Traditionally, statistical inference is taught in the context of probability samples and the nature of sampling error. This model is the basis for the test of significance. However, there are other ways to model systematic departures from chance and it turns out that our parametric (sampling based) tests tend to be good approximations of these alternatives.
Parametric tests of hypotheses rely on sampling theory to produce estimates of likely error. If a sample of a given size is taken from a population, knowledge of the systematic
nature of sampling makes testing and confidence intervals meaningful. With a population, sampling theory is simply not relevant and tests are not meaningful in the traditional sense. Inference is useless, there is nothing to infer to, there is just the thing...the
Some get around this by appealing to super-populations that the current census represents. I find these appeals unconvincing--parametric tests are premised on probability sampling and its characteristics. A population at a given time may be a sample of a larger population over time and place. However, I don't see any way that one could legitimately argue that this is a random (or more generally any form form of a probability) sample. Without a probability sample, sampling theory and the
traditional logic of testing simply do not apply. You may just as well test on the basis of a convenience sample.
Clearly, to accept testing when using a population, we need to dispense with the basis of those tests in sampling procedures. One way to do this is to recognize the close connection between our sample-theoretic tests--such as t, Z, and F--and randomization procedures. Randomization tests are based on the sample at hand. If I collect
data on the income of males and females, the probability model and the basis for our estimates of error are repeated random allocations of the actual data values. I could compare observed differences across groups to a distribution based on this randomization. (We do this all the time in experiments, by the way, where the random sampling from a population model is rarely appropriate).
Now, it turns out that sample-theoretic tests are often good approximations of randomization tests. So, ultimately, I think tests from populations are useful and meaningful within this framework and can help to distinguish systematic from chance variation--just like with sample-based tests. The logic used to get there is a little different, but it doesn't have much affect on the practical meaning and use of tests. Of course, it might be better to just use randomization and permutation tests directly given they are easily available with all our modern computing power.