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Unformatted text preview: If pvalue , reject H If pvalue ≥ , fail to reject H ∙ The pvalue is the smallest significance level at which we can reject H . For example, if pvalue .075, we can reject at the 10% level but not the 5% level. 51 ∙ For testing the mean in a Normal ,1 population, the pvalue is computed using the Normal 0,1 distribution. ∙ pvalues can be computed using other distributions, too. Later we will see how pvalues are computed in the case where the population variance is unknown. 52 pvalue = .075 .1 .2 .3 .4 phi(z) 1.44 z OneSided pvalue 53 ∙ If the alternative is H 1 : 0, the pvalue is pvalue P T t  P T t  H which means it is only an interesting calculation when t 0. ∙ For a twosided alternative, we must compute pvalue P  T   t  H and this is usually called a two sided p value and requires calculating the probality of being in either tail of the distribution under the null. 54 area = .063 area = .063 pvalue = .126 .1 .2 .3 .4 phi(z) 1.531.53 z TwoSided pvalue 55 ∙ Statistical packages often compute pvalues routinely, but in some cases only twosided pvalues are computed (for example, with regression analysis). ∙ A onesided pvalue is easily obtained from a twosided pvalue by dividing the latter by two, but only if the estimate is in the direction of the stated onesided alternative. 56 EXAMPLE : We can compute exact pvalues in some cases without a normal population. Consider the plebiscite example, where the null is H : ≥ .5, which is operationally the same as H : .5, against the alternative H 1 : .5. In the sample n 500, we obtained 241 yes votes, so x ̄ .482. Let Y Binomial 500,.5 – the distribution of the number of yes votes if .5. Then the pvalue against .5 is P Y ≤ 241 . di binomial(500,241,.5) .22356475 57 ∙ This pvalue is probably not small enough to overturn the election because it would entail a chance of Type I error of greater than 22%. ∙ There is substantially more evidence against the dictator’s claim that the population vote was 52.6%. . di binomial(500,241,.526) .02715266 So we would reject H : .526 against H 1 : .526 at the 5% level but not at the 2.5% level. ∙ If the dictator cheated, he would arouse less suspicion by reporting a percentage just a little higher than 50%. 58 8 . Testing Hypotheses About the Mean from a Normal Population ∙ Suppose now that the population has the Normal , 2 distribution, and we would like to test hypotheses about . The variance, 2 , can be any positive number....
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 Fall '12
 Jeff
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, alternative hypotheses

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