95%๎Interval๎for๎Difference๎in๎Population๎Proportions,๎p1โp2๎
Agresti๎โ๎Caffo๎
Returning๎to๎the๎independent๎two๎sample๎case,๎suppose๎now๎the๎data๎are๎from๎a๎
binomial๎setting๎with๎๎๎Y1๎~๎Bin(n1,p1)๎๎independent๎of๎๎๎Y2๎~๎Bin(n2,p2)๎
Of๎interest๎is๎the๎difference๎p1๎โ๎p2.๎
A๎good๎point๎estimator๎for๎p1๎โ๎p2๎is๎the๎difference๎in๎sample๎proportions๎๎
p๏1โp๏2๏ Y1
n1โY1
n2๎
But,๎again๎we๎will๎use๎the๎previous๎AC๎strategy๎for๎this๎confidence๎interval.๎
Presented๎here๎is๎a๎95%๎confidence๎interval.๎๎
Point๎Estimator๎๎Add๎one๎success๎and๎one๎failure๎to๎each๎sample.๎
๎p๏ค1โp๏ค2๏ Y1๏
1
n1๏
2โ Y2๏
1
n2๏
2๎
๎
Critical๎Point๎๎๎๎๎Zฮฑ/2๎=๎1.96๎
๎
Standard๎Error๎of๎Point๎Estimator๎
๎
SE๓ฐp๏ค1โp๏ค2๓ฐ ๏๏จp๏ค1๓ฐ1โp๏ค1๓ฐ
n1๏
2 ๏
p๏ค2๓ฐ1โp๏ค2๓ฐ
n2๏
2 ๎
๎
95%๎Confidence๎Interval๎for๎p1๎โ๎p2๎
๎
p๏ค1โp๏ค2 ๏ Zฮฑ2
โ๏จp๏ค1๓ฐ1โp๏ค1๓ฐ
n1๏
2 ๏
p๏ค2๓ฐ1โp๏ค2๓ฐ
n2๏
2 where p๏ค1โp๏ค2๏ Y1๏
1
n1๏
2โY2๏
1
n2๏
2๎
๎
Note:๎๎Presented๎here๎is๎a๎95%๎confidence๎interval๎โ๎generalizations๎exist๎for๎other๎levels๎of๎
confidence.๎
๎
