The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 21 Comparing two proportions Section 8 2 11 28 06 Lecture 21 1 Two populations an extension of Lecture 20 Population 1 with proportion p 1 Population 2 with proportion p2 Interested in the difference p 1 p2 Sample 1 size n1 count X1 proportion Sample 2 size n2 count X2 proportion Consider the difference p 1 X1 n1 p 2 X 2 n 2 D p 1 p 2 Assume the two samples are independent and both n1 and n2 are large 11 28 06 Lecture 21 2 Useful probability facts The random variable D has approximately a normal distribution with mean p1 p2 and standard deviation q SD D p1 1 p1 n1 p2 1 p2 n2 An estimate of SD D q SED p 1 1 p 1 n1 p 2 1 p 2 n2 11 28 06 Lecture 21 3 Confidence Interval for p 1 p 2 Expression D m D m m z SED where the margin of error Confidence level C determines 11 28 06 Lecture 21 z 4 Hypothesis testing for p1 p2 We want to test H0 p 1 p 2 versus some 1 sided or 2 sided alternative Recall the 4 steps Step 1 need to specify the alternative Ha 11 28 06 Lecture 21 5 Hypothesis testing for p1 p2 continued Step 2 Test statistic z D SE D p where q SEDp p 1 p 1 n1 1 n2 and p X 1 X 2 n 1 n 2 Note The pooled standard error SEDp is different from SED on page 3 11 28 06 Lecture 21 6 Hypothesis testing continued Step 3 The P value will be equal to P Z z for 1 sided upper tail Ha p1 p2 P Z z for 1 sided lower tail Ha p1 p2 2 P Z z for 2 sided Ha p16 p2 Step 4 Compare the P value with the significance level and draw your conclusion 11 28 06 Lecture 21 7 Gender difference in frequent binge drinking Proportion of frequent binge drinkers Population 1 male college students p 1 Population 2 female college students p2 Sample 1 n 1 7180 X 1 1630 p 1 0 227 Sample 2 n 2 9916 X 2 1684 p 2 0 170 Total n1 n2 17096 X 1 X 2 3314 p 0 194 11 28 06 Lecture 21 8 Gender difference continued Test H0 p 1 p 2 vs Ha p 1 p 2 Test statistic p z 0 227 0 170 0 194 0 806 1 7180 1 9916 9 34 P value P Z 9 34 0 00 Reject H0 95 CI for p1 p2 is 0 045 0 069 where p SED 0 227 0 773 7180 0 170 0 830 9916 0 00622 m z SE D 1 96 0 00622 0 012 11 28 06 Lecture 21 9 Take Home Message CI for the difference p 1 p 2 Hypothesis testing for comparing p1 and p2 4 steps Note different standard errors are used SED in CI SEDp 11 28 06 in testing Lecture 21 10
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