Formulas for t-statistic and degrees of freedom in Welch's t-test

1. Formula for the t-statistic: The formula for the t-statistic (often denoted as $t'$ or $t$) when comparing two independent samples with unequal variances (Welch's t-test) is given by:
$$t = \frac{\mu_1 - \mu_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
Here, $\mu_1$ and $\mu_2$ represent the population means, $s_1^2$ and $s_2^2$ represent the sample variances, and $n_1$ and $n_2$ represent the sample sizes for the two groups.

2. Formula for the degrees of freedom (d.f.): For Welch's t-test, the degrees of freedom are not a simple integer but are calculated using the Welch-Satterthwaite equation. This equation accounts for the unequal sample sizes and variances, providing a more accurate approximation of the t-distribution:
$$d.f. = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}$$
This formula calculates an effective number of degrees of freedom, which is often a non-integer value.