Course Notes - Department of Mathematics and Statistics

6.4.3 Confidence Interval for Difference Between Two ProportionsCONFIDENCE INTERVAL FOR DIFFERENCE BETWEEN TWO PROPORTIONS√p1 (1−pInterval for the difference π 1 − π 2 : (p 1 −p 2 )±1.961 )n 1+ p 2(1−p 2 )n 2EXAMPLETo study the effectiveness of a drug for arthritis, two samples ofpatients were randomly selected. One sample of 100 was injected withthe drug, the other sample of 60 receiving a placebo injection. Aftera period of time the patients were asked if their arthritic conditionhad improved. Results were:Drug PlaceboImproved 59 22Not improved 41 38TOTAL 100 60SolutionThe proportions improved are: p 1 =100 59 p 2 =60 22 √The confidence interval is: (0.59 − 0.37)±1.96= 0.22 ± 0.16 = (0.06, 0.38)We can write:0.06 < π 1 − π 2 < 0.38(0.59×0.41)100+ (0.37×0.63)60Since 0 is excluded from the interval and the interval is positive, thereis evidence that the difference π 1 − π 2 > 0. That is, we conclude theproportion improved is higher when the drug is used.114