Which null hypothesis can be tested by a $\chi^2$ test?
could you please write the description of the relevant null hypothesis or at least give an example.
There are lots of things to test with chi-square. Perhaps the most common, in my experience anyway, is whether two categorical variables are associated. But also whether one variable fits a certain distribution. It can also come up in testing of various models.
It is used to check if a given sample is of a certain law (goodness-of-fit test), and it's also used to check whether to samples are independent. Here's an example using R