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# chapter8 - Chapter 8 Inference Concerning Proportions...

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Chapter 8 Inference Concerning Proportions

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Inference for a Single Proportion ( p ) Goal: Estimate proportion of individuals in a population with a certain characteristic ( p ). This is equivalent to estimating a binomial probability Sample: Take a SRS of n individuals from the population and observe X that have the characteristic. The sample proportion is X / n and has the following sampling properties: ) 15 , : thumb of (Rule samples large for normal ely approximat : Shape 1 : Error Standard Estimated ) 1 ( : on distributi sampling of Dev. Std. and Mean : proportion Sample ^ ^ p ^ ^ ^ ^ - - = - = = = X n X n p p SE n p p p n X p p p σ μ
Large-Sample Confidence Interval for p Take SRS of size n from population where p is true (unknown) proportion of successes. Observe X successes Set confidence level C and choose z * such that P (- z * Z z * )= C ( C = 90% z * =1.645 C = 95% z * =1.96 C = 99% z * =2.576) m p p C z m n p p n X p p p ± = - = = ^ * ^ ^ ^ : for interval confidence % SE : error of Margin 1 SE : Error Standard Estimated : Estimate Point ^ ^

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Example - Ginkgo and Azet for AMS Study Goal: Measure effect of Ginkgo and Acetazolamide on occurrence of Acute Mountain Sickness (AMS) in Himalayan Trackers Parameter: p = True proportion of all trekkers receiving Ginkgo&Acetaz who would suffer from AMS. Sample Data: n= 126 trekkers received G&A, X =18 suffered from AMS ) 204 ,. 082 (. 061 . 143 . : for CI % 95 061 . ) 031 (. 96 . 1 : %) 95 ( error of Margin 031 . 126 ) 86 )(. 14 (. SE 143 . 126 18 ^ ^ ± = = = = = = = p m C p p
Wilson’s “Plus 4” Method For moderate to small sample sizes, large-sample methods may not work well wrt coverage probabilities Simple approach that works well in practice ( n 10) : Pretend you have 4 extra individuals, 2 successes, 2 failures Compute the estimated sample proportion in light of new

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chapter8 - Chapter 8 Inference Concerning Proportions...

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