ON THE COMPARISON OF SOME METHODS OF ALLOCATION IN STRATIFIED RANDOM SAMPLING FOR SKEWED POPULATION

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1 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( O THE COMPARISO OF SOME METHODS OF ALLOCATIO I STRATIFIED RADOM SAMPLIG FOR SKEWED POPULATIO Lawal M 1., Salami A., Obisesa K.O., usuff K.M 3. ad Owolabi A.A 1 1 Matematical & Computer Sciece Departmet, Foutai Uiversity Osogbo igeria Departmet of Statiics Uiversity of Ibada, igeria 3 Departmet of Statiics Federal Uiversity of Agriculture Abeokuta, igeria ABSTRACT: A udy to evaluate ad compare some metods of allocatio i ratified radom samplig suitable for te eimatio of populatio total of a skewed populatio was carried out i tis paper. We looked at tree metods of allocatio i te above sceme amely; Optimum allocatio, -proportioal allocatio, ad variable () proportioal allocatio metods. We iveigate te coditio uder wic oe metod of allocatio is better ta te oter usig tree sets of real life data o aff ad udet erolmet, collected from te record of te Teacig Service Commissio (TESCOM) Oyo State igeria. Te tird set of data is o Icome ad expediture of Idurial ad Geeral Isurace (IGI) Plc. We foud out tat optimum allocatio is te lea ad te be despite variatio observed i te sizes of witi te rata. KEWORDS: Stratificatio, Allocatio, Skewess, Eimatio, Samplig ITRODUCTIO Samplig is te process for selectig a part or a fractio of a populatio ad observig te selected part wit respect to some properties of itere ad te drawig some coclusios about te populatio. Samplig is carried out every day by makig decisio cocerig te caracteriics of a large umber of items based o aalysis of a small umber selected from tem. Samplig plays a vital role i researc desig ivolvig uma populatio ad commads icreasig attetio from social scietis, cemi, egieers, accoutats, biologi ad medical practitioers (Kis 19) ad (Hut & Tyrell 00). Stratified radom samplig is a tecique wic attempts to rerict te possible samples to tose wic are ``less extreme'' by esurig tat all parts of te populatio are represeted i te sample i order to icrease te efficiecy. Rarely, is a survey carried out witout ratificatio, were a populatio is divided ito a set early omogeous subset called rata ad idepedet samples are draw from eac ratum. I ratified samplig, te values of sample size i te respective rata are cose by te sampler, wic may be carried out wit eiter te aim of miimizig te variace for a specified co or to miimize te co for a specified value of variace. Adebola et al (01). A differet approac of miimizatio of geeral variace ad maximizatio of sample sizes usig ratificatio of skewed populatio as bee suggeed by differet autors amog wom are Daleius ad Hodges (199), Glasser (19), Hidiroglou (198) ad Morga et al (00). Morga et al (00) said if ratificatio boudaries are take i geometric progressio, te coefficiet of variatio of rata usig skewed diributio are equal owever, tis metod Prit ISS: 03-9, Olie ISS: ISS

2 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( does ot satisfy Daleius ad Hodges (199) coditios of variace miimizatio of te sample mea but compares favourably wit te commoly used cumulative root frequecy approximatio of Daleius ad Hodges i terms of precisio. I te practical applicatios of eyma s metod of allocatio, oe is faced wit certai limitatios especially we it is cocered wit problem of eimatio of several populatio caracteriics. It is ofte foud tat tese caracteriics make coflictig demads o te desig ad terefore, allocatio of sample to te differet rata based o oe caracter usig eyma s allocatio may vary, ad as well lead to loss i precisio of eimates of oter caracters as compared to proportioal allocatio, Sukatme (19). It was discovered, by Cocra (1977) tat te precisio of te eimate is icreased if eyma s approac is adopted i preferece to metods of proportioal samplig, be it - proportioal or variable () proportioal samplig. If te allocatio of te sample amog te rata is far from optimum, proportioal metods ad ratified samplig may ave a iger variace, Sukatme (193). However, proportioal allocatio will be very useful if te rata averages differ from oe aoter, Raj (198). We skewed populatios are used i te eimatio of -proportioal allocatio, a cosiderable variace results due to te fact tat te ratum cotaiig te very large uits will be foud to be muc greater ta oter rata. It is believed tat a more cosiderable eimate of populatio totals will be yielded if -proportioal allocatio is used i te eimatio of skewed populatio as its eimate makes use of ratum averages wic is expected to be may times greater ta te geeral average. I eimatig populatio totals ad percetages, Hase, Hurwitz ad Madow (193) foud out tat tere are cases were a kid of combied procedure migt be useful if optimum is used i eimatig populatio totals. It is our aim i tis paper to examie ad compare some of tese metods of allocatio i ratified radom samplig to kow te oe suitable for eimatig populatio totals of a skewed populatio METHODOLOG Te aalysis is based o ratificatio of skewed populatio were tree sets of real life data were used. Te fir set of data was sourced from te record of Oyo State Teacig Service Commissio o umber of Staff i eac scool of te tirty tree (33) local govermets i Oyo State for 008/009 sessio, wic represet our auxiliary variable (), as we allow te variable of itere (), to be aff erolmet for 009/010 sessio. Also from te record of Oyo State Teacig Service Commissio is te secod set of data o udets erolmet for te year 008/009 sessio (), ad 009/010 sessio (). Te la data set is o Icome () ad Expediture () of Idurial ad Geeral Isurace Plc. Defiitio Of Populatio Total Te eimate of populatio i ratified radom samplig is; Prit ISS: 03-9, Olie ISS: ISS

