at a 5% significance level.

12. **Problem 12**: For a population with density function \( f(x; \theta) = \frac{1}{2\theta^3} x^2 e^{-x/\theta} \), find the most powerful test for \( H_0: \theta = 3 \) vs \( H_1: \theta > 3 \).

Would you like a solution for a specific problem or further details on hypothesis testing concepts? 

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**Exploration Questions:**

1. How is the Monotone Likelihood Ratio (MLR) property used in constructing hypothesis tests?
2. What steps are involved in finding a uniformly most powerful (UMP) test?
3. How does sample size influence the power of a hypothesis test?
4. Why is it important to set the correct significance level \( \alpha \) when constructing a test?
5. How do UMP tests differ when applied to discrete versus continuous distributions?

**Tip**: In hypothesis testing, a lower significance level \( \alpha \) reduces the likelihood of rejecting a true null hypothesis, but may also reduce the test's power to detect a false null. Balance is key for reliable conclusions.

A problem set on hypothesis testing from Egerton University's STAT 432 course, including tasks such as deriving uniformly most powerful (UMP) tests for different distributions and hypotheses.

Explore advanced hypothesis testing problems from Egerton University's STAT 432 course, including tasks like deriving UMP tests for Bernoulli, normal, Poisson, geometric, and exponential distributions. This set covers MLR, critical regions, and power functions, suitable for advanced undergraduate students.

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