After reading this article you should identify with one of these arguments.

Statistical inference or inferential statistics refers to a set of methods by which we can make statements regarding an entire population from the observation of only a part of it. Two basic ways to perform statistical inference are estimation and hypothesis testing, also called "hypothesis testing." A statistical hypothesis is a statement about a probability distribution (for example, we might say that a certain phenomenon behaves in a way that can be explained by a binomial distribution). In particular, a statistical hypothesis can be a statement with respect to a parameter (if we know that the distribution is binomial, then we could hypothesize that the probability of success is p = 0.5). A statistical hypothesis test is a procedure by which what is proposed by a hypothesis is compared against the empirical evidence provided by the observation of data from the population on which the hypothesis is made. The title that has been given to this discussion has to do with a basic issue in hypothesis testing, for which it could be considered that the name "proof" is not very appropriate. The foregoing, because this noun could give the reader the impression that the procedure implies certainty, which in statistics, of course, is hardly possible. Additionally, a situation that often causes doubts in students learning the statistical hypothesis testing method with the Neyman-Pearson approach for the first time is why it is said: The null hypothesis is not rejected. and it can't just be said The null hypothesis is accepted Let us consider the following situation, very simplified, but that helps us to clarify ideas: Suppose we are presented with a closed, opaque box, inside which we know there are 100 marbles that can be red, white, or a mixture of both. We are interested in saying something about all the marbles inside the box (are they all red, all white, or how many of each are there). What would be a completely safe way to do it? If we were able to empty the box, for example, and examine the entire contents, then we would know for sure what conditions exist inside the box; But, what happens then if for some reason we cannot examine all the content, although we can examine a part of it? One way to deal with the impossibility of examining all the content is to bring in probability. Suppose we are told that the box contains only white marbles, but our assumption is that there are actually some red ones inside. We can pose our first test of prototype hypotheses as follows: H0: In the box there are only white marbles Ha: In the box there is at least one red marble We now need to test our null hypothesis against the evidence we get from looking at data by taking a small number of marbles out of the box (without being able to look at the others) and examining their color. Our test statistic, which we will call X, in this case is the number of red marbles among those drawn. Since the appearance of at least one red marble would make it completely clear that the null hypothesis is not true, the rejection region is R = {X 1}. Therefore, we would reject the null hypothesis if X 1. Suppose that resource constraints allow us to only draw five marbles, which we do, and we observe that they are ALL white. Now two important questions arise: Do the observed data contradict the null hypothesis? They certainly do not, because the test statistic did not take a value that was within the rejection region, that is, none of the marbles drawn were red. At this point we are unable to verify our assumption that at least some of the marbles in the box are red, so the most we can say is: "There is no statistical evidence against H0", with which we just indicate that the small portion of reality that resource limitations allowed us to observe is not in disagreement with what the null hypothesis says. So far, then, "We cannot reject H0". On the other hand, does the fact that none of the marbles we draw are red PROVE that all the marbles in the box are white? Of course not. The only way our observation would show that all the marbles in the box are white would be if we observed all of them, which we initially said was not possible. Therefore, although our observation does not contradict the null hypothesis, it does not irrefutably prove it either. Consequently, it would not be correct to say "We accept that H0 is true", because this is not really known to us.