In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1≠ μ2; α=0.01. Assume (σ1)^2 = (σ2)^2

Sample statistics: x̅1= 61, s1= 3.3, n1= 5 and x̅2= 55.1, s2= 1.2, n2= 7

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Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \) based on the claim. Since the claim is \( \mu_1 \neq \mu_2 \), set \( H_0: \mu_1 = \mu_2 \) and \( H_a: \mu_1 \neq \mu_2 \).

Since the population variances are assumed equal (\( \sigma_1^2 = \sigma_2^2 \)), use the pooled variance formula to estimate the common variance. Calculate the pooled variance \( s_p^2 \) as: \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]

Calculate the test statistic \( t \) using the formula for two independent samples with equal variances: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] where \( s_p = \sqrt{s_p^2} \).

Determine the degrees of freedom for the test, which is \( df = n_1 + n_2 - 2 \).

Find the critical value(s) from the \( t \)-distribution table for a two-tailed test at significance level \( \alpha = 0.01 \) with \( df \) degrees of freedom. Compare the calculated \( t \)-statistic to the critical value(s) to decide whether to reject or fail to reject the null hypothesis.

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주요 개념

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Hypothesis Testing for Two Population Means

This involves testing whether there is a statistically significant difference between the means of two populations. The null hypothesis typically states that the means are equal (μ1 = μ2), while the alternative reflects the claim (μ1 ≠ μ2). The test uses sample data to decide whether to reject the null hypothesis at a given significance level.

Pooled Variance and Equal Population Variances Assumption

When the population variances are assumed equal (σ1² = σ2²), a pooled variance estimate combines the sample variances to improve the accuracy of the test statistic. This pooled variance is a weighted average of the sample variances, accounting for different sample sizes, and is used in the calculation of the t-test statistic.

Level of Significance (α) and Critical Values

The level of significance, α, is the probability of rejecting the null hypothesis when it is true (Type I error). For a two-tailed test with α = 0.01, critical values define the rejection regions in the t-distribution. If the test statistic falls beyond these critical values, the null hypothesis is rejected in favor of the alternative.