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.