An equivalence hypothesis test formally tests if two population means are equivalent,
that is, practically the same.
An equality hypothesis test can never prove that the means are equal, it can only ever disprove
the null hypothesis of equality. It is therefore of interest when comparing say a new treatment
against a placebo, where the null hypothesis (assumption of what is true without evidence to the
contray) is that the treatment has no effect, and you want to prove the treatment produces a
useful effect. By contrast, an equivalence hypothesis test is of interest when comparing say a
generic treatment to an existing treatment where the aim is to prove that they are equivalent,
that is the difference is less than some small negligible effect size. A equivalence hypothesis
test therefore constructs the null hypothesis of non-equivalence and the goal is to prove the
means are equivalent.
The null hypothesis states that the means are not equivalent, against the alternative
hypothesis that the difference between the means is within the bounds of the equivalence
interval, that is, the effect size is less than some small difference that is considered
practically zero. The hypothesis is tested as a composite of two one-sided t-tests (TOST),
H01 tests the hypothesis that mean difference is less than the lower bound of the
equivalence interval, test H02 that the mean difference is greater than the upper
bounds of the equivalence interval. The p-value is the greater of the two one sided t-test
p-values. When the test p-value is small, you can reject the null hypothesis and conclude the
samples are from populations with practically equivalent means.