I am on the hypothesis testing for two populations unit. I need some intuitive explanation as to why this formula is used. My statistics professor put this up on the board but he didn't explain why its true.
For a T distribution, the formula for the degrees of freedom is:
$ \large \mathrm{df} = \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}}$
Here $s_1, s_2$ are the sample standard deviations and $n_1,n_2$ are the sample sizes.