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=0.10 , n1=30 , n2=32

Verified step by step guidance

1. Identify the type of test based on the alternative hypothesis. Since the alternative hypothesis is \(H_a: \mu_1 < \mu_2\), this is a left-tailed test.
2. Determine the level of significance \(\alpha = 0.10\). This means the critical region is in the left tail of the distribution with an area of 0.10.
3. Since the population variances are assumed equal, use the pooled variance \(s_p^2\) and the \(t\)-distribution with degrees of freedom \(df = n_1 + n_2 - 2\).
4. Calculate the degrees of freedom: \(df = 30 + 32 - 2 = 60\).
5. Find the critical value \(t_{\alpha, df}\) from the \(t\)-distribution table corresponding to \(\alpha = 0.10\) in the left tail and \(df = 60\). This critical value will be negative because it is a left-tailed test.

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) specifies the direction of the test, indicating that the mean of population 1 is less than that of population 2. Understanding the alternative hypothesis guides the selection of the critical region for the test.

Level of Significance (α)

The level of significance, denoted by α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). In this question, α = 0.10 means there is a 10% risk of a false positive. It determines the critical value(s) that define the rejection region for the test statistic.

Two-Sample t-Test with Equal Variances

When population variances are assumed equal, a pooled two-sample t-test is used to compare means from two independent samples. The test statistic follows a t-distribution with degrees of freedom based on sample sizes (n1 + n2 - 2). Critical values are found from this distribution to decide whether to reject the null hypothesis.