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The two-sample t-test is a statistical method used to compare the means of two independent groups to determine if there is a significant difference between them. This test is particularly useful when you have two sets of data and want to know if they come from populations with equal means. In our exercise, the hypothesis tested is whether the population means \(\mu_1\) and \(\mu_2\) are equal or not.

• The null hypothesis \((H_0)\) assumes that the means of the two groups are equal: \(\mu_1 = \mu_2\).
• The alternative hypothesis \((H_1)\) assumes that the means are not equal: \(\mu_1 eq \mu_2\).
• The formula for the two-sample t-test statistic, assuming equal variances, is: \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{2}{n}}} \]

In our exercise solution, the computed t-statistic was \(-2.978\), with degrees of freedom calculated as \(28\) (\(n_1 + n_2 - 2\)). Using a significance level \(\alpha = 0.05\), we compare the t-statistic to the critical value from the t-table to decide whether to reject the null hypothesis. Since our calculated p-value of approximately 0.006 is less than 0.05, there is strong evidence against \(H_0\), suggesting that the population means are indeed different.

A confidence interval provides a range of values that likely contain the true difference in means between two populations. It is another approach, besides hypothesis testing, to examine whether two sample means differ significantly. Specifically, a confidence interval helps us understand the reliability and precision of our sample estimate of the difference between population means.
For this exercise, the confidence interval for the difference between the means \((\bar{x}_1 - \bar{x}_2)\) is given by:\[ (\bar{x}_1 - \bar{x}_2) \pm t_{\alpha/2, df} \cdot s_p \sqrt{\frac{2}{n}} \]

• This provides the lower and upper bounds within which the true difference in means is likely to lie, with a certain level of confidence (usually 95%).
• In our exercise, the computed interval was approximately \((-5.27, -0.93)\).

Because zero is not included in this interval, it reinforces the conclusion of the t-test that the null hypothesis \(H_0: \mu_1 = \mu_2\) should be rejected. Hence, it confirms that there is a significant difference between the population means.

The power of a test refers to the probability that the test correctly rejects a false null hypothesis, indicating that the test is effective at detecting an actual effect when it exists. High power in a test means a lower chance of making a Type II error, which occurs when a test fails to detect a significant effect. The formula to calculate the power involves the size of the effect, sample size, and variance.
In the context of our exercise, we calculate the power of the test for detecting a true mean difference of \(3\) using the given variances and sample sizes. Here's the general process:

• Convert the true effect size into a standardized measure using this formula: \[ z_{\beta} = \frac{|\Delta| - \Delta_0}{s_p \sqrt{\frac{2}{n}}} \]
• Where \(\Delta_0 = 0\), and \(|\Delta|\) is the actual difference in means.

Using this calculation, we determined \(z_{\beta} = 3.88\), which exceeded the critical value of about 2.048, indicating high power. The estimated power was close to 1, meaning the test is very likely to correctly detect a difference of \(3\) in population means. This high power suggests that our t-test is very reliable for this particular true difference.