Comparing Proportions: Confidence Intervals and Significance Tests, Exams of Statistics

Download Comparing Proportions: Confidence Intervals and Significance Tests and more Exams Statistics in PDF only on Docsity! Comparing 2 proportions BPS chapter 21 © 2006 W.H. Freeman and Company Objectives (BPS chapter 21) Comparing two proportions The sampling distribution of a difference between proportions Large Sample confidence intervals for comparing two proportions Using technology Accurate confidence intervals for comparing two proportions Significance tests for comparing proportions Cholesterol and heart attacks How much does the cholesterol-lowering drug Gemfibrozil help reduce the risk of heart attack? We compare the incidence of heart attack over a 5-year period for two random samples of middle-aged men taking either the drug or a placebo. So the 90% CI is (0.0414−0.0273) ± 1.645*0.00746 = 0.0141 ± 0.0125 We are 90% confident that the percentage of middle-aged men who suffer a heart attack is 0.16% to 2.7% lower when taking the cholesterol-lowering drug. Standard error of the difference p1− p2: SE = ˆ p 1(1− ˆ p 1) n1 + ˆ p 2(1− ˆ p 2) n2 SEzpp *)ˆˆ( is interval confidence The 21 ±− SE = 0.0273(0.9727) 2051 + 0.0414(0.9586) 2030 = 0.00764 4.14%203084Placebo 2.73%205156Drug nH. attack p̂ Using technology p̂ Here are some examples of output for a different set of data. “Plus four” CI for two proportions The “plus four” method again produces more accurate confidence intervals. We act as if we had four additional observations: one success and one failure in each of the two samples. The new combined sample size is n1 + n2 + 4, and the proportions of successes are: 2 1~ and 2 1~ 2 2 2 1 1 1 + + = + + = n Xp n Xp An approximate level C confidence interval is: Use this when C is at least 90% and both sample sizes are at least 5. 2 )~1(~ 2 )~1(~*)~~(: 2 22 1 11 21 + − + + − ±− n pp n ppzppCI