1. Let's start by stating the problem: Understanding what a two-sample t-test is and when it is used. 2. A two-sample t-test is a statistical method used to determine whether the means of two independent groups are significantly different from each other. 3. It is commonly used when you want to compare two sets of data collected from two different groups. 4. The test calculates the t-statistic, which measures the difference between the group means relative to the variability in the data. 5. The formula for the two-sample t-test statistic (assuming equal variances) is: $$ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} $$ where: - $\bar{x}_1$ and $\bar{x}_2$ are the sample means, - $n_1$ and $n_2$ are the sample sizes, - $s_p$ is the pooled standard deviation given by $$ s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}} $$ and $s_1^2$, $s_2^2$ are the sample variances of the two groups. 6. After calculating the t-statistic, you compare it to a critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom. 7. If the calculated t is greater than the critical value (or if the p-value is less than the significance level), you conclude that the two group means are significantly different. In summary, a two-sample t-test helps you decide if two independent group means differ statistically, which is very useful in research and experiments.