Suppose the situation in which a researcher, using, by choice, to let alpha=0.05 in the pack of tests just he (she) is performing, did find p=0.012. Consequently one thinks that something?s going wrong: or the data is very anomalous in comparison with the usual, or this time H0, the null hypotheses, doesn?t fulfil. The test statistics direct me, by pure induction only, to reject H0, because p is the larger value I expect to find EVEN THAT the null is true. If not true is very likely to find p-values that are more or less smaller because p-values, under H0 true, follows a uniform distribution. Then, evidently we must have something as prob. (p-value <0.05 | real world) at most = 0.012. By other words: I never will commit a Type I error (reject H0 true) larger than this latter value. We are full confident about our decision to reject the Null. What else could be? If, on contrary, p>alpha we stay at a shady situation: the most we can say is that there is no sufficient evidence to reject H0. ____________
Not every body have confidence in this strictly inductive approach. For example R. Hubbard, J. Scott Armstrong, Incompatible Measures of ?Statistical Significance? . . . Dec. 22, 2004.
I have anything against the critics that a statistical significant occurrence could be completely unworthy in practical/economic terms, but this´ a complete different problem, of course.