Sampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING

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1 Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR

2 he smple random samplng scheme provdes a random sample where every unt n the populaton has equal probablty of selecton. Under certan crcumstances, more effcent estmators are obtaned by assgnng unequal probabltes of selecton to the unts n the populaton. hs type of samplng s known as varyng probablty samplng scheme. If y s the varable under study and x s an auxlary varable related to y, then n the most commonly used varyng probablty scheme, the unts are selected wth probablty proportonal to the value of x, called as sze. hs s termed as probablty proportonal to a gven measure of sze (pps) samplng. If the samplng unts vary consderably n sze, then SRS does not take nto account the possble mportance of the larger unts n the populaton. A large unt,.e., a unt wth large value of y contrbutes more to the populaton total than smaller unts, so t s natural to expect that a selecton scheme whch assgns more probablty of ncluson n a sample to larger unts than to smaller unts would provde more effcent estmators than the estmators based on equal probablty scheme. hs s accomplshed through pps samplng. ote that the sze consdered s the value of auxlary varable x and not the value of study varable y. For example n an agrculture survey, the yeld depends on the area under cultvaton. So bgger areas are lkely to have larger populaton and they wll contrbute more towards the populaton total, so the value of the area can be consdered as the sze of auxlary varable. Also, the cultvated area for a prevous perod can also be taken as the sze whle estmatng the yeld of crop. Smlarly, n an ndustral survey, the number of workers n a factory can be consdered as the measure of sze whle studyng the ndustral output from the respectve factory.

3 Dfference between the methods of SRS and varyng probablty scheme: In SRS, the probablty of drawng a specfed unt at any gven draw s the same. In varyng probablty scheme, the probablty of drawng a specfed unt dffers from draw to draw. It appears n pps samplng that such procedure would gve based estmators as the larger unts are over-represented and the smaller unts are under-represented n the sample. hs wll happen n case of sample mean as an estmator of populaton mean where all the unts are gven equal weght. Instead of gvng equal weghts to all the unts, f the sample observatons are sutably weghted at the estmaton stage by takng the probabltes of selecton nto account, then t s possble to obtan unbased estmators. In pps samplng, there are two possbltes to draw the sample,.e., wth replacement and wthout replacement. Selecton of unts wth replacement: he probablty of selecton of a unt wll not change and the probablty of selectng a specfed unt s same at any stage. here s no redstrbuton of the probabltes after a draw. Selecton of unts wthout replacement: he probablty of selecton of a unt wll change at any stage and the probabltes are redstrbuted after each draw. PPS+WOR s more complex than PPS + WR. We consder both the cases separately. 3

4 PPS samplng wth replacement (WR): Frst we dscuss the two methods to draw a sample wth PPS and WR.. Cumulatve total method: he procedure of selecton of a smple random sample of sze n conssts of assocatng the natural numbers from to unts n the populaton and then selectng those n unts whose seral numbers correspond to a set of n numbers where each number s less than or equal to drawn from a random number table. In selecton of a sample wth varyng probabltes, the procedure s to assocate wth each unt a set of consecutve natural numbers, the sze of the set beng proportonal to desred probablty. If x, x,..., x are the postve ntegers proportonal to the probabltes assgned to the unts n the populaton, then a possble way to assocate the cumulatve totals of the unts. hen the unts are selected based on the values of cumulatve totals. hs s llustrated n the followng table: 4

5 Unts Sze Cumulatve szes = = + Select a random number R between and by usng random number If R, then th unt s selected wth probablty, =,,,. = j= j table. = j= j Repeat the procedure n tmes to get a sample of sze n. = j= = j= 5

6 In ths case, the probablty of selecton of th unt s P = = P. ote that s the populaton total whch remans constant. Drawback: hs procedure nvolves wrtng down the successve cumulatve totals. hs s tme consumng and tedous f the number of unts n the populaton s large. hs problem s overcome n the Lahr s method. 6

7 Lahr s method: Let M. M = Max =,,...,,.e., maxmum of the szes of unts n the populaton or some convenent number greater than he samplng procedure has followng steps:. Select a par of random number (, j) such that, j M.. If j then th, unt s selected otherwse rejected and another par of random number s chosen. 3. o get a sample of sze n, ths procedure s repeated tll n unts are selected. ow we see how ths method ensures that the probabltes of selecton of unts are varyng and are proportonal to sze. Probablty of selecton of th unt at a tral depends on two possble outcomes ether t s selected at the frst draw or n subsequent draws preceded by neffectve draws. Such probablty s gven by P( ) P( j M / ) = = M. * P, say. Probablty that no unt s selected at a tral = = M = M = = Q, say. M 7

8 Probablty that unt s selected at a draw (all other prevous draws result s non selecton of unt ) = P + QP + Q P +... * * * * P = Q / M = = =. / M total hus the probablty of selecton of unt s proportonal to the sze. So ths method generates a pps sample. Advantages:. It does not requre wrtng down all cumulatve totals for each unt.. Szes of all the unts need not be known before hand. We need only some number greater than the maxmum sze and the szes of those unts whch are selected by the choce of the frst set of random numbers to for drawng sample under ths scheme. Dsadvantages: It results n wastage of tme and efforts f unts get rejected. he probablty of rejecton =. M M he expected numbers of draws requred to draw one unt =. hs number s large f M s much larger than. 8

9 Example: Consder followng data set of 0 number of workers n the factory and ts output. Factory umber of Industral Cumulatve total of szes no. workers producton () (n metrc tons) (Y) (n thousands) 30 = 5 60 = + 5= = = = = = + 7 = = 8 + = = = = = = 47 + = = = 64 9

10 Selecton of sample usng cumulatve total method:. Frst Draw: Draw a random number between and Suppose t s 3-4 < 3 < 5 - Unt Y s selected and Y 5 = 8 enters n the sample... Second Draw: Draw a random number between and 64 Suppose t s 38 7 < 38 < 8 Unt 8 s selected and Y 8 = 7 enters n the sample and so on. hs procedure s repeated tll the sample of requred sze s obtaned. Selecton of sample usng Lahr s Method In ths case M = Max = 4 =,,...,0 So we need to select a par of random number (, j) such that 0, j 4. Followng table shows the sample obtaned by Lahr s scheme: 0

11 Random Random Observaton Selecton of number number unt 0 j j = 7 < = 0 tral selected 3 ( y3) 6 3 j = 3 > 6 = tral rejected 4 7 j = 7> 4 = 4 tral rejected 9 j = 9> = 5 tral rejected 9 j = < = tral accepted 9 ( y9) and so on. Here ( y, y ) 3 9 are selected nto sample.

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