t-prime and degrees of freedom formulas for two-sample t-test

1. Understanding the Formulas: The image displays two formulas related to statistical hypothesis testing, specifically for comparing two independent samples. The first formula calculates a t'-statistic, which is a variation of the t-statistic used when population variances are not assumed to be equal (Welch's t-test). The second formula calculates the degrees of freedom (d.f.) associated with this t'-statistic, using the Welch-Satterthwaite equation.

2. t'-statistic Formula Explanation: The formula for the t'-statistic is given by:
$$t' = \frac{\mu_1 - \mu_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$$
Where:
- $\mu_1$ and $\mu_2$ are the population means of the two groups.
- $s_1^2$ and $s_2^2$ are the sample variances of the two groups.
- $n_1$ and $n_2$ are the sample sizes of the two groups.
This statistic measures the difference between the two sample means relative to the variability within the samples. It's used to test the null hypothesis that the population means are equal.

3. Degrees of Freedom (d.f.) Formula Explanation: The formula for the degrees of freedom is:
$$d.f. = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(s_1^2/n_1\right)^2}{n_1 - 1} + \frac{\left(s_2^2/n_2\right)^2}{n_2 - 1}}$$
This formula, known as the Welch-Satterthwaite equation, provides an approximation for the degrees of freedom when population variances are unequal. It's a more complex calculation than the standard pooled variance t-test degrees of freedom (which is simply $n_1 + n_2 - 2$). The calculated d.f. is used to find the critical value from the t-distribution for hypothesis testing.