11-2 Contingency Tables 593 TABLE 11-1 Results From a Study of a Link Between the MMR Vaccine and Autism Unvaccinated Vaccinated Total Autism 25 1E = ?2 64 89 No Autism 362 1427 1789 Total 387 1491 1878 TABLE 11-6 Expected Frequencies from Table 11-1 Unvaccinated Vaccinated Autism 251E = 18.3402 641E = 70.6602 No Autism 3621E = 368.6602 14271E = 1420.3402 Table 11-1 from the Chapter Problem is shown below. Also shown are the row totals and column totals. Find the expected frequency E for the cell with an observed frequency of 25. CP YOUR TURN. Exercise 1: “Dogs Detecting Malaria.” EXAMPLE 1 Finding Expected Frequency SOLUTION The cell with the frequency of 25 lies in the first row (with a total frequency of 89) and in the first column (with total frequency of 387). The “grand total” is the sum of all frequencies in the table, which is 1878. For that first cell with an observed frequency of 25, the expected frequency is E = 1row total21column total2 1grand total2 = 189213872 1878 = 18.340 Using this formula, the expected frequency for the remaining three cells can be found. These expected frequencies are shown in Table 11-6. INTERPRETATION We know that the first cell has an observed frequency of O = 25 and an expected frequency of E = 18.340. We can interpret the expected value by stating that if we assume that autism is independent of vaccine, then we expect to find that 18.340 of the subjects would be unvaccinated and they would have autism. There is a discrepancy between O = 25 and E = 18.340, and such discrepancies are key components of the test statistic that is a collective measure of the overall disagreement between the observed frequencies and the frequencies expected with independence between the row and column variables. Use the same sample data from Example 1 with a 0.05 significance level to test the claim that autism is independent of the MMR (measles, mumps, rubella) vaccine. CP EXAMPLE 2 Is There a Link Between the MMR Vaccine and Autism? SOLUTION REQUIREMENT CHECK (1) On the basis of the study description, we will treat the subjects as being randomly selected and randomly assigned to the different treatment groups. (2) The results are expressed as frequency counts in Table 11-6 in Example 1. (3) Table 11-6 shows that the expected frequencies are all at least 5. (The lowest expected frequency is 18.340.) The requirements are satisfied. continued Go Figure $178,201: The annual salary that a mother staying at home would earn if she were being paid for her work. 96: The number of hours worked each week by the typical mother who stays at home.
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