Chapter 5: Inference on a Population Median Recall for well-behaved samples, i.e., samples that are smooth, symmetric and roughly bell-shaped that even for moderate sample sizes we can be confident that the Central Limit Theorem has kicked in and we can safely assume that the sample distribution of the mean is normal. Sometimes, however, the sample is so skewed or irregular that it is highly questionable that the Central Limit Theorem can save us. What do we do then? We can perform tests about the population median instead. Order Statistics Let represent a random sample and let represent the order statistics. That is, let represent the sample in order from smallest to largest. Equivalently,nxxx K,,21)()2()1(,nxxx K)()2()1(,nxxx K)()2()1( nxxx≤≤≤ K. Let Mˆrepresent the sample median as our estimate of the population median. How do we construct a confidence interval forMˆ? What do we need to know? We can estimate the variability in Mˆ using the binomial distribution with 5.0=π. A 100(1-a)% confidence interval for the median is given by ()()2/2/,,ααULULyyMM = Where and 1),2(2/+=nCLααnCnU),2(2/αα−=. Table 5 in the appendix of your text gives the values for , which are the percentiles from the binomial(0.5) distribution. nC),2(α So for example, if we have a sample size of 20 and wish to conduct a test at the 95% confidence interval, we check the table to find C(.05),20=5. Hence our confidence interval is formed by taking ()()15)6(, yy as our confidence interval. See examples in the text.Large sample approximation When the sample size is large we can use the normal approximation to the binomial. Here the confidence interval is given by 422/),2(nznCnαα−= When n is larger than 30, often the exact value will be the same integer as that given by the large sample approximation. Example - ATL/HNL Flight Times • n = 31, • Small-Sample: C.05(2),31 = 9 Day Time Ordered15675312582538359254246015455567553658555875695588568567956956710 553 56711 531 56812 538 56913 545 56914 542 57615 558 57716 558 57917 579 58018 584 58219 583 58220 582 58221 577 58222 582 58323 583 58324 596 58425 589 58526 586 58627 567 58928 582 59229 627 59630 580 60131 576 627105.55.1543196.123131),2(05.=−=−≈CSample Size Lα/2 Uα/2 Y(L) Y(U)Small 10 22 567 583 Large 11 21 568
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