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If p-value reject h if p-value ≥ fail to reject h

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Unformatted text preview: If p-value , reject H If p-value ≥ , fail to reject H ∙ The p-value is the smallest significance level at which we can reject H . For example, if p-value .075, we can reject at the 10% level but not the 5% level. 51 ∙ For testing the mean in a Normal ,1 population, the p-value is computed using the Normal 0,1 distribution. ∙ p-values can be computed using other distributions, too. Later we will see how p-values are computed in the case where the population variance is unknown. 52 p-value = .075 .1 .2 .3 .4 phi(z) 1.44 z One-Sided p-value 53 ∙ If the alternative is H 1 : 0, the p-value is p-value P T t | P T t | H which means it is only an interesting calculation when t 0. ∙ For a two-sided alternative, we must compute p-value 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 p-value = .126 .1 .2 .3 .4 phi(z) 1.53-1.53 z Two-Sided p-value 55 ∙ Statistical packages often compute p-values routinely, but in some cases only two-sided p-values are computed (for example, with regression analysis). ∙ A one-sided p-value is easily obtained from a two-sided p-value by dividing the latter by two, but only if the estimate is in the direction of the stated one-sided alternative. 56 EXAMPLE : We can compute exact p-values 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 p-value against .5 is P Y ≤ 241 . di binomial(500,241,.5) .22356475 57 ∙ This p-value 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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If p-value reject H If p-value ≥ fail to reject H ∙ The...

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