ABSTRACT
We consider a well-known statistical methodology for tests of hypotheses. Suppose that we have a fixed number of independent observations Xl, ... ,Xm from a single population with a continuous probability density function (p.d.f.) f(x) and the distribution function (d.f.) F(x). We assume that this population has a finite mean /-L and a finite variance a2(> 0). In order to test a null hypothesis Ho : /-L = /-Lo against a two-sided alternative hypothesis HI : /-L =I-/-Lo at a given size or level a,O < a < 1, a customary two-sided test
would reject Ho in favor of Hi if and only if Ivm(~-j.lQ) I> tm-i ,o./2 (14.1.1)
where X == X"m = m.....I~~lXi is the sample mean, 52 == 5; = (m - 1)-1~~1 (Xi - X m )2 is the sample variance, and tm - l ,o./2 is the
upper 50Q% point of the Student's t distribution with m - 1 degrees of freedom. If the population happens to be normal, that is if
f(x) == ~ exp {- (X2-J1.2)2} ,-00 < x < 00, (14.1.2)ay21r a then the size of the test procedure (14.1.1) will be exactly Q. In statistical literature, this is a widely used and recommended test procedure to decide between Ho versus HI even when f is not given by (14.1.2). When f is not given by (14.1.2), that is when the normality of the population distribution can not be justified from practical perspectives, then many standard books and manuals on statistical methodologies recommend that the sample size m should preferably be "large", customarily m ~ 30. The wisdom behind this comes from the following well-known result:
Under Ho, the test statistic vm(~m-J.lQ) !:t N(O, 1) as m -+ 00. In other words, the distribution of J.:n(~:-J.lQ) can be approximated
by the standard normal distribution if the sample size m is "large". Then, the percentage point tm - I ,a/2 may be replaced in (14.1.1) by
Za/2, the upper ~Q point of the standard normal distribution. (14.1.3)
Since a population distribution is rarely exactly normal in real applications, practitioners often invoke the idea summarized in (14.1.3) and then follow up with the test procedure (14.1.1). But, then, the size or level of the resulting test may no longer be Q.
