If I am conducting a difference in means hypothesis test, when do we use the pooled variance and why?
Lets say the population variance was unknown for two samples, the sample sizes for the two means were small (around 20) and they follow a normal distribution. Therefore I would be using a t distribution. In this case, can't we just do a difference of means hypothesis test by adding the variances together e.g Var(X)/n1 + Var(Y)/n2 and then square rooting, (as shown below).
$t = \frac{(\bar{X}_{x}- \bar{X}_{y})-(\mu_{x}-\mu_{y})}{\sqrt{\left(\frac{\sigma^{2}_{x}}{n_x}+ \frac{\sigma^{2}_{y}}{n_y} \right)}}$
Since we can add variances for independent random variables, why is it necessary to pool? Similarly, for difference of means tests where a Z statistic is calculated (samples size large and true sample variances are known), why is it that the variances are never pooled and are added instead?
Please can someone explain what I am missing here.