Chapter 5

Testing Hypotheses

In this chapter, we develop improved methods for testing hypotheses by means of the bootstrap, introduce permutation testing methods, and apply these and the t-test to experiments involving one and two samples. We then address the obvious but essential question: How do we choose the method and the statistic that is best for the problem at hand?

# 5.1 TESTING A HYPOTHESIS

Suppose we were to pot a half-dozen tomato plants in ordinary soil and a second half-dozen plants in soil enriched with fertilizer. If we wait a few months, we can determine whether the addition of fertilizer increases the resulting yield of tomatoes, at least as far as these dozen plants are concerned. But can we extend our findings to all tomatoes? Or, at least, to all tomatoes of the same species that are raised under identical conditions apart from the use of fertilizer.

To ensure that we can extend our findings, we need to proceed as follows: First, the 12 tomato plants used in our study must be a random sample from a nursery. If we choose only plants with especially green leaves for our sample, then our results can be extended only to plants with especially green leaves. Second, we have to divide the 12 plants into two treatment groups at random. If we subdivide by any other method, such as tall plants in one group and short plants in another, then the experiment would not be about fertilizer but about our choices.

I performed just such a randomized experiment a decade or so ago. ...

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