Functional Anatomy of the Null Hypothesis and of Tests of It

The author compared simulations of the "true" null hypothesis (z) test, in which ò was known and fixed, with the t test, in which s, an estimate of ò, was calculated from the sample because the t test was used to emulate the "true" test. The true null hypothesis test bears exclusively on calculating the probability that a sample distance (mean) is larger than a specified value. The results showed that the value of t was sensitive to sampling fluctuations in both distance and standard error. Large values of t reflect small standard errors when n is small. The value of t achieves sensitivity primarily to distance only when the sample sizes are large. One cannot make a definitive statement about the probability or "significance" of a distance solely on the basis of the value of t.