Hypothesis Testing

We have seen so far two types of statistical estimation frameworks: confidence intervals and point estimation. Hypothesis testing is a third inference framework that is concerned with choosing a hypothesis supported by available data out of a number of competing alternatives. We start with the basic definitions and then proceed to describe a few important cases. We assume that we have data X1, . . . , Xn sampled from a distribution characterized by a parameter θ. A hypothesis is a set of possible values for θ. We will consider cases where there are two competing hypothesis: the null hypothesis H0 and the alternative hypothesis HA, with H0 ∩ HA = ∅. The two hypothesis are not treated in a symmetric manner. The null hypothesis usually is used to describe a set of standard or believable values. The alternative hypothesis is an alternative to the null. It is sometimes called the research hypothesis since it may describe a research statement that one wishes to examine. A hypothesis test is executed by observing the values of a test statistic T (X1, . . . , Xn). If it lies in a set called the rejection region (RR) then the null hypothesis is rejected and the alternative is accepted. Otherwise, the null is accepted and the alternative hypothesis is rejected