To plan a statistical hypothesis test, specify the model you will use to test the null hypothesis and the parameter of interest. Of course, all models require assumptions, so you will need to state them and check any corresponding conditions.
When the conditions fail, you might proceed with caution, explicitly stating your concerns. Or you may need to do the analysis with and without an outlier, or on different subgroups, or after re-expressing the response variable. Or you may not be able to proceed at all.
–Intro Stats (De Veaux, Velleman, Bock)