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Unformatted text preview: meanmean(cscrap)t0.3620Ho: mean1.5degrees of freedom18Ha: mean1.5Ha: mean !1.5Ha: mean1.5Pr(Tt)0.6392Pr(Tt)0.7215Pr(Tt)0.360894The“Rule of Two”∙When the df are reasonably large – saydf30 or larger – a simpleruleofthumb is often used to reject a null hypothesis: rejectHif theabsolute value of thetstatistic is greater than two. This rule comesfrom (1) Using a 5% size test and (2) Specifying a twosidedalternative. For such a test, the cv in thetdistribution is about 2.04whendf30 and reaches 2.00 atdf60. For largerdf, the criticalvalue is less then two, and converges to 1.96 – the critical value for thestandard normal distribution.95∙In fact,tdf→Normal0,1asdf→. With even modest samplesizes, one is safe obtaining critical values or computingpvalues fromthe standard normal distribution.96Practical versus Statistical Significance∙In almost all applications it is important to distinguish between thesize of the estimated effect – say, of a job training program – andwhether the estimate is statistically significant.∙Consider the job training grant application. Usingclscrap, the changein the log of the scrap rate, we estimated the mean policy effect to beabout−.374. Crudely, this means an average fall in the scrap rate ofabout 37.4% – a very large economic effect.97∙We also found that, even with a small number of observations, we cansoundly rejectH:≥0 againstH1:0 at the 1% significancelevel.∙But suppose the estimated sample average had been−.040 withn1,600 andS.637 (the same estimate obtained withn19). Thismeans a standard error of .637/40≈.016. Thetstatistic is thent−.040/.016−2.50, which is still a very strong rejection ofH.But the estimated policy effect is much smaller – roughly 4% ratherthan 37%.98∙The point is that a policy effect can be modest, or even small, but stilllead to statistical significance, particularly with large samples. So donot just fixate on test statistics and statistical significance!99∙A more difficult scenario is when an estimated policy effect is largebut it is not statistically significant – possibly because of small samplesize. Unfortunately, while it is tempting to draw conclusions, the lackof statistical signficance means that the estimators are noisy, and adifferent random sample might produce a very different estimate.∙With small sample sizes one often uses a larger significance level;with larger sample sizes, one often requires rejection at a smallersignificance level. But the important thing is to be sure to interpret themagnitude of the estimate as well as how to test hypotheses.100...
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 Fall '12
 Jeff
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, alternative hypotheses

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