When creating this question, StackExchange told me that this question appears subjective and is likely to be closed, but I'm going to ask it anyway to see if anyone can help me answer this. Sorry if this is not a valid question for Math StackExchange
In previous courses (and looking online) the method I've always been taught to find a confidence interval with $c\%$ for $\mu_1 - \mu_2$ when both $\sigma_1$ and $\sigma_2$ are unknown is $$(\bar{x}_1 - \bar{x}_2) \pm t_c \sqrt{\frac{s^2_p}{n_1}+\frac{s^2_p}{n_2}} $$ With degrees of freedom $n_1+n_2-2$ and $s^2_p = \frac{(n_1-1)(s^2_1)+(n_2-1)(s^2_2)}{n_1+n_2-2}$
However in my class this year my professor told me that the formula is: $$(\bar{x}_1 - \bar{x}_2) \pm t_c \sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}} $$ With degrees of freedom being the smaller of $n_1-1$ and $n_2-1$
My question is this: Are these two formulas interchangeable? Or is one better than the other? Thank you.