The basic approach

The previous chapter on estimation showed how it is possible to use sample statistics to make estimates of population parameters that include estimates of sampling uncertainty. However, sometimes we would like to go further and use sample statistics to test the binary true/false validity of certain hypotheses (assumptions) about population parameters. In other words, we want to make true/false decisions about specified hypotheses based on the evidence provided by the sample of data. This ``Sherlock Holmes'' detective approach is obviously more risky than simply estimating parameters, but is often used to clarify conclusions from scientific work. In fact, the radical idea underlying the whole of natural science is that hypotheses and theories are not only judged by their intrinsic beauty but can also be tested for whether or not they explain observed data. This is exemplified by the Royal Society's <sup>6.1</sup> revolutionary famous motto ``nullis in verba'', which is taken from a poem by the roman poet Horace and means do not (unquestioningly) accept the words (or theories) of anyone !

Suppose, for example, we suspect there might be a non-zero correlation between two variables (e.g. sunspot numbers and annual rainfall totals in Reading). In other words, our scientific hypothesis $H_1$ is that the true (population) correlation between these two variables is non-zero. To test this hypothesis, we examine some data and find a sample correlation with, let's imagine, a quite large value. Now did this non-zero sample correlation arise because the hypothesis $H_1$ is really true, or did it just happen by chance sampling ? The hypothesis that the large sample value happened just by chance is known as the null hypothesis $H_0$. In order to claim that the alternative hypothesis $H_1$ is true, we must first show that the large value would be very unlikely to happen by pure chance. In other words, we must use the data to reject the null hypothesis $H_0$ in favour of the more scientifically interesting alternative hypothesis $H_1$. By using the data to reject $H_0$, we are able to make a binary decision about which of the two hypotheses is least inconsistent with the data.