[Question] How important are assumptions in hypothesis tests?
Certain statistical tests, such as the Z-test for an equality of a mean, chi squared test for cont. tables and the significance of the correlation coefficient are often based on certain assumptions, such as data that is normally distributed. However, often i seem to not see any visual description of the data that is being tested (for example histograms) or any tests (like the Kolmoforov-Smirnov test) being showcased for the distribution of the data. I understand that the test assumptions might be sattisfied or differ insignificantly when the data follows a distribution similar to a normal one, such as the student distribution, however, why are these tests often preformed even on data that is not shown to be normaly distributed? Are these assumptions strict enough that even when a non normaly distributed data satisfies or rejects the null hypothesis, we can be satisfied with the result and accept it as a probable fact? The same question follows on other statistical tests, when they are being preformed without testing whether these assumptions are satisfied.