OK Mr Anonymous Unverifiable PhD, what if you do a different study? In this new study, thousands of studies are done, and show results with p-values p-1, p-2, p-3, .... However, the results of most of these studies mysteriously vanish, lets say a bunch of them aren't reported by the researcher, and a bunch more are not accepted by any journal of note, and vanish into obscurity. The vanishing studies aren't random, but the vast majority of them are ones where the null hypothesis were not rejected. Besides this, the journals select for the most "interesting" studies, which for this example let's say it means that the results were surprising and in contradiction with either conventional wisdom or previous studies. Let's call the remaining p-values tainted-p-1, tainted-p-2, tainted-p-3. These p-values are nothing new, but rather a subset of the original set. Let's say only 8% (the acceptance rate of Nature) of the original studies make it to this second group, biased towards the more "interesting" studies with mostly positive results.

So my question to you is, are you equally as confident in the hypotheses from the smaller subset of studies as you are in the hypotheses from the second set? And what, if any, is your mathematical reasoning behind this answer?