Why are standard frequentist hypotheses so uninteresting? In almost any textbook introducing the topic of frequentist statistics, null hypotheses of the form $H_0: \mu=\mu_0$ or similar are presented (the coin is unbiased, two measurement devices have identical behavior, etc.). Classic statistical tests such as the $Z$ or $T$ tests are then based on rejecting these null hypotheses.
As I see it, these types of equality hypotheses are uninteresting for a number of reasons:

*

*In "real life" one is always only interested in some finite accuracy $\epsilon$, meaning the hypothesis of interest is actually of the form $H_0: |\mu-\mu_0|<\epsilon$.

*The equality hypothesis is a priori known to be wrong when considering continuous variables (no coin is perfectly unbiased in reality!), and as a corollary,

*The fact that the null hypothesis cannot be rejected is by definition temporary, and is an artifact of not enough data. Given enough data, any type of equality hypothesis on continuous variables will be rejected in a real world use case.

So, why are these hypotheses still used, both in text books and in applications, while it is difficult to find formulas for more interesting* "real" hypotheses?

* e.g. a question I recently asked regarding these types of hypotheses 



*

*In "real life" one is always only interested in some finite accuracy ϵ, meaning the hypothesis of interest is actually of the form H0:|μ−μ0|<ϵ.


Indeed, it is an important practical issue that a significant effect turn out to be small and therefore "insignificant" for practical purposes. As an anecdotal example: bilingual children start speaking later than their peers in monolingual families... but this discrepancy is smaller than that between boys and girls (boys start speaking later than girls.) - The effect is statistically significant, but of too little importance to deprive your child from learning a second language from their birth.
There are procedures for testing whether the effect is bigger than some pre-specified value, see Equivalence testing. However these testing procedures are based on simpler testing procedures, which therefore must be learned beforehand.


*

*The equality hypothesis is a priori known to be wrong when considering continuous variables (no coin is perfectly unbiased in reality!), and as a corollary,

*The fact that the null hypothesis cannot be rejected is by definition temporary, and is an artifact of not enough data. Given enough data, any type of equality hypothesis on continuous variables will be rejected in a real world use case.


The seeming paradox arises here, because we abandon some idealizations, while preserving the other: thus, while taking into account that no coin is perfect, we still assume that a) we can make an infinite number of trials, and b) the effect is important (greater than some $\epsilon$.) A well known adagio about models is that "All models are wrong, but some are useful" and statistical testing is a case in point.
In short story On Exactitude in Science Jorge Luis Borges has given a well-known poetic description of why any useful model/theory is necessarily approximate, whereas the one that takes into account everything is totally useless. (See here for the full text.)
