In all hypothesis tests, you end up with a test statistic that follows a specific distribution, and you have to decide on a probability based on that.
My question is: how do we know that a given statistic follows a given distribution?
- was the distribution created to fit experimental results
- or is the distribution directly derived from theory
I'm asking because, contrary to some probability distributions (eg. binomial, geometric) which are directly derived from probability rules, for test statistic distributions most textbooks just state the fact (at least, most textbooks I have seen).
Where do the $t$, $F$ and $\chi^2$ distributions come from?