"Accept null hypothesis" or "fail to reject the null hypothesis"? I'm trying to conduct a Student's t-test for a table of values while trying to follow the explanation and details found on this website. I understand that if the p-value is


*

*<.01 then it's really significant

*>.05 it's not significant

*in between then we need more data


But on that page they seem to accept their null hypothesis no matter what the p-value is. So I'm really not understanding now when to accept or reject the null hypothesis. 


*

*When you do you accept or reject the null-hypothesis?

*Is it true that you are never supposed to accept your null hypothesis, but rather reject or fail to reject the null?

 A: I would suggest that it is much better to say that we "fail to reject the null hypothesis", as there are at least two reasons we might not achieve a significant result: Firstly it may be because H0 is actually true, but it might also be the case that H0 is false, but we have not collected enough data to provide sufficient evidence against it.  Consider the case where we are trying to determine whether a coin is biased (H0 being that the coin is fair).  If we only observe 4 coin flips, the p-value can never be less than 0.05, even if the coin is so biased it has a head on both sides, so we will always "fail to reject the null hypothesis".  Clearly in that case we wouldn't want to accept the null hypothesis as it isn't true.  Ideally we should perform a power analysis to find out if we can reasonably expect to be able to reject the null hypothesis when it is false, however this isn't usually nearly as straightforward as performing the test itself, which is why it is usually neglected.
Update: The null hypothesis is quite often known to be false before observing the data.  For instance a coin (being asymmetric) is almost certainly biased; the magnitude of this bias us undoubtedly negligible, but not precisely zero, which is what the H0 for the usual test of the bias of a coin asserts.  If we observe a sufficiently large number of flips, we will eventually be able to detect this miniscule deviation from exact unbiasedness.  It would be odd then to accept the "null hypothesis" in this case as we know before performing the test that it is certainly false.  The test is certainly still useful though as we are generally interested in whether the coin is practically biased. 
