4.5 Hypothesis Testing
In statistics, a hypothesis is a probabilistic assertion. A hypothesis might assert that a random variable’s mean is 1—or perhaps that its variance is less than 5. A hypothesis might state that a random variable is normally distributed. It might assert that two random variables are independent—or that they have the same mean.
With hypothesis testing, we assess how reasonable a hypothesis is in light of available empirical data. The hypothesis to be tested is called the null hypothesis and is denoted . We would like to answer the question “how plausible is it that is true given the data we obtained?” That question is generally not amenable to statistical analysis, so we turn it around and ask “how plausible would it have been for us to obtain the data we did, assuming is true?” If we conclude that it would have been implausible, we reject .
In this section, we formalize techniques of hypothesis testing. We limit ourselves to hypotheses asserting that some parameter of a random variable has a certain value. Many hypotheses—all we will consider in this book—can be formulated in this manner. This will allow us to keep definitions simple.