Test the claim about the difference between two population means and 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=52, s1=4.8, n1=32 and x̅2=50, s2=1.2, n2=40

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Identify the null and alternative hypotheses based on the claim. Since the claim is \( \mu_1 > \mu_2 \), set the hypotheses as: \( H_0: \mu_1 \leq \mu_2 \) and \( H_a: \mu_1 > \mu_2 \).

Determine the significance level \( \alpha = 0.01 \) and note that this is a right-tailed test because the alternative hypothesis is \( \mu_1 > \mu_2 \).

Since the population variances are assumed unequal (\( \sigma_1^2 \neq \sigma_2^2 \)), use the two-sample t-test for unequal variances (Welch's t-test). Calculate the test statistic using the formula: \[ t = \frac{\overline{x}_1 - \overline{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} \]

Calculate the degrees of freedom for the test using the Welch-Satterthwaite equation: \[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} \]

Find the critical t-value from the t-distribution table corresponding to \( \alpha = 0.01 \) and the calculated degrees of freedom. Compare the test statistic to this critical value to decide whether to reject \( H_0 \).

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주요 개념

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Hypothesis Testing for Two Population Means

This involves comparing the means of two independent populations to determine if there is statistical evidence supporting a claim about their difference. The null hypothesis typically states no difference (μ1 = μ2), while the alternative reflects the claim (μ1 > μ2). The test uses sample data to decide whether to reject the null at a given significance level.

Unequal Variances (Welch's t-test)

When population variances are not assumed equal, Welch's t-test is used instead of the pooled variance t-test. It adjusts the degrees of freedom and test statistic to account for different sample variances, providing a more reliable inference about the difference between means under heteroscedasticity.

Significance Level and p-value

The significance level (α) is the threshold for rejecting the null hypothesis, representing the probability of a Type I error. A p-value is calculated from the test statistic and compared to α; if p-value ≤ α, the null is rejected, supporting the claim. Here, α=0.01 indicates a strict criterion for evidence.