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Even Statisticians Love Geometry Charles Burd, April 16, 2014 Advisor: Dr. Chauhan

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Objective There may be multiple ways of estimating an unknown value. The results obtained from multiple methods may not be the same. In such situation, is there a way to determine which method may be more appropriate, and under what conditions?

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Background Estimating proportion Population proportion :unknownRandom sample proportion - known

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Comparing Proportions of two populations Unknown Objective: Estimate

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Overlap: Compute CI for each sample Decision rule: If the two intervals overlap, population proportions may be the same. Standard: Decision rule: If the interval contains zero, the proportions may be the same. 0

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Example Overlap Approach Population 1Population 2 0.30.40.50.60.7 Overlap Approach: The intervals overlap, so the proportions may be the same.

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Example Continues Standard Approach Population 1 – Population 2 00.10.20.3 Standard Approach: Interval does not contain zero, so the proportions are not the same. Result: The overlap method concludes the population proportions not different while the standard method finds a difference.

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A Closer Look Equal population proportions by overlap method implies equal by standard method, but not vice-versa (ratio greater than 1). Overlap method is more conservative and less powerful. If two populations differ, standard method will detect it. Only difference between the two methods then is the width of the intervals. narrower width less chance zero included proporotions different

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A Closer Look What does this geometric relationship tell us about overlap method’s deficiencies?

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Simulation Do simulation results confirm analytical expectations? Percentage of time finding a difference between populations OverlapStandard

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Conclusion We can always get better results with the standard method. Overlap method is at its worst when the two margin of errors are close. Overlap is simple, convenient to use, but for formal testing, use standard method.

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Reference Nathaniel Schenker and Jane F. Gentleman : On judging the significance of differences by examining the overlap method between confidence intervals, The American Statistician 55 (Aug., 2001) 182-186. Charles Burd, April 16, 2014 Advisor: Dr. Chauhan

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