3 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( y L 1 y wile te variace is V L 1 y 1 1 (Cocra 1977) Allocatio Of Sample To Strata Stratified radom samplig does ot specify a particular size of sample to be attaced to a give ratum. Te sample ca be selected i order to ave same size i eac ratum or it ca be allocated i some oter ways. As log as we select at lea oe elemet i a ratum te specificatio of a ratified radom samplig is satisfied ad wit two uits i a ratum, we ca eimate bot its mea ad adard error. -Proportioal Allocatio I tis allocatio metod, sample uits are selected from witi rata i proportio to teir rata sizes. Its umber of sample i t ratum ca be eimated by Its correspodig variace of total eimate witout replacemet is V prop) L 1 y (Raj 198) Optimum Allocatio For optimum allocatio, te umber of samples to be selected i te t ratum (eyma 193) is give by y L 1 y Its correspodig variace of total eimate for witout replacemet terefore, V opt 1 L L y y (Raj 198) Prit ISS: 03-9, Olie ISS: ISS

4 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( -Proportioal Allocatio Te sample size per t ratum, ca be foud as a proportio of rater ta to i.e. L 1 suc tat; Te variace of populatio total for witout replacemet is V x prop L 1 / y Raj (198) COMPUTATIOAL FRAMEWORK Statiics O Teacers Populatio Data i x i1 Skewess () 3(Mea Media) i i1 Skewess () 3(Mea Media) 1.3 x Prit ISS: 03-9, Olie ISS: ISS

5 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( Sample Size Allocatio to Strata for -Proportioal Allocatio 1,, 3,,, i.e. 1 1 y V prop) 1 y S.E prop) S. E C.V prop x % 1% Sample Size Allocatio to Strata for -Proportioal Allocatio 97, ad 1, Prit ISS: 03-9, Olie ISS: ISS

6 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( Tus, y V x prop) 1 y / x S.E x prop) C.V. prop S.E x % 1% Sample Size Allocatio to Strata for Optimum Allocatio 97, 1 1 y y, ad y 1 yi y 1 H Hece, 1 1 y Prit ISS: 03-9, Olie ISS: ISS

7 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( V opt 1 1 y 1 y opt S.E C.V. opt S.E x % 1% Statiics O Studet Populatio Data i1 1 i Skewess () 3 (Mea Media) i1 1 i Skewess () 3(Mea Media) 1.3 Sample Size Allocatio to Strata for -Proportioal Allocatio, 1,, 3, Prit ISS: 03-9, Olie ISS: ISS

8 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( Tus, 1 1 y 1.9 V prop) 1 y S.E prop) S. E C.V prop x % 1% Sample Size Allocatio to Strata for -Proportioal Allocatio 97, ad 1, Tus, y Prit ISS: 03-9, Olie ISS: ISS

9 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( V x prop) 1 y / x S.E x prop) C.V. prop S.E x % 1% Sample Size Allocatio to Strata for Optimum Allocatio 97, 1 y 1 y, ad y 1 yi y Hece, 1 1 y V opt 1 1 y 1 y opt S.E Prit ISS: 03-9, Olie ISS: ISS

10 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( C.V. opt S.E x % 1% Statiics o Icome Ad Expediture Data x i 1 1 i Skewess () 3(Mea Media) i1 1 1 x i 100. Skewess () 3(Mea Media) 1.0 Sample Size Allocatio to Strata for -Proportioal Allocatio, 1,, 3,, Prit ISS: 03-9, Olie ISS: ISS

