Power of a binomial drug test when true rate is 60 percent

Question

A regulator tests a drug claimed to help $80\%$ of patients by treating $20$ patients and counting responders $X$ ; it tests $H_0:p=0.8$ against $H_a:p<0.8$ , rejecting when $X\le12$ . What is the power of this test if the true response rate is $0.6$ ?

Short answer

About $0.584$ . Power is the chance of rejecting under the alternative: $\mathbb{P}(X\le12)$ for $X\sim\text{Bin}(20,0.6)$ , about $0.5841$ (so $\beta\approx0.416$ ).

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
The Hypothesis Testing Framework
Pivotal Quantities and Exact Confidence Intervals

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