A SRS of sixe n = n1 + n2 with mean y- is drawn from a finite population, subsam

Question

A SRS of sixe n = n1 + n2 with mean y- is drawn from a finite population, subsample of size n, is drawn with mean y-1, y-2, is the mean of the remaining ne in the sample. Show v(y-1, - y-2) S2 [(1/n1) + (1/n2)] by using the basic definition put and into , we can get , but I dont know how

Explanation / Answer

Given that y(bar) is the mean of sample size n, y1(bar) is the mean of sample size n1 and y2(bar) is the mean of the sub sample size n2. var(y1bar-y2bar)=s^2((1/n1)+(1/n2)) var(y1bar-y2bar)=E(y1bar-y2bar)^2-(E(y1bar-y2bar))^2 E(y1bar-y2bar)^2=E(y1bar^2-2y1bar*y2bar+y2bar^2)-------(1) (E(y1bar-y2bar))^2=(E(y1bar))^2+(E(y2bar))^2-2*E(y1bar)E(y2bar)-----(2) (1)-(2) ==>E(y1bar^2-2y1bar*y2bar+y2bar^2)-[(E(y1bar))^2+(E(y2bar))^2-2*E(y1bar)E(y2bar)] ==>E(y1bar^2)+E(y2bar^2)-(E(y1bar))^2-(E(y2bar))^2 ==>E(y1bar^2)-(E(y1bar))^2+E(y2bar^2)-(E(y2bar))^2 ==>var(y1bar)+var(y2bar) ==>s^2/n1 +s^2/n2 ==>s^2((1/n1)+(1/n2))

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