Take a sample of size $n$ from any distribution with mean $\mu_X$ and standard deviation $\sigma_X$

If $n$ is large enough (≥ 30 ish),

$\bar{X}$ ~ Normal $( \mu_{\bar{X}} = \mu_X, \sigma_{\bar{X}} = \frac{\sigma_X}{\sqrt{n}})$

So, if $n$ is big enough, the sampling distribution of $\bar{X}$ is Normal, and 95% of the time we should get an $\bar{x}$ within $\pm 2 \frac{\sigma_X}{\sqrt{n}}$ of $\mu_X$

Therefore, 95% of the time, if we have an $\bar{x}$, $\mu_X$ should be within $\pm 2 \frac{\sigma_X}{\sqrt{n}}$ of $\bar{x}$