# Is it possible to test the converse of the null and alternative hypotheses?

Consider a simple t-test.

$$H_{0} : \mu_{1}=\mu_{2}$$

$$H_{1} : \mu_{1}\ne\mu_{2}$$

Is it possible to instead assume:

$$H_{0} : \mu_{1}\ne\mu_{2}$$

and test if:

$$H_{1} : \mu_{1}=\mu_{2}$$?

Random example: a guinea pigs and giraffes are different heights. But a shrink ray can make them the same size. Does the shrink ray make every organism significantly the same size?

However, you can do some kind of equivalence testing where you use two one-sided tests to put bounds on the extent to which a value differs from some theorized value. Perhaps $$\mu_1$$ and $$\mu_2$$ are practically equivalent if they are within $$0.1$$ of each other, and you can test that they are.
• Testing a null hypothesis relies on having some distribution of the test statistic under the null hypothesis. <...> When the null hypothesis is not an equals sign, we cannot specify a distribution of the test statistic under the null hypothesis. Yet we routinely run one-sided tests, e.g. one-sided $t$-test with $H_0$ covering one half of the real line. You may want to briefly explain how that is possible. Aug 3 at 6:11