I got some feedback that my definitions of Type I errors and Type II errors would be clearer if I specified what the actual hypothesis are. I wanted to avoid symbols like mu or pi, so here is what I wrote.
Consider a new drug that we will put on the market if we can show that it is better than a placebo. In this context, H0 would represent the hypothesis that the average improvement (or perhaps the probability of improvement) among all patients taking the new drug is equal to the average improvement (probability of improvement) among all patients taking the placebo.
as well as
Suppose we are comparing two groups of patients, one with a possibly dangerous exposure (e.g., non-ionizing radiation), and the other unexposed. In this context, H0 would represent the hypothesis that the average level of harm (or perhaps the probability of harm) among those with exposure is equal to the average level (probability) of harm among those without the exposure.
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