In the conclusion of the section What's... uhhh... the deal with the bootstrap?, I briefly touched on two important points. The first was an ominous and unexplained implication that a parametric distribution describing the sampling distribution of a statistic of interest may not exist. The second was a promise that even if, for example, the bootstrap distribution of means were identical to the t-distribution in all cases, there would still be great merit in learning how to wield the bootstrap. In this section, I hope to make clear these two points.
First, let's think back to all the tests of means we performed in Chapter 6, Testing Hypotheses. Let's ask ourselves why we wanted to test equality of means. It is certainly true that the arithmetic mean is one of the most common, if not the most common measures of central tendency and, indeed, in all of statistics. But why is it that we are always testing means? May it not be useful to ask (and test...