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Unformatted text preview: Ch. 8 Hypothesis Testing 1 Introduction In this chapter we will consider the problem of testing whether a given hypothesis regard- ing the true value of a parameter is supported by the data. We have seen that confidence intervals provide a set of plausible (i.e. compatible with the data) values for the true value of . Thus, confidence intervals can be (and are) used to conduct hypothesis testing. For example, consider testing a hypothesis of the form H : = , (1.1) where is a specified value. A sensible way of testing this hypothesis is to construct a CI for and check if the specified value is one of the plausible values for the true . If does not belong in the CI then the hypothesis H is refuted, or rejected , and if belongs in the CI then H is not rejected. As an illustration, suppose we are interested in testing whether or not the mean value of a population equals 9 . 8 (so = and = 9 . 8), which is written as H : = 9 . 8 . Suppose, further, that the data yield a 95% CI for the true of (9 . 3 , 9 . 9). Since 9 . 8 belongs in the 95% CI, we say that H is not rejected (it is not refuted by the data) at level of significance = 0 . 05. Even though there is a close connection between CIs and hypothesis testing, there are a number of specific questions and issues that arise in hypothesis testing and those deserve separate treatment. These issues are: 1. The null hypothesis and the alternative hypothesis. In every hypothesis testing situation, there is a null hypothesis and an alternative hypothesis. Typically, the statement of the alternative hypothesis is the complement of the statement of the null hypothesis. For example, the alternative to the null hypothesis H : = is H a : 6 = . 1 Other common null hypotheses are of the form H : , or H : , (1.2) with corresponding alternative hypotheses H a : &gt; , or H a : &lt; . (1.3) Testing procedures, however, do not treat the null and the alternative hypotheses equally. Basically, test procedures treat the null hypothesis in a manner similar to manner in which the presumption of innocence is treated in a court of law. One of the learning objectives of this chapter is to learn which of the two complementary statements should be designated as the null hypothesis. 2. Rejection rules. The intuitive, confidence-interval-based, procedure for rejecting the null hypothesis in relation (1.1), is not suitable for testing the one-sided null hypotheses in (1.3). While it is possible to define one-sided CIs and base test procedures for one-sided hypotheses on them, it is more common to present rules for rejecting a null hypothesis without making explicit reference to a CI....
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This note was uploaded on 01/11/2012 for the course STAT 401 taught by Professor Akritas during the Fall '00 term at Penn State.
- Fall '00