To conduct a two-sample t-test, follow these steps to ensure clarity and precision in your calculations.

A two-sample t-test is employed to compare the means of two independent samples, helping to ascertain whether they differ significantly. The first step involves calculating the means and standard deviations of both samples.

1. Calculate the Means and Standard Deviations: Begin by calculating the mean ($\bar{x}_1$ and $\bar{x}_2$) and standard deviation ($s_1$ and $s_2$) for both samples.

2. Compute the Pooled Standard Deviation: Next, you will calculate the pooled standard deviation ($s_p$) using the following formula:

   $$s_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}}$$

   In this formula, $s_1$ and $s_2$ represent the standard deviations of the two samples, while $n_1$ and $n_2$ denote the respective sample sizes.

3. Calculate the T-Statistic: Following this, compute the t-statistic ($t$) using the formula:

   $$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \cdot \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$$

   Here, $\bar{x}_1$ and $\bar{x}_2$ are the means of the two samples, $n_1$ and $n_2$ are the sample sizes, and $s_p$ is the pooled standard deviation.

4. Determine the Degrees of Freedom: Next, calculate the degrees of freedom ($df$) using the formula:

   $$df = n_1 + n_2 - 2$$

5. Find the P-Value: Finally, consult a t-distribution table or utilize a calculator to determine the p-value corresponding to your calculated t-statistic and degrees of freedom. If the p-value is less than your predetermined significance level (typically set at $0.05$), this indicates that the means of the two samples are significantly different from one another.

By following these steps, you will effectively perform a two-sample t-test and interpret its results accurately.