This question is intentionally written in a very abstract way, because I am interested in a generic solution, or for pointers to different possible concrete cases in different fields:
I am measuring a particular quantity in a sample of subjects taken from a given population, and I want to apply a null hypothesis test to check whether there is evidence for "an effect" in that population. Under most circumstances, I would apply a one-sample t-test to test whether the population mean is different from zero.
In this case, however, I know that the underlying quantity can only be zero or positive, but never negative. In my view this invalidates the null hypothesis of a population mean of zero, because as soon as there is any variation across subjects in the population that mean has to be non-zero anyway. Or to put it differently: A t-test applied in such a situation effectively tests whether the variance in the population is non-zero, which however we usually assume to be the case anyway, because no person is exactly like the other.
The situation is complicated by the fact that I can only obtain estimates of that underlying non-negative quantity, where the sampling distribution can be safely assumed to be normal, so that in contrast to the true value these estimates can and will often be below zero.
Is this a known situation with a generic approach? I couldn't find anything in standard textbooks on hypothesis testing.
Are you aware of instances of this situation? It might help me get to the core of the problem if I can compare my concrete case to other concrete cases in different disciplines.