Editing Power (statistics) (section)

==Description==
{{see also|Type I and type II errors}}
[[File:PowerOfTest.png|thumb|Illustration of the power of a statistical test, for a two sided test, through the probability distribution of the test statistic under the null and alternative hypothesis. ''α'' is shown as the <span style="color:blue">blue area</span>, the probability of rejection under null, while the <span style="color:red">red area</span> shows power, 1 − ''β'', the probability of correctly rejecting under the alternative.]]

Suppose we are conducting a hypothesis test. We define two hypotheses <math>H_0</math> the null hypothesis, and <math>H_1</math> the alternative hypothesis. If we design the test such that ''α'' is the significance level - being the probability of rejecting <math>H_0</math> when <math>H_0</math> is in fact true, then the power of the test is 1 - ''β'' where ''β'' is the probability of failing to reject <math>H_0</math> when the alternative <math>H_1</math> is true.

{| class="wikitable"
!   !! Probability to reject <math>H_0</math> !! Probability to not reject <math>H_0</math>
|-
| If <math>H_0</math> is True  || α || 1-α
|-
| If <math>H_1</math> is True || 1-β (power) || β
|}

To make this more concrete, a typical statistical test would be based on a [[test statistic]] ''t'' calculated from the sampled data, which has a particular [[probability distribution]] under <math>H_0</math>. A desired [[significance level]] ''α'' would then define a corresponding "rejection region" (bounded by certain "critical values"), a set of values ''t'' is unlikely to take if <math>H_0</math> was correct. If we reject <math>H_0</math> in favor of <math>H_1</math> only when the sample ''t'' takes those values, we would be able to keep the probability of falsely rejecting <math>H_0</math> within our desired significance level. At the same time, if <math>H_1</math> defines its own probability distribution for ''t'' (the difference between the two distributions being a function of the effect size), the power of the test would be the probability, under <math>H_1</math>, that the sample ''t'' falls into our defined rejection region and causes <math>H_0</math> to be correctly rejected.

Statistical power is one minus the type&nbsp;II error probability and is also the [[Sensitivity and specificity|sensitivity]] of the hypothesis testing procedure to detect a true effect. There is usually a trade-off between demanding more stringent tests (and so, smaller rejection regions) and trying to have a high probability of rejecting the null under the alternative hypothesis. Statistical power may also be extended to the case where [[multiple testing|multiple hypotheses]] are being tested based on an experiment or survey. It is thus also common to refer to the '''power of a study''', evaluating a scientific project in terms of its ability to answer the [[research question]]s they are seeking to answer.