Is it possible to test the converse of the null and alternative hypotheses? Consider a simple t-test.
Traditionally,
$H_{0} : \mu_{1}=\mu_{2}$
$H_{1} : \mu_{1}\ne\mu_{2}$
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?

NO
Testing a null hypothesis relies on having some distribution of the test statistic under the null hypothesis. For example, when we do a t-test, we can show that, when the null is true, the test statistic has a t-distribution with some degrees of freedom.
When the null hypothesis is not an equals sign, we cannot specify a distribution of the test statistic under the null hypothesis.
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.
