Why the Null Hypothesis is Not Accepted (1 of 5)
A null hypothesis is not accepted just because it is not rejected. Data not sufficient to show convincingly that a difference between means is not zero do not prove that the difference is zero. Such data may even suggest that the null hypothesis is false but not be strong enough to make a convincing case that the null hypothesis is false. For example, if the probability value were 0.15, then one would not be ready to present one's case that the null hypothesis is false to the (properly) skeptical scientific community. More convincing data would be needed to do that. However, there would be no basis to conclude that the null hypothesis is true. It may or may not be true, there just is not strong enough evidence to reject it. Not even in cases where there is no evidence that the null hypothesis is false is it valid to conclude the null hypothesis is true. If the null hypothesis is that µ1 - µ2 is zero then the hypothesis is that the difference is exactly zero. No experiment can distinguish between the case of no difference between means and an extremely small difference between means. If data are consistent with the null hypothesis, they are also consistent with other similar hypotheses.