why there different standard error form in Hypothesis testing

Question

In Hypothesis testing,
Comparison of a sample mean with an assigned population mean: $$Z=\frac{\overline x-\mu_0}{\frac{\sigma}{\sqrt n}}\text{ Or }Z=\frac{\overline x-\mu_0}{\frac{s}{\sqrt n}}$$ But in Comparison of two independent sample means(when $\sigma_1^2$ and $\sigma_2^2$ are known): $$d=\frac{(\overline x_1-\overline x_2)-\mathbb E(\overline x_1-\overline x_2)}{\sqrt{Var(\overline x_1-\overline x_2)}}=\frac{(\overline x_1-\overline x_2)}{\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}}$$

I was wondering why $\sqrt n$ is missing in the second case$?$ Even it's missing for other cases( when $\sigma_1^2$ and $\sigma_2^2$ are unknown but equal, when $\sigma_1^2$ and $\sigma_2^2$ are unknown but unequal ).
I heartily thank if anyone explain this.

Answer 1

The sample size term is not missing. It is implicit in the expression (Variance of the sample mean). This variance is equal to the variance of X divided by sample size, as the latter expression shows. With comparison of two sample means, it is possible that sample sizes differ. That is why there are two sample size terms in the latter case.