Find the critical value(s) for the alternative hypothesis, level of significance , and sample sizes and . Assume that the samples are random and independent, the populations are normally distributed, and the population variances are (a) equal
Ha:μ1≠μ2 , α=0.01 , n1=19 , n2=22

Verified step by step guidance

1. Identify the type of test based on the alternative hypothesis. Since the alternative hypothesis is \(H_a: \mu_1 \neq \mu_2\), this is a two-tailed test.
2. Determine the degrees of freedom for the test. Because the population variances are assumed equal, use the pooled degrees of freedom formula: \(df = n_1 + n_2 - 2\).
3. Find the level of significance for each tail. Since \(\alpha = 0.01\) and the test is two-tailed, divide \(\alpha\) by 2 to get \(\alpha/2 = 0.005\) for each tail.
4. Use the \(t\)-distribution table or a calculator to find the critical \(t\)-value corresponding to \(df\) degrees of freedom and the upper tail probability of \$0.005$.
5. The critical values will be \(\pm t_{\alpha/2, df}\), representing the rejection regions in both tails of the \(t\)-distribution.

Key Concepts

Hypothesis Testing and Alternative Hypothesis

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. The alternative hypothesis (Ha: μ1 ≠ μ2) indicates a two-tailed test, meaning we are checking if the two population means are different in either direction.

Level of Significance (α)

The level of significance, α, represents the probability of rejecting the null hypothesis when it is actually true (Type I error). Here, α = 0.01 means there is a 1% risk of a false positive, which affects the critical value and the rejection region in the test.

Two-Sample t-Test with Equal Variances

When population variances are assumed equal, a pooled variance estimate is used in the two-sample t-test. The test statistic follows a t-distribution with degrees of freedom calculated from the sample sizes (n1 + n2 - 2), which determines the critical values for the test.