ABSTRACT
A hypothesis test in the Neyman-Pearson paradigm is a decision criterion that allows practitioners of statistics to select between two complementary hypotheses. Before conducting the hypothesis test, define the null hypothesis, H0, which is assumed to be true prior to conducting the hypothesis test. The null hypothesis is compared to another hypothesis, called the alternative hypothesis, and denoted H1. The alternative hypothesis is often called the research hypothesis since the theory or what is believed to be true about the parameter is specified in the alternative hypothesis. Both hypotheses define complementary subsets of the parameter space ⇥ where the parameter ✓ is defined. The null hypothesis defines the region [✓ 2 ⇥0] and the alternative hypothesis defines the region [✓ 2 ⇥1]. The subsets⇥ 0 and⇥ 1 are mutually exclusive by definition, and they are complementary since ⇥0 [⇥1 = ⇥. When a hypothesis uniquely specifies the distribution of the population from which the sample is taken, the hypothesis is said to be simple. For a simple hypothesis,⇥ 0 is composed of a single element. Any hypothesis that is not a simple hypothesis is called a composite hypothesis. A composite hypothesis does not completely specify the population distribution. Of the various combinations of hypotheses that could be examined, the case where the null hypothesis is simple and the alternative hypothesis is composite will be the focus of this text. Hypothesis tests will generally take a form similar to those in Table 9.1, where ✓0 is a single numerical value. For alternative hypotheses (A) and (B), which are lower one-sided and upper one-sided, respectively, the hypothesis test is called a one-tailed test. For the alternative hypothesis in (C), a two-sided alternative, the hypothesis test is called a two-tailed test.
