Question

1. A stratified simple random sample (STSI) is being planned. Information is available from a pilot study on the variance and cost of data collection in stratum h. The size of stratum h is known to be . The cost equation is $Ym+LPWfQbYwAAAAASUVORK5CYII=$ . The total sample size is . Derive the optimal allocation of units () to the strata.

Answer 1

A stratified simple random sample (STSI) is being planned Information is available from a pilot study on the variance Syun The size of stratum h is known to be na Stratum divided into it non-overlapping group or strata haring sizes Ne , No ... NH : Now , Thy : joth element in the fith stratum h=1CDH. Th = I Thi= Mean of fith stratum. syon an - 1 h (v hj - Fnje - Variance of hith stratum Y = t Non In a Population mean : (N-1) 5² = 2 (n-1) sy on the Min ( Fn = Fj? "h: No of sample units selected from Ath stratum . = n = Total sample size ni = jth sample element selected from both hai n=1 stratum 22152 I Thi = ih h=1 Sample mean Stratum of the ith = 1CDH Inn shop I (Wn; - Un ga = Sample variane of both stratum &

Our onbiased estimator of Ist = I wn In T is where wn- Nu SRS WOR - vare (5 st) = 3 wir vare (in) Here, var (in) = Suun (Na-na); h=1CDA Vare Cust) = I went Syon (Nhanh) A hal it nnnn - 3 winsson (1-hh ) hal na If Nn's are large compared to na's for each h, then we can finore ignore the terms may on f. Var (5 st) ~ I went Synon ret) ~ ne na Now, for estimating the population total Y = NY We can use ist = N 5 st as an enbrand estimator Vare (ist ) = N² vare (5 st) = 2 Nh (Nn-nn Sie na bet South nu Unbiased estimator Vare (Est) = N of vare (5 st) (Nn-nn) No son hal na

nh h=1 nn h ! N Under SRSDOR, sin is an unbiased estimator of Syun V = Vare (Est) = (Nn-on) Na susun Now, ve n huon – 2 An son - Wit Suun. I wasyon We want to minimize v subject to a given east consider the following cost function... c=lo & I en na Co = Over head cost and cn = cost for drawing one unit from Stratum h (h= ICDH Robin Geiven C=o*, to minimize v f(x) subject to g(x) = e . Here we minimize the Lagrangian function z = vet (e-c*) worthh ; h= ICDH 2= whsyon # waste hal na Ona so s ha (0H hal 25 c 1 hel 7 – Wn Syon & sch=0 - nna Wn Syun x rren

Geiven (= e* to men & et_co Wn Syun .cn = c *- co Wn syun Sen = c*_ca hol I wa sogon den put the value of h is og we get nha Whsyun (t_co . Ten 3 ansy unren Consider the special (n= ef* & ease when h=1CDH. n = Whsyon & Je** (et-Co) Wasyon re* = (ex- co ) Wns yun e** & Wn such hal We have n H n=1 1 Wh Sy on = n.

» C&-lo This ninn n Wnsyon 5 Wh Syun h=1 = n Nn Syun | 5 b= [( ) + & Nh Sy Un hal (Neyman's This is of units called (ne optimal alloe allocation the strata to