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As title states, why is not rejecting null hypothesis not equivalent to accepting it? I can't find any information about a 'state' other than accepting or rejecting null hypothesis.

ziobo
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    Generally speaking, in these matters one is interested in the question "can the observed data be explained by some sort of neutral hypothesis?" The test may then show that you can't rule that out within the pre-specified confidence band. That, however, is radically different from showing that the neutral hypothesis must be applicable. Maybe there is some other explanation you haven't considered, or at least one that the test did not consider. – lulu Feb 11 '21 at 18:12
  • Your first sentence goes directly against what your title says. Could you please tidy that up for us? – Arthur Feb 11 '21 at 18:22
  • @Arthur sorry for that, somehow I missed the "not" before "equivalent". I believe your answer is what I'm looking for. – ziobo Feb 11 '21 at 18:24

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There are three states an hypothesis can be in: accepted, rejected, or not concluded.

Before we do an experiment and analyze the data, we can't conclude anything about the null hypothesis. Then after we do the experiment and the statistics, if the results are far away from what the null hypothesis would imply, we say we reject the null hypothesis.

However, if the analysis reveals that the result was rather close to what the null hypothesis predicts, then we can't reject it. However, this does not mean we immediately accept it. It could be that the null hypothesis is wrong, but that the true solution is too close to the null hypothesis for our experiment to distinguish.

This is how failing to reject the null hypothesis does not mean accepting it. It just means we can't conclude.

Arthur
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