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Unformatted text preview: MTH/STA 562 ELEMENTS OF A STATISTICAL TEST The main objective of statistics is to make inferences about unknown population para- meters on the basis of sample information. That is, how can we reach some conclusion about the population based upon the sample evidence? One of the major areas of statistical infer- ences is the estimation of population parameters, which has been discussed in the previous chapters. Another important area of statistical inferences is the hypothesis testing about population parameters. The objective of this chapter is to develop the theory and methods for hypothesis testing. Statistical Hypotheses Often statisticians are not interested solely in estimating unknown population parame- ters, but in deriving the formulation of rules or procedures that leads to a decision culmi- nating in the acceptance or rejection of some statement or hypothesis about the population parameters. In light of sample observations, sometimes the problem is to decide which of several assertions or hypotheses is most supported by the sample evidence. For example, an educator might be interested in deciding on the basis of sample data whether or not male students are in general doing better than female students in mathematics; a medical researcher might have to decide on the basis of sample evidence whether or not a new drug reduces the blood pressure to normal levels; an environment advocator might wish to collect appropriate sample data to decide whether or not the level of benzene in a speci&c oil re&nery is systematically above the compliance standard set by government for benzene levels. The area of hypothesis testing comprises a major area of statistical inference in developing rules or procedures that lead to the decision on accepting or rejecting assertions or hypotheses such as those presented in the above example. First, let us de&ne precisely the terms of statistical hypothesis and hypothesis testing. De&nition 1. A statistical hypothesis is an assertion or conjecture concerning one or more populations, speci&cally about one or more parameters of such population distribu- tions. The hypothesis testing is a statistical procedure that is designed to test a statistical hypothesis. Because the statistical hypothesis is being made about a population parameter that tends to be an unknown quantity, the truth or falsity of a statistical hypothesis is never known with sure certainty unless the entire population has been thoroughly examined. Obviously, it is absolutely impractical to survey the whole population in most situations. Just as in the theory of estimation, a random sample is drawn from the population of interest and the acceptance or rejection of a statistical hypothesis will be decided on the basis of sample evidences. Intuitively speaking, we will reject the given hypothesis if the sample evidence seems to be inconsistent with the hypothesis, whereas the acceptance of the hypothesis seems appropriate if the evidence appears to be somewhat in concordance with the hypothesis....
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This note was uploaded on 04/20/2008 for the course MTH 562 taught by Professor Cheng during the Spring '08 term at Creighton.
- Spring '08