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CH 12 – INFERENCE FOR PROPORTIONS YMS 12.1 Inference for a Population Proportion Ch 9 Sampling Distributions p is an unbiased estimator of population proportion p Standard deviation of is if the population p(1 p) p is at least 10 times n n Sampling distribution of is approximately normal if np and n(1-p) are at least p10 Use z-scores to standardize Conditions for Inference To be representative: Data are from an SRS from the population of interest To accurately calculate standard deviation: Population is at least 10 times n To use normal calculations: Counts of successes/failures must be at least 10 Standard Error Replacing p with p in standard deviation formula Test of Significance Ho: p = po Verify that npo and n(1-po) are at least 10 p po Formula z po (1 p o ) n Confidence Interval and n(1- p) are at least 10 Verify that n p Form p (1 p ) p z * n Choosing the Sample Size Margin of error p *(1 p*) z* m n Use a guess for p* Based on previous data Use the conservative estimate of 0.5 (unless you believe is closer to p0 or 1 because then p* = 0.5 will give you a much larger sample size than necessary) Which to use in formulas and conditions? Hypothesis Tests Use po because that is the distribution you’re comparing your result to Confidence Intervals because you don’t have any other values Use p (remember you’re using the CI to estimate the true proportion p) p698 #12.14-12.15, 12.17 YMS 12.2 Comparing Two Proportions Sampling Distribution of p1 p 2 When samples are large, the sampling distribution is approximately normal. Mean Variance p p p p p1 p2 1 2 2 p1 p 2 1 2 2 p1 2 p 2 p1 (1 p1 ) p2 (1 p2 ) n1 n2 Confidence Intervals for Comparing Two Proportions Same form as for two means and standard error is replacing p with p Conditions are still SRS, a population at least 10 times n, but now n1 , n1(1p1 ), n2 p1 , andpn2 2(1 2 greater than 5 ) are pall p706 #12.22-12.23 Pooled Sample Proportion Because both samples actually come from one huge population, we combine the sample results to estimate the unknown population proportion p Formula X1 X 2 n1 n2 Significance Tests for Two Props 1and pwith standard error formula and Replace p pooled p in 2 the conditions for count of successes and failures Other conditions remain the same! Test Stat z pˆ1 pˆ 2 1 1 pˆ (1 pˆ ) n1 n2 p707 #12.24-12.26 A Civil Action – text, video and article