Introduction

As an alternative to estimating the values of one or more parameters of a probability distribution, as was the objective in Chapter 2, we may test hypotheses about such parameters. Both estimation and hypothesis testing may be viewed as different aspects of the same general problem of reaching decisions on the basis of data. Explicit formulation as well as important basic concepts on the theory of testing statistical hypotheses are due to J. Neyman and E.S. Pearson, who are considered pioneers in the area.

Although the notation of H and A for hypotheses and alternatives has been used, we will now use that of Hogg and Craig where the terms null hypothesis and alternative hypothesis are used, with corresponding notation $H_0$ and $H_1$ . The null hypothesis is always a statement of either `no effect' or the `status quo'. If the statistical hypothesis completely specifies the distribution, it is called simple; if it does not, it is called composite.

After experimentation (for example, taking a sample of size $n$, ( $X_1, \ldots, X_n$) or X), some reduction in data is used, resulting in a test statistic, T$=$ t(X), say. We may consider a subset of the range of possible values of T as a rejection (or critical) region. Previously this has been denoted by $R$, but to be consistent with HC, we will now use $C$. That is, $C$ is the subset of the sample space of $T$, which leads to the rejection of the hypothesis under consideration. The region $C$ can refer to either the X-values or the T-values.

The case where both $H_0$ and $H_1$ are simple, where the size of the Type I and Type II error is easily determined, is assumed known. You may find it helpful to read HC 7.1. In the case of composite hypotheses and alternatives, the power function of the test is an important tool for evaluating its performance, and this is examined in the following section.