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By: A. Renwik, M.B. B.CH. B.A.O., M.B.B.Ch., Ph.D.

Associate Professor, Michigan State University College of Human Medicine

If each student in a sample of 100 is classified according to treatment yellow jacket sting purchase 40mg zerit overnight delivery whether he or she is enrolled full-time or parttime medicine park oklahoma purchase 40mg zerit fast delivery, data on a categorical variable with two categories result 4 medications generic zerit 40mg visa. Each airline passenger in a sample of 50 might be classified into one of three categories based on type of ticket-coach, business class, or first class. Each registered voter in a sample of 100 selected from those registered in a particular city might be asked which of the five city council members he or she favors for mayor. The first few observations might be Probably will Probably will not Definitely will not Definitely will Probably will not Definitely will not Counting the number of observations of each type might then result in the following one-way table: Outcome Definitely Will Frequency Probably Will Probably Will Not Definitely Will Not 14 12 24 50 For a categorical variable with k possible values (k different levels or categories), sample data are summarized in a one-way frequency table consisting of k cells, which may be displayed either horizontally or vertically. In this section, we consider testing hypotheses about the proportion of the population that falls into each of the possible categories. For example, the manager of a tax preparation company might be interested in determining whether the four possible responses to the tax credit card question occur equally often. If this is indeed the case, the long-run proportion of responses falling into each of the four categories is 1/4, or. The test procedure to be presented shortly would allow the manager to decide whether the hypothesis that all four category proportions are equal to. Notation k p1 p2 pk o number of categories of a categorical variable true proportion for Category 1 true proportion for Category 2 true proportion for Category k (Note: p1 p2 p pk 1) hypothesized proportion for Category 1 hypothesized proportion for Category 2 the hypotheses to be tested have the form H0: p1 p2 o pk hypothesized proportion for Category k Ha: H0 is not true, so at least one of the true category proportions differs from the corresponding hypothesized value. To decide whether the sample data are compatible with the null hypothesis, we compare the observed cell counts (frequencies) to the cell counts that would have been expected when the null hypothesis is true. The expected cell counts are Expected cell count for Category 1 Expected cell count for Category 2 np1 np2 and so on. The expected cell counts when H0 is true result from substituting the corresponding hypothesized proportion for each pi. The paper "The Effect of the Lunar Cycle on Frequency of Births and Birth Complications" (American Journal of Obstetrics and Gynecology : 1462­1464) classified births according to the lunar cycle. Data for a sample of randomly selected births occurring during 24 lunar cycles consistent with summary quantities appearing in the paper are given in the accompanying table. Since there are a total of 699 days in the 24 lunar cycles considered and 24 of those days are in the new moon category, if there is no relationship between number of births and lunar cycle, p1 24 699. If the differences between the observed and expected cell counts can reasonably be attributed to sampling variation, the data are considered compatible with H0. On the other hand, if the discrepancy between the observed and the expected cell counts is too large to be attributed solely to chance differences from one sample to another, H0 should be rejected in favor of Ha. Thus, we need an assessment of how different the observed and expected counts are. The goodness-of-fit statistic, denoted by X 2, is a quantitative measure of the extent to which the observed counts differ from those expected when H0 is true. In using X 2 rather than x2, we are adhering to the convention of denoting sample quantities by Roman letters. A small value of X 2 (it can never be negative) occurs when the observed cell counts are quite similar to those expected when H0 is true and so would be consistent with H0. As with previous test procedures, a conclusion is reached by comparing a P-value to the significance level for the test. The P-value is computed as the probability of observing a value of X 2 at least as large as the observed value when H0 is true. A chi-square curve has no area associated with negative values and is asymmetric, with a longer tail on the right. There are actually many chi-square distributions, each one identified with a different number of degrees of freedom. For a test procedure based on the X 2 statistic, the associated P-value is the area under the appropriate chi-square curve and to the right of the computed X 2 value. Appendix Table 8 gives upper-tail areas for chi-square distributions with up to 20 df. Our chi-square table has a different appearance from the t table used in previous chapters. In the t table, there is a single "value" column on the far left and then a column of P-values (tail areas) for each different number of degrees of freedom. A single column of t values works for the t table because all t curves are centered at 0, and the t curves approach the z curve as the number of degrees of freedom increases. Diseases

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