# Computing from the observations the observed value tobs of the test statistic T.

## s

In the statistics literature, statistical hypothesis testing plays a fundamental role The usual line of reasoning is as follows:

- There is an initial research hypothesis of which the truth is unknown.
- The first step is to state the relevant
**null**and**alternative hypotheses**. This is important, as mis-stating the hypotheses will muddy the rest of the process. - The second step is to consider the statistical assumptions being made about the sample in doing the test; for example, assumptions about the statistical independence or about the form of the distributions of the observations. This is equally important as invalid assumptions will mean that the results of the test are invalid.
- Decide which test is appropriate, and state the relevant
**test statistic**T. - Derive the distribution of the test statistic under the null hypothesis from the assumptions. In standard cases this will be a well-known result. For example, the test statistic might follow a Student’s t distribution or a normal distribution.
- Select a significance level (
*α*), a probability threshold below which the null hypothesis will be rejected. Common values are 5% and 1%. - The distribution of the test statistic under the null hypothesis partitions the possible values of T into those for which the null hypothesis is rejected—the so-called
*critical region*—and those for which it is not. The probability of the critical region is*α*. - Compute from the observations the observed value t
_{obs}of the test statistic T. - Decide to either reject the null hypothesis in favor of the alternative or not reject it. The decision rule is to reject the null hypothesis H
_{0}if the observed value t_{obs}is in the critical region, and to accept or “fail to reject” the hypothesis otherwise.

An alternative process is commonly used:

- Compute from the observations the observed value t
_{obs}of the test statistic T. - Calculate the
*p*-value. This is the probability, under the null hypothesis, of sampling a test statistic at least as extreme as that which was observed.