Suppose, regardless the data, we reject the null hypothesis ($H_{0}$) by tossing a fair coin (i.e. reject $H_{0}$ with probability 0.5). What is the type I error rate and power of this test.
2026-03-31 23:53:30.1775001210
Hypothesis testing for type I error rate and power
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Either the null hypothesis is true or it isn't. If the null hypothesis is true, then it is rejected with probability $0.5$ so the Type I error rate is $\alpha = 0.5$. If the null hypothesis is false (and the alternative hypothesis is instead true), then the null hypothesis is accepted with probability $0.5$ so the Type II error rate is $\beta = 0.5$. So the power is $1 - \beta = 0.5$.