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An example of this problem would be to examine the number of students that do not pass an exam. In a school district, say that 300 out of 1,000 students that take the same test do not pass (3 do not pass per 10 testtakers). One could ask whether a Class A of 20 students performed differently than the overall population on this test (note we are assuming passing or not passing the test is independent of being in Class A for the sake of this simplified example). Say Class A had 10 out of 20 students that did not pass the exam (5 do not pass per 10 test takers). Class A had a not pass rate that is double the rate of the school district. When we use a Poisson confidence interval, however, the rate of not passing in the class of 20 is not statistically different from the school district average at the 95% confidence level. If we instead compare Class A to the entire state of 100,000 students (with the same 3 not pass per 10 test takers rate, or 30,000 out of 100,000 to not pass), the 95% confidence intervals of this comparison are almost identical to the comparison to the county (300 out of 1000 test takers). This means that for this comparison, the uncertainty in the small number of observations in Class A (only 20 students) is much more than the uncertainty in the larger population. Take another class, Class B, that had only 1 out of 20 students not pass the test (0.5 do not pass per 10 test takers). When applying the 95% confidence intervals, this Class B does have a statistically different pass rate from the county average (as well when compared to the state). This example shows that when comparing rates of events in two populations where one population is much larger than the other (measured by test takers, or miles driven), the two things that drive statistical significance are: (a) the number of observations in the smaller population (more observations = significance sooner) and (b) bigger differences in the rates of occurrence (bigger difference = significance sooner).

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