47 7 computing p values classical hypothesis testing

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7 . Computing p - values Classical hypothesis testing is somewhat cumbersome because it requires specifying a size ahead of time. Yet different researchers may have different tolerances for Type I errors. Moreover, in complicated settings, classical testing allows one to ignore a “close” rejection and just report that H 0 was not rejected at, say, the 5% significance level. But it could have been rejected at the 6% level. 48
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We can avoid choosing a size ahead of time by reporting a p - value with any test. This approach requires all the steps of classical testing except choosing a size. For illustration, consider testing H 0 : 0 against H 1 : 0 in a Normal ,1 population under random sampling. Let T n X ̄ be the test statistic so that, under H 0 , T Normal 0,1 . Now suppose we obtain the data and the value of the test statistic is t . If t 0 (which means x ̄ 0 the test provides no evidence in favor of H 1 ; we fail to reject H 0 in favor of H 1 at any reasonable size. 49
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If t 0, for classical testing we would compare it with a relevant critical value one we have chosen the size of the test. Instead, consider the following calculation: P T t | 0 P T t | H 0 That is, if the null hypothesis is true, what is the probability of obvserving a statistic at least as large as the one we did observe? This is called the p -value of the test. Sometimes it is called a one - sided p - value . 50
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The p -value nicely summarizes the evidence against the null hypothesis. It allows us to carry out a test at any significance level. If p -value , reject H 0 If p -value , fail to reject H 0 The p -value is the smallest significance level at which we can reject H 0 . For example, if p -value .075, we can reject at the 10% level but not the 5% level. 51
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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
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p-value = .075 0 .1 .2 .3 .4 phi(z) 0 1.44 z One-Sided p-value 53
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If the alternative is H 1 : 0, the p -value is p -value P T t | 0 P T t | H 0 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 0 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
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