Power (statistics)

The power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H₀) when a specific alternative hypothesis (H₁) is true. The Statistical power ranges from 0 to 1, as statistical power increases, the probability of making an error decreases (specifically a type 2 error), type 2 error is =β, statistical power is =1-β. Therefore, as an example, if experiment 1 has a statistical power of 0.7, and experiment 2 has a statistical power of 0.95, then there is a stronger probability that experiment 1 had a type 2 error than experiment 2, and experiment 2 is more reliable than experiment 1 due to the reduction in probability of a type 2 error. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H₁) when it is true—that is, the ability of a test to detect a specific effect, if that specific effect actually exists. That is,

If ${\displaystyle H_{1}}$ is not an equality but rather simply the negation of ${\displaystyle H_{0}}$ (so for example with ${\displaystyle H_{0}:\mu =0}$ for some unobserved population parameter ${\displaystyle \mu ,}$ we have simply ${\displaystyle H_{1}:\mu \neq 0}$) then power cannot be calculated unless probabilities are known for all possible values of the parameter that violate the null hypothesis. Thus one generally refers to a test's power against a specific alternative hypothesis.

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