Results from a statistical tests will fall into one of two regions: the rejection region— which will lead you to reject the null hypothesis, or the acceptance region, where you provisionally accept the null hypothesis.

The acceptance region is basically the complement of the rejection region; If your result does not fall into the rejection region, it must fall into the acceptance region. Therefore, it’s important to understand what a rejection region is. If you aren’t familiar with the rejection region, this short video explains it:

## More Formal Definition of Acceptance Region

According to the Concise Encyclopedia of Statistics, the acceptance region is “…the interval within the sampling distribution of the test statistic that is consistent with the null hypothesis H_{0} from hypothesis testing.”

In more simple terms, let’s say you run a hypothesis test like a z-test. The results of the test come in the form of a z-value, which has a large range of possible values. Within that range of values, some will fall into an interval that suggests the null hypothesis is correct. That interval is the acceptance region.

## Why is it “Provisionally” Acceptance?

You *provisionally* accept the null hypothesis because a hypothesis test doesn’t tell you which hypothesis is true (the null or alternate hypothesis), or even which is probably true. The only thing is tests is whether there’s enough evidence in your data to reject the null hypothesis. Failure to accept the alternate hypothesis doesn’t make the null hypothesis true.

Let’s say our “experiment” is where you caught a child red-handed with a stolen cookie:

- Null hypothesis (H
_{0}): The child didn’t steal the cookie (innocent until proven guilty!). - Alternate hypothesis (H
_{1}): The child did steal the cookie.

You’re pretty certain the child stole the cookie. But after gathering all evidence, you don’t find enough evidence to say for sure that the child is guilty. Therefore, **there isn’t enough evidence in support of the alternate hypothesis that the child is guilty**. In other words, you can’t reject the null hypothesis that the child is innocent in favor of the hypothesis that the child is guilty. That doesn’t mean the child *is* innocent. You just didn’t have enough evidence to prove them guilty. Although your result fell into the acceptance region, you don’t actually “accept” the null hypothesis of innocence. You just provisionally (perhaps begrudgingly) accept it and let the child off without punishment. Later on you might find crumbs in their bed, leading you to revisit your findings.

This subtle difference may seem pedantic, and in elementary statistics it’s usually not an important matter to stress. However, if you plan to publish your results you should never say you “accept the null hypothesis”. You can say that you provisionally accept it, or that you **failed to reject it**. Check with your intended publication (or with your professor) to see what wording they prefer.

## References

Dodge, Y. (2008). The Concise Encyclopedia of Statistics. Springer.

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