I'm currently teaching basic probability and after that I would like to do some simple hypothesis testing. Frankly, probability theory and statistics have never really interested me but I feel like my students like how it can very often be applied to everyday life, so I want to teach them as much as possible.
I've only learned about hypothesis testing at university level, and there it was just at the end of a very theoretical course on probability. I know some parametric and non-parametric tests from my studies (and from talking with friends), for example $t$-tests, sign-tests, $\chi^2$-tests. I usually have to look up what they are about, when I can apply them and how they are applied. This is a huge disadvantage when trying to answer the question:
Which statistical tests should I thoroughly teach?
Also, it is very important to me that my students do not only blindly follow some algorithm and have no idea why (as seems to be the rule for high schoolers doing statistics). I want to teach them why exactly a certain test works the way it does. This obviously requires myself to understand why certain tests work the way they do. The problem is, most sources I've found have rather little detail on "why" and much more on "how".
Even worse, some reasonings behind certain tests seem to be out of scope for most high schoolers (within a normal curriculum). I looked up what the idea for the $t$-test is and I think it would be rather confusing to point out that by $z$-transforming the sample mean, the random variable is no longer normally distributed and then to introduce the Student's $t$-distribution. So maybe, a more accurate question would be:
Which statistical tests can I thoroughly teach while not skipping the idea behind the test?
This is high school level, 12th grade. They already know single-variable calculus, so at least integrals can be understood properly.