https://file.aiquickdraw.com/mathbot/file/1730140799516p2b5kzns.pdf

The document contains a series of hypothesis testing problems involving the Generalized Likelihood Ratio (GLR) tests, specifically focused on various distributions (e.g., normal, exponential, Poisson, and Weibull). Here’s a quick breakdown of a few selected problems to illustrate the types of tasks:

1. **Problem 1**: Derive a GLR test of size \( \alpha \) for testing \( H_0: \theta = \theta_0 \) vs \( H_1: \theta \neq \theta_0 \) using a normal distribution with unknown mean \( \theta \).
2. **Problem 3**: Demonstrate that the GLR test for \( H_0: \theta = \theta_0 \) vs \( H_1: \theta \neq \theta_0 \) for an exponential distribution has a critical region and explore its modification for chi-square table application.
3. **Problem 5 (b)**: For a given sample size and observed mean, determine if \( H_0 \) is accepted at \( \alpha = 0.05 \).
4. **Problem 11**: Apply the GLR test on a sample of waiting times to evaluate if the true mean is different from a hypothesized value.

Would you like a solution for a specific problem, or a general explanation of GLR test derivation? 

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### Related Questions

1. How is the GLR test derived for normal distributions with unknown mean and variance?
2. What adjustments are necessary for using chi-square tables with exponential distributions in hypothesis testing?
3. How can you interpret the critical region in hypothesis testing?
4. What are the implications of sample size on the acceptance of \( H_0 \) at a specified significance level?
5. How is the likelihood ratio principle applied in tests involving non-standard distributions?

### Tip
To solve these types of problems, start by writing down the likelihood function and then derive the likelihood ratio.

The document contains a series of hypothesis testing problems involving the Generalized Likelihood Ratio (GLR) tests, specifically focused on various distributions (e.g., normal, exponential, Poisson, and Weibull).

Explore a comprehensive set of hypothesis testing problems using Generalized Likelihood Ratio (GLR) tests, covering normal, exponential, Poisson, and Weibull distributions. Each problem includes detailed steps for deriving GLR tests, determining critical regions, and modifying tests for applications such as chi-square tables.

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