## 4. The statistical null hypothesis

The use of inferential statistics to test the predictions arising originally from a research hypothesis typically refers to a so-called 'null hypothesis' (written H_{0}). As its name suggests, a null hypothesis sets out to establish that nothing unusual is present, and by inference the prediction is untrue. The alternative hypothesis (H_{1}) posits that something unusual is present, and that the prediction is likely to be true.

We use the null hypothesis, because it is mathematically and logically easier to construct the test. Essentially, all of the inferential statistical tests in the NuMBerS toolkits are concerned with establishing the likelihood that particular pattern suggested by the prediction arose simply by chance.

The procedure comprises four steps, that are used in each of the toolkits:

- Construct a null hypothesis that is appropriate to your prediction, eg if the prediction is that there is a difference between the means of samples from two populations, the null hypothesis is that the means are the same.
- Decide on a critical significance level (denoted by α, the Greek letter alpha). As we indicated in the introduction, we are rarely in a position to reject a hypothesis categorically, and in these tests we set a level of uncertainty that we are prepared to accept in testing the null hypothesis.
- Calculate the statistic that is appropriate to your null hypothesis and is consistent with your data. For instance, when testing for difference between two unrelated samples where data fulfil parametric criteria, calculate the t-statistic.
- Reject or accept your null hypothesis, either by comparing the value of the calculated statistic with published values for the statistic for given critical significance level and degrees of freedom, or by directly using a probability (or significance level). The probability, P, is the likelihood of obtaining data equal to or more extreme than the observations were the null hypothesis to be true. So a very low probability (P ≤ α) indicates that you should reject the null hypothesis that the pattern in the data arose purely by chance, and that the prediction is probably correct.