Slide 1

Comparing 2 Proportions 2006 W.H. Freeman and Company

16/4/2015ObjectivesComparing two proportionsComparing two independent samplesLarge-sample CI for two proportions Test of statistical significance

3Sample 1 Sample size n1Number of successes x1Sample proportion

Two random samples are drawn from two populations.The number of successes in each sample is recorded.The sample proportions are computed.Sample 2 Sample size n2Number of successes x2Sample proportion

xn11=p1

Point Estimator of p1 p2 :

Comparing two independent samples

We often need to estimate the difference p1 p2 between two unknown population proportions based on independent samples. We can compute the difference between the two sample proportions and compare it to the corresponding, approximately normal sampling distribution for

Large-sample CI for two proportionsFor two independent SRSs of sizes n1 and n2 with sample proportion of successes 1 and 2 respectively, an approximate level C confidence interval for p1 p2 is

Use this method whenC is the area under the standard normal curve between z* and z*.

Cholesterol and heart attacksHow 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 = (0.016, 0.0266)We are 90% confident that the interval 0.16% to 2.66% captures the true percentage difference in heart attack rates for middle-aged men when taking a placebo and the cholesterol-lowering drug. Standard error of the difference p1 p2:

H. attacknDrug p25620512.73%Placebo p18420304.14%

7Example: 95% confidence interval for p1 p2

The age at which a woman gives birth to her first child may be an important factor in the risk of later developing breast cancer. An international study conducted by WHO selected women with at least one birth and recorded if they had breast cancer or not and whether they had their first child before their 30th birthday or after.CancerSampleSizeAge at First Birth > 30683322021.2%

Age at First Birth 30p2 = cancer rate when age at 1st birth 30 is between .05 and .082 higher than when age 0

pgf = proportion that receive relief from gastric freezingpplacebo = proportion that receive relief using a placebo