We study the problem of testing the hypothesis on the “approximate normality” formulated in terms of large values of the shape parameter of an asymptotically normal underlying distribution. Considering the examples of gamma-and generalized Birnbaum—Saunders distributions, we propose one way to obtain the asymptotic of the necessary sample size for testing the mentioned hypothesis. Our approach differs from those based on contiguous alternatives or on the use of the large deviations theory for distributions of sums of independent random variables. Our method yields remarkably precise approximate formulas, what is illustrated by numerical data.