N x ̄ n if 0 then by the clt t n d → normal 0,1

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Unformatted text preview: n X ̄ n . If 0 then, by the CLT, T n d → Normal 0,1 . ∙ If 0 then T n n X ̄ n − n O p 1 n p → − because n X ̄ n − d → Normal 0,1 and 0 so n → − . It follows that the asymptotic size of the test for 0 is zero. 8 ∙ Therefore, just as in the finite sample case, we can focus on the size of the test under H : 0. ∙ Because of the CLT, we can use the same approach to choosing critical values that we did in the case of a normal population: given asymptotic size , we choose critical value c such that P Z c where Z Normal 0,1 . 9 ∙ Theoretically, the important difference is that (when 0) we can only conclude P T n c → as n → and not P T n c for all n . This is why we now call c an asymptotic critical value , and is the asymptotic size. 10 ∙ In the same way, we compute p-values using the standard normal distribution. So, if (say) r n is the outcome of the test statistic (and r n 0) and Z denotes a standard normal random variable, we compute P Z r n and report this as the asymptotic p- value . ∙ The language of hypothesis testing is the same as before, except we often use the qualifier “asymptotic” or “approximate.” ∙ The approach to two-sided tests is exactly the same as with finite-sample analysis. 11 ∙ What happens if we consider the general population assumption X , 2 and use the usual t statistic, T n n X ̄ n S n , where S n is the sample standard deviation? Generally, we cannot find the distribution of T n for any n because we cannot even determine the distribution of X ̄ n , S n in general. 12 ∙ We do know that, under H : 0, T n n X ̄ n 1 S n − 1 n X ̄ n n X ̄ n o p 1 O p 1 n X ̄ n o p 1 13 ∙ Therefore, by the asymptotic equivalence lemma, T n n X ̄ n S n d → Normal 0,1 when 0. ∙ As an exercise, you can show that it still holds that T n p → − if 0. 14 ∙ This means that we can use the standard normal critical values for testing hypotheses when the variance is generally unknown; and we can use it to compute asymptotic p-values. ∙ Asymptotically, replacing with S n has no effect on the limiting distribution, and so, when conducting asymptotic tests, it is as if we know . 15 ∙ What about using the t n − 1 critical values rather than the standard normal critical values? Because the t n − 1 crtical values converge to those of the standard normal as n → , there is nothing wrong with it. In fact, it is usually preferred (for n 120 or so) because it includes exact inference for the normal population as a special case....
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n X ̄ n If 0 then by the CLT T n d → Normal 0,1 ∙ If 0...

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