[themadryaner]

You cannot prove the null hypothesis; you can only disprove it. The p-value is the probability that the null hypothesis is true. So if the null hypothesis is that there is no difference, and there is a low probability of that being true, then you have shown that there is a difference between the groups. If the p-value is high, you do not show that the null hypothesis is true. Instead, you show that you did not find a statistically significant difference. This can happen when the difference between the values is small or there is not enough data to make the difference clear, which is why a high p-value is not enough to reject the alternative hypothesis.

[jostylr]

> The p-value is the probability that the null hypothesis is true.

No, the p-value is the probability that the given data generated is as far away or farther by random chance given the null hypothesis is true. That is to say, we assume the null hypothesis to make some predictions and see if the data is a likely occurrence under those assumptions.

This is not the same as the probability that the null hypothesis is true. If that is what you want (and most of us do want this), then Bayesian methods are more appropriate though they are more complicated and more sensitive to initial assumptions.

[nonbel]

>"You cannot prove the null hypothesis; you can only disprove it."

Sure, but thats why the null hypothesis should be predicted by your theory. Then you are checking your theory. Most people could care less about whether there is a miniscule difference between groups or not.

>"The p-value is the probability that the null hypothesis is true."

No, as pointed out by jostylr this is wrong.