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= 44.5, s1= 5.85, n1= 17 and x̅2= 49.1, s2= 5.25, n2= 18

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Identify the null and alternative hypotheses based on the claim. Since the claim is \( \mu_1 \geq \mu_2 \), the hypotheses are: \( H_0: \mu_1 - \mu_2 \geq 0 \) and \( H_a: \mu_1 - \mu_2 < 0 \). This is a left-tailed test.

Since the population variances are assumed equal (\( \sigma_1^2 = \sigma_2^2 \)), calculate the pooled sample variance 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 \( t \) using the pooled variance: \[ t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{S_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \] Note that the hypothesized difference under \( H_0 \) is 0.

Determine the degrees of freedom for the test, which is \( df = n_1 + n_2 - 2 \). Then, find the critical value \( t_{\alpha} \) from the t-distribution table corresponding to \( \alpha = 0.01 \) and the calculated degrees of freedom for a left-tailed test.

Compare the calculated test statistic \( t \) to the critical value. If \( t \) is less than the critical value, reject the null hypothesis; otherwise, do not reject it. This will help you decide whether there is sufficient evidence to support the claim \( \mu_1 \geq \mu_2 \).

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주요 개념

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Hypothesis Testing for Two Population Means

This involves formulating null and alternative hypotheses about the difference between two population means (μ1 and μ2). The goal is to use sample data to decide whether to reject the null hypothesis at a given significance level (α), based on evidence from the test statistic.

Assumption of Equal Population Variances

When the population variances are assumed equal (σ1² = σ2²), a pooled variance estimate is used to calculate the test statistic. This assumption simplifies the test and affects the degrees of freedom, leading to a more precise inference about the difference between means.

Significance Level and Decision Rule

The significance level (α = 0.01) defines the probability of rejecting the null hypothesis when it is true (Type I error). It sets the critical value(s) for the test statistic, guiding whether the observed data provide enough evidence to support the claim that μ1 ≥ μ2.