Power of a Test about a Binomial Parameter

The left plot shows the power function of the selected test with the value of marked in purple. The power function is a function of the true value of the parameter . Its value at is called the significance of the test. The right image shows the critical region as a subset of the range of the test statistic. The expected value of the test statistic if , is marked in purple.


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To test the hypothesis or the hypothesis about a Bernoulli parameter , it is decided that if the number of successes in independent trials is above (for the first test) or below (for the second test) a certain threshold, then the hypothesis will be accepted. This threshold is the boundary of what is called the critical region—whether the number of successes falls in this region determines the outcome of the test. The probability that the number of successes falls in the critical region is a function of the true value of . In the case of the test , the power function gives this probability for . If the number of successes falls outside the critical region, then a type II error (false negative) occurs. The power function gives the opposite of this probability, that is, for each value of covered by , the power function gives the probability that the test does not yield a false negative.
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