For the t test we have $D = \frac{\bar{X}-\bar{Y}-\delta}{\sqrt{\frac{S_c^2}{n_1}+\frac{S_c^2}{n_2}}}$, where $\delta = \mu_1-\mu_2$ and $S_c^2$ is the combined sample variance.
When finding the confidence interval for $\delta$ we do: $$(t_{n_1+n_2-2, 0.025}<D<t_{n_1+n_2-2,0.975})$$
However when solving this I get $$(\bar{X}-\bar{Y}-t_{n_1+n_2-2,0.975}\sqrt{\frac{S_c^2}{n_1}+\frac{S_c^2}{n_2}}, \bar{X}-\bar{Y}-t_{n_1+n_2-2, 0.025}\sqrt{\frac{S_c^2}{n_1}+\frac{S_c^2}{n_2}})$$
Whereas my lecturer gets $$(\bar{X}-\bar{Y}+t_{n_1+n_2-2,0.025}\sqrt{\frac{S_c^2}{n_1}+\frac{S_c^2}{n_2}}, \bar{X}-\bar{Y}+t_{n_1+n_2-2, 0.975}\sqrt{\frac{S_c^2}{n_1}+\frac{S_c^2}{n_2}})$$
I've tried this multiple times, and it seems to me that my solution is correct. However my lecturer uses the other one throughout the text. So I wonder how mine can be wrong, even though it is just algebra. Which one is correct and why?