Date: May 17, 2014 5:40 PM
Author: Luis A. Afonso
Subject: Fail to reject H0: to accept it?  NO WAY !

Fail to reject H0: to accept it?  NO WAY !

(I place no obstacle to leave Jack Tomsky to accept H0 . . . anyway ).
Because we, by convention, do ascribe a large amount of probability, at least 95%( 5% alpha), before one decide to reject the Null Hypothesis it is clear that we are not allowed to state that H0 is proved true, even it is accepted.
Inside the wrongly named ?acceptance interval? lies a long tail of the test statistics with decreasing probability the parameter p0 under inspection can be found. But it is not the exclusive dweller, of course, at least the chosen Alternative Hypothesis, Ha, must be put in consideration, and it is more likely to be there as alpha decreases. Consequently Type I decreasing leads to increase Type II error. For example a Critical Value moving to right will cut a larger slice of the Ha test statistics slice to wrongly choose the Alternative against the true Null. Consequently when we intend not to reject at all the true H0, it is very alike not to reject a false Ha, i.e. to make a Type II error, then the Power= 1- beta is low and the Test Conclusion is dubious. On contrary if we want to increase the power we must increase alpha and in consequence we run a strong risk to reject a true Null.

Luis A. Afonso