# Statistical hypothesis testing

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A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study (not controlled). In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold probability, the significance level. The phrase "test of significance" was coined by Ronald Fisher: "Critical tests of this kind may be called tests of significance, and when such tests are available we may discover whether a second sample is or is not significantly different from the first."[1]

Hypothesis testing is sometimes called confirmatory data analysis, in contrast to exploratory data analysis. In frequency probability, these decisions are almost always made using null-hypothesis tests (i.e., tests that answer the question Assuming that the null hypothesis is true, what is the probability of observing a value for the test statistic that is at least as extreme as the value that was actually observed?)[2] One use of hypothesis testing is deciding whether experimental results contain enough information to cast doubt on conventional wisdom.

A result that was found to be statistically significant is also called a positive result; conversely, a result whose probability under the null hypothesis exceeds the significance level is called a negative result or a null result.

Statistical hypothesis testing is a key technique of frequentist statistical inference. The Bayesian approach to hypothesis testing is to base rejection of the hypothesis on the posterior probability.[3] Other approaches to reaching a decision based on data are available via decision theory and optimal decisions.

The critical region of a hypothesis test is the set of all outcomes which, if they occur, will lead us to decide that there is a difference. That is, cause the null hypothesis to be rejected in favor of the alternative hypothesis. The critical region is usually denoted by the letter C.