Math Problem Statement

STAT 432 PROBLEMSET 4a (2).pdf

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Solution

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 α\alpha for testing H0:θ=θ0H_0: \theta = \theta_0 vs H1:θθ0H_1: \theta \neq \theta_0 using a normal distribution with unknown mean θ\theta.
  2. Problem 3: Demonstrate that the GLR test for H0:θ=θ0H_0: \theta = \theta_0 vs H1:θθ0H_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 H0H_0 is accepted at α=0.05\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?


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 H0H_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.

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Math Problem Analysis

Mathematical Concepts

Generalized Likelihood Ratio (GLR) tests
Hypothesis testing
Normal distribution
Exponential distribution
Poisson distribution
Weibull distribution

Formulas

GLR test statistic
Likelihood function
Critical region determination

Theorems

Likelihood ratio principle
Chi-square distribution for critical region adjustment

Suitable Grade Level

Undergraduate