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=0.05. Assume (σ1)^2 = (σ2)^2

Sample statistics: x̅1=228, s1=27, n1= 20 and x̅2=207, s2=25, n2= 13

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Identify the null hypothesis \( H_0 \) and the alternative hypothesis \( H_a \). Since the claim is \( \mu_1 = \mu_2 \), we have \( H_0: \mu_1 = \mu_2 \) and \( H_a: \mu_1 \neq \mu_2 \) because this is a two-tailed test.

Since the population variances are assumed equal (\( \sigma_1^2 = \sigma_2^2 \)), calculate the pooled sample variance \( s_p^2 \) using the formula: \[ s_p^2 = \frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2} \]

Calculate the test statistic \( t \) for the difference between two means using the pooled variance: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] 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 \( \alpha = 0.05 \) and \( df \) degrees of freedom. Then 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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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 (H0) typically states that the means are equal (μ1 = μ2), while the alternative hypothesis suggests they are different. The test uses sample data to decide whether to reject H0 at a given significance level (α).

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 two sample variances, reflecting the assumption of homogeneity of variance, which affects the calculation of the standard error.

Level of Significance (α) and Decision Rule

The level of significance, α, is the threshold probability for rejecting the null hypothesis, commonly set at 0.05. It represents the risk of a Type I error (false positive). The test statistic is compared to critical values from the t-distribution, and if it falls in the rejection region, H0 is rejected, indicating a significant difference between means.