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• The power of a significance test measures its ability to detect an alternative hypothesis. The power to detect a specific alternative is calculated as the probability that the test will reject H0 when that alternative is true. This calculation requires knowledge of the sampling distribution of the test statistic under the alternative hypothesis. Increasing the size of the sample increases the power when the significance level remains fixed.
• An alternative to significance testing regards H0 and Ha as two statements of equal status that we must decide between. This decision theory point of view regards statistical inference in general as giving rules for making decisions in the presence of uncertainty.
• In the case of testing H0 versus Ha, decision analysis chooses a decision rule on the basis of the probabilities of two types of error. A Type I error occurs if H0 is rejected when it is in fact true. A Type II error occurs if H0 is accepted when in fact Ha is true.
• In a fixed level α significance test, the significance level α is the probability of a Type I error, and the power to detect a specific alternative is 1 minus the probability of a Type II error for that alternative.