ABSTRACT
Hypothesis testing procedures are a common way of comparing random variables, but in many situations these procedures provide only the first step in addressing the issues that are important. For instance, suppose you have three methods for treating schizophrenia, and that the average or expected effectiveness of the jth method is μj. Further suppose that you test and reject H0:μ1=μ2=μ3. In fact, suppose you conclude that μ1≠μ2≠μ3. What do you do now? A likely response might be to choose the method that appears to be most effective. If the higher the μj the better the method, the natural thing to do is to choose the method associated with the largest sample mean, and this is exactly what is done in practice. However, this procedure leaves an important issue unresolved—how certain can you be that the best treatment yielded the highest sample mean? Put another way, how certain can you be that the best treatment was indeed selected? Ifμ2>μ1 μ2>μ3, and if X̄2>X𰌄1, and X̄2>X𰌄3, a correct selection is made because you would choose method 2. But of course there is some chance that you might observe X̄2<X𰌄1 or X̄2<X𰌄3, in which case an inferior method would be selected.
