# Null hypothesis of same group means

I have 4 (different sized) groups, each group having more than 20 values. Now I have made the following hypothesis:

The mean between these groups is the same and I have tested it with ANOVA getting a p-value of 0.12, i.e. I cannot reject the hypothesis.

Does that mean the mean between these groups is the same? Or else how can I test the null hypothesis that the mean between the groups is not the same, so that I can reject these null hypothesis to prove that the mean is the same?

• Conclusions drawn from hypothesis tests are properly with respect to the alternate hypothesis in terms of (a) evidence for the alternative hypothesis (if you rejected the null), or (b) lack of evidence for the alternative hypothesis (if you failed to reject the null). – Alexis Aug 8 '18 at 17:59

You cannot say that the means are the same (you can see that they're not the same, if you look at them).

You can say that you have failed to find evidence that there is a difference between the means in the population.

You cannot reject the null hypothesis. You've run the test.

You cannot prove that the means are the same.

• Might I suggest "You can say that you have failed to find evidence that there is a significant difference..." – Alexis Aug 8 '18 at 17:58
• "significant" would be best qualified with something like statistically before it, significance has meanings beyond statistics. – Heteroskedastic Jim Aug 8 '18 at 18:38
• @Alexis - I think that's what a significant difference means. You have failed to find evidence of a difference in the population. A statistically significant result is evidence of a difference. – Jeremy Miles Aug 8 '18 at 19:35
• "You cannot prove that the means are the same." You most certainly can provide evidence that the means are equivalence. (I'll leave out the word "prove", as one cannot prove difference with statistics, either. ;) – Alexis Aug 8 '18 at 19:42
• When you say "You cannot say that the means are the same (you can see that they're not the same", you have not captured the fact that the statistical inferences concern population means but the means available for inspection are the sample means. – Michael Lew Aug 8 '18 at 20:35