I have a game with somewhat intrincate rules regarding its prizes (it's vide-bingo). Thankfully, we managed to find out the expected mean $\mu_0$, for the liniarity of the Expected Value. Even though, we can't do the same with variance, so we don't know the standard deviation $\sigma_0$
We have $n$ simulations of the random variable $X$, which returns the amount won, with unkown distribution, with mean $\mu_x$ and standard deviation $\sigma_x$. Sadly, we have no access to individual values $x_i$ so we can't calculate $Var(X)$ nor $sd$, we only have $\tilde x$.
How would you do an hypothesis test for $H_0: \mu_x = \mu_0$?
My first approach was to look for $P(Z>\mid\frac{\tilde x-\mu_0}{\sqrt{\mu_0}}\mid)$ but seems a bit off, since when I change the scale, the probability changes.
Edit: I know how to do the Hypothesis Test if I'd knew the $sd$ and I know also how to calculate it. The problem is that due to the nature of the simulations, that's not something I can keep track of. So the question is whether it's possible or not to make an Hypothesis Test without using Standard Deviation of any. Or making some kind of estimate using only means (which would seem weird)
Long, hopefully helpful, Comment. Not even close to an Answer.
First, $H_0 : \mu = \bar x$ is not a reasonably posed null hypothesis. The null hypothesis is expressed in terms of parameters, never anything derived from data. (And in a continuous distribution there is essentially 0 chance that $\mu = \bar x.$)
Second, I do not see how it is possible to test a null hypothesis about a population mean $\mu$ based on $\bar X,$ but with no idea about the variance of the population or sample. If this is an exercise that refers back to an example, perhaps the population standard deviation (SD) $\sigma$ or variance $\sigma^2$ is given there. Or maybe the actual data are there so you can estimate $\sigma$ as the sample SD $S$. (You would not compute a population variance $Var(X)$ from data. From data you might compute an estimate of it.)
There are enough misconceptions about terminology and notation in your Question, that I think you need to read the accompanying material very carefully and try to edit your question accordingly.
I see you have been active on this site with probability and combinatorics Q & A. Guess you may be just starting with statistics. Lots of new ideas, terminology and notation to absorb.