|So do the normality tests figure out whether the data are close enough to Gaussian to use one of those tests?|
1 RB D'Agostino, "Tests for Normal Distribution" in Goodness-Of-Fit Techniques edited by RB D'Agostino and MA Stepenes, Macel Decker, 1986.
Parts of this page are excerpted from Chapter 24 of Motulsky, H.J. (2010). Intuitive Biostatistics, 2nd edition. Oxford University Press. ISBN=978-0-19-973006-3.