Let X1, X2, …, X10 be a random sample of size 10 from a...

Let $X_{1}, X_{2}, \ldots, X_{10}$ be a random sample of size 10 from a normal distribution $N\left(0, \sigma^{2}\right) .$ Find a best critical region of size $\alpha=0.05$ for testing $H_{0}: \sigma^{2}=1$ against $H_{1}: \sigma^{2}=2 .$ Is this a best critical region of size $0.05$ for testing $H_{0}: \sigma^{2}=1$ against $H_{1}: \sigma^{2}=4 ?$ Against $H_{1}: \sigma^{2}=\sigma_{1}^{2}>1 ?$

Key Concepts

Chi-Square Distribution

In the context of testing hypotheses about the variance of a normal distribution, the sum of squared deviations from the mean (or sample variance multiplied by a factor) follows a chi-square distribution. This distribution is crucial for determining the critical values associated with a specified significance level in variance tests.

Neyman-Pearson Lemma

The Neyman-Pearson Lemma is a fundamental result that establishes the framework for finding the most powerful test for a given size when testing between two simple hypotheses. It does so by comparing the likelihoods under the null and alternative hypotheses, thereby identifying the rejection region that maximizes the power subject to a fixed significance level.

Likelihood Ratio Test

A Likelihood Ratio Test is a statistical method that involves computing the ratio of the likelihood under the alternative hypothesis to the likelihood under the null hypothesis. This approach is used to derive critical regions that decide whether to reject or fail to reject the null hypothesis, particularly in settings where variance parameters are being tested.

Test Size (Significance Level)

The size of a test, often represented by alpha (?), is the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. It is a crucial concept when designing tests, ensuring that the rejection region is calibrated so that the probability of a false positive does not exceed the predetermined level, such as 0.05.

Uniformly Most Powerful Test

A Uniformly Most Powerful (UMP) Test is one that provides the highest power among all possible tests of a certain size for all values of the parameter under the alternative hypothesis. This concept is especially important when the alternative hypothesis is composite or when comparing tests under different specific parameter values to ensure the test remains optimal across possible scenarios.

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