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# Lecture10 - HYPOTHESIS TESTING We have already described...

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HYPOTHESIS TESTING We have already described how the sample information is used to estimate the parameters of the underlying population distributions. Now, we intend to use our sample information to test existing presumptions regarding the values of parame- ters. More generally, we are concerned with testing hypotheses regarding the state of nature. The initial presumptions, which are formed without the benefit of the statistical evidence, constitute the null or prior hypothesis, denoted by H 0 . The alternative hypothesis, denoted by H 1 , is what we shall assert if we reject H 0 in the light of the statistical evidence. Together, H 0 and H 1 ought to comprises all possibilities. Thus, the set Ω of all states of nature is partitioned as Ω = { Ω 0 Ω 1 0 Ω 1 = ∅} in a manner corresponding the hypotheses. The decision to maintain the null hypotheses after the evidence has been reviewed will be denoted by d 0 ; and the decision to reject it in favour of the alternative hypothesis will be denoted by d 1 . The procedure for testing an hypothesis will depend upon our forming a test statistic x from a random sample. The decision will depend upon the value of the statistic. Let S = { C C C ; C C C = ∅} be the sample space comprising all possible values of x . This is partitioned into the critical region C and its complement, which is the non-critical region C C . Then, the decision rule is as follows: x C C = d 0 , x C = d 1 .

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Lecture10 - HYPOTHESIS TESTING We have already described...

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