Suppose that X1, X2, …, Xn1, Y1, Y2, …, Yn2, and W1, W2, …, Wn3 are independent random samples f

step 1 For exercise 108a, we reject H .0 when n -sub 1 times natural log of t, sub 1, close parentheses

Suppose that X1, X2, …, Xn1, Y1, Y2, …, Yn2, and W1, W2, …,...

Suppose that $X_{1}, X_{2}, \ldots, X_{n_{1}}, Y_{1}, Y_{2}, \ldots, Y_{n_{2}},$ and $W_{1}, W_{2}, \ldots, W_{n_{3}}$ are independent random samples from normal distributions with respective unknown means $\mu_{1}, \mu_{2},$ and $\mu_{3}$ and variances $\sigma_{1}^{2}, \sigma_{2}^{2},$ and $\sigma_{3}^{2}$ a. Find the likelihood ratio test for $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}$ against the alternative of at least one inequality. b. Find an approximate critical region for the test in part $(\mathrm{a})$ if $n_{1}, n_{2},$ and $n_{3}$ are large and $\alpha=.05$.

Suppose that $X_{1}, X_{2}, \ldots, X_{n_{1}}, Y_{1}, Y_{2}, \ldots, Y_{n_{2}},$ and $W_{1}, W_{2}, \ldots, W_{n_{3}}$ are independent random
samples from normal distributions with respective unknown means $\mu_{1}, \mu_{2},$ and $\mu_{3}$ and variances $\sigma_{1}^{2}, \sigma_{2}^{2},$ and $\sigma_{3}^{2}$
a. Find the likelihood ratio test for $H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}=\sigma_{3}^{2}$ against the alternative of at least one inequality.
b. Find an approximate critical region for the test in part $(\mathrm{a})$ if $n_{1}, n_{2},$ and $n_{3}$ are large and $\alpha=.05$.

Key Concepts

Likelihood Ratio Test

The likelihood ratio test (LRT) is a statistical method for comparing the fit of two models: one under the null hypothesis and one under the alternative hypothesis. In this context, the LRT is used to test whether the variances of multiple normal distributions are equal. The test involves computing the ratio of the maximum likelihood under the restricted model (where the variances are assumed equal) to the maximum likelihood under the unrestricted model (where the variances are allowed to differ), and then assessing whether this ratio is too small, which would indicate evidence against the null hypothesis.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) is a method for estimating the parameters of a statistical model by maximizing the likelihood function, which measures how likely it is to obtain the observed data for given parameter values. In the context of testing for equality of variances, MLE is used to obtain estimates of the variances under both the null and alternative hypotheses. Under the null hypothesis, a pooled variance estimate is derived, whereas under the alternative hypothesis, separate variance estimates are calculated for each group.

Hypothesis Testing for Equality of Variances

Hypothesis testing for equality of variances involves comparing the variances from different samples to determine if they can be assumed to be equal. This is crucial in many statistical analyses, particularly in analysis of variance (ANOVA) and other inferential procedures. The null hypothesis is that all group variances are equal, and rejection of the null indicates that at least one pair of variances is different. The test statistic derived from the likelihood ratio can be compared to a reference distribution to make this decision.

Approximate Critical Region

When the sample sizes are large, the distribution of the test statistic from the likelihood ratio test can be approximated by a known distribution, such as a chi-square distribution. This allows for the construction of an approximate critical region for the test at a given significance level, typically ? = 0.05. The critical region defines the set of values of the test statistic that would lead us to reject the null hypothesis. This asymptotic approach simplifies the testing procedure when exact distributions are complex or intractable.

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