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 contrary) 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.
