Typically, I am curious about the process of proving the hypotheses wrong. If I am not mistaken, no hypothesis can be proven correct, we can only prove hypothesis wrong.
The hypothesis that you refer to is its statistical definition. As someone else pointed out, any scientific statement is a hypothesis. When you reject the null-hypothesis you are accepting the alternative hypothesis.
Now, what statistical tests cannot say is which of the infinite alternate hypotheses is actually true; in other words our alternative hypothesis is always general such as m ≠ µ. There are one-tailed tests that are more stringent than this and address the direction of the inequality (either m > µ or m < µ).
This is so for statistical tests because each of them can only test if a certain observation fits a given model. There are many popular models which are the basis of the common statistical tests. Note that if you intend to test if a certain observation follows Poisson process, then you are trying to validate an observation as a case of a certain model instead of the rejection based tests.
However you should note that it is difficult to prove that a certain data follows a model because you need to verify each and every parameter. Two distributions are considered same only if all of their moments match. This is essential because there can be an infinite number of functions that produce a certain shape (a function can be expressed as McLaurin series and if all moments are same the functions ought to be the same).
Now what we regularly do is to determine if something that we observed is not just because of a random measurement error and that is why we aim to reject the null hypothesis which assumes that the data is an outcome of a Gaussian distribution model (You can see this answer in in Biology.SE for details).
I would say that you can validate a possibility instead of reject infinite possibilities if you know the underlying model and have enough data. This is how the prediction-validation-correction method works. In certain cases the model can be built using basic principles instead of inferring from the data. Finally, all models have assumptions and you need to be sure if your experiments satisfy these assumptions or not; if not you should revise the model.
Intuition or "gut feeling" is no alternative method
Intuition cannot be called a method because there is no set protocol for it. And there is no way to replicate it. Biologically speaking, intuitive guess is basically a case of applying multiple statistical tests (sub-consciously, we can say). Intuition works only when you have a good deal of prior information (sub-consciously or consciously).
