Question

Q5 Consider a problem of estimating the difference of proportions for two populations. In sample 1, out of nį subjects, Si of them are “successes” and the rest are “failures”. In sample 2, out of n2 subjects, S2 of them are “successes” and the rest are "failures”. It is known that Si ~ B(n1, p1) and S2 ~ B(n2, P2). We are interested in estimating p1 – P2. Si and 2 1. Denote ſi S. Show that Ôi – P2 is an unbiased estimator for pı - P2. And what is the mean squared error of it? ni n2 2. Explain how CLT would be applied to approximate the distribution of P1 - Ô2 so that 01 - 02 * N (: - pı(1 – pı), p2(1 – p2) - P2, + ni 1-P2)) approximately

Answer 1

el bo-be) = prope 1. S~ Blno, p.) and S₂ ~ B{"2, p.) Pa = Se P = Si and Elp - A) = Si E Els S. n2 Elpi - P2) nobi n252 . piope ng :. ho - B is an unbiased estimates of pi-pr. Essor is mean square given by 2 .: to E[161-x) - (pı-pu)]" M5€ • E [ cm –to) - l$-pu)] E [ 14-p)* + (9-4)'. 2161 – bu) (pap] El fi- 1ņu)]* + E [ fe- E (px)] v (8) + VIB) nipuli-pu) + repell-p.) Pe prlı-pu) - 0 central limit theorem states that if we have prop" with meas u and stardard deviation we take sufficiently large random samples, the seamples mean will be approximately M, Galo normally dist? samples & independent & mean deviat ion about in men 0,2 o. 2. o, if dist of samples and as distributed vợ (ệ -... ? (1-2) + ba (1-5) na Hence, proved that pi-pz, N pilt-bu) + bali-pe) Le Å - Pr ก.