Arguably, one of the most important contributions of the theory of maximum-likelihood estimation is that it provides us with a systematic approach to find estimators. Similar considerations apply to interval estimation and hypothesis testing. We have very simple and intuitive ways of computing confidence intervals and testing hypotheses about the mean of a normal population, but we might well be clueless when dealing with less trivial problems. Ad hoc methods may be difficult to find, and we need general strategies. Furthermore, a sound theory is useful to assess desirable properties of a test. For instance, we just considered the probability of a type I error, but we disregarded type II errors completely. Actually, we cannot just focus on type I errors. Indeed, it is quite easy to obtain a test with zero probability of a type I error; we have just to choose an empty rejection region, C = ø. Unfortunately, by doing so, we obtain a test where the probability of a type II error is 1. We should also keep an eye on type II errors, which leads us to considering both the size and the power of a test. In the next two subsections, we first introduce these two concepts, and then we get a glimpse of a more general strategy to obtain hypothesis testing procedures.
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