11 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( Tus, 1 1 y V prop) 1 y x S.E prop) S. E C.V prop x % 1% Sample Size Allocatio to Strata for -Proportioal Allocatio 7, ad 81, Tus, y Prit ISS: 03-9, Olie ISS: ISS

12 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( V x prop) 1 y / x S.E x prop) C.V. prop S.E % % Sample Size Allocatio to Strata for Optimum Allocatio 7, 81 y 1 y, ad y 1 yi y Hece, 1 1 y V opt 1 y 1 1 y x opt S.E Prit ISS: 03-9, Olie ISS: ISS

13 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( C.V. opt S.E x % 1% RESULTS AD DISCUSSIO I te aalysis, -Proportioal, -proportioal ad optimum metod of allocatio i ratified radom samplig of populatio totals of a skewed populatio were demorated o tree sets of real life data. Tese demoratios are sow usig te tree sets of data wit four, five ad six rata, ad two varieties draw from differet secodary sources. Te fir ad secod sets of data are o teacers populatio ad udets erolmet respectively i eac cosidered scool from te tirty tree Local Govermet Area of Oyo State igeria. Wile te tird set of data is o icome ad expediture of IGI. Summary of te results is as sow i te followig table; TABLE 1: SUMMAR OF THE RESULTS OBTAIED S/ DATA 1. Staff Erolmet (008/ 009(), 009/ 0010(). Studet Erolmet 008/009(), 009/ 010() 3. Icome () ad Expediture () POPULATIO SIZE () SAMPLE SIZE () V (opt) V (-prop) V (-prop) SKEWE SS () SKEWE SS () , 1, , ,079, x , ,9, x x10 13 DISCUSSIO Te table above sows te summary of te results obtaied from te aalysis usig ratificatio of skewed populatios for tree metods of allocatio i ratified radom samplig. From te table, it was observed tat; V opt < V () < V(x) at skewess rate of > 1 ad > 1 It te follows tat; 1. V opt domiates te oter two metods. V () domiates V (x) i all te cases we skewess rate of auxiliary variable is greater ta oe. Based o te results obtaied ad our observatios i table 1, we could see tat, a optimum allocatio metod is te be for eimatig populatio total of a skewed populatio. Also, as Prit ISS: 03-9, Olie ISS: ISS

14 Iteratioal Joural of Matematics ad Statiics Studies Vol., o.3, pp.9-, September 018 _Publised by Europea Cetre for Researc Traiig ad Developmet UK ( it ca be observed i table.1, tat < V() < V(x), iferrig tat - proportioal allocatio provided tat te skewess rates of auxiliary variable ad variable of itere are bot greater ta oe. COCLUSIO Results from table 1 sow tat optimum allocatio procedure attracted te lea variace as compared to oter procedures, despite variatio observe i te sizes of witi te rata. It is terefore evidet i tis researc work, tat for eimatig te average ad variaces of parameters uder ratified radom samplig of skewed populatio, te performace of Optimum Allocatio tecique is te be we compared wit te two oter cosidered allocatio procedures. REFERECES Abraamso, I.L., elso, C.R., ad Affleck, D.L.R. (011). Assessig te performace of Samplig Desigs for measurig te aboudace of uderory plats. Ecological Apllicatio 1(): - Adebola, F. B. & Ajayi O. S. (01) Compariso of allocatio procedures i ratified radom samplig of skewed populatios uder differet diributios ad sample sizes. Iteratioal Joural of iovative Sciece, Egieerig & Tecology, Vol. 1 (8); pp 18 - Adegboye, O. S. & Ipiyomi R.A. (199), Statiical Tables for class work & Examiatios Tertiary Publicatio igeria Limited. Amaia G.. (009) STA 7 ote o Desig ad Aalysis of Sample Survey Upublised ote. Cocra, W. G. (1977) Samplig Tecique ( d Editio) Wiley Publicatio Ltd. Daleius T. (190) Problem of optimum ratificatio skadiavsk Aktuarietidskrift Vol 33, pp Des Raj (198), Samplig Teory (3 rd Editio) MC Graw Hall Book Compay. Hidiroglou, M.A. & Lavellee (1988) O te ratificatio of skewed populatios A joural o survey metodology. Vol. 1 o 1, pp Horga, J.M. (00), Stratificatio of skewed populatio Iteratioal atiical review Vol. 7. Hut,. & Tyrell, S. (00) Stratified Samplig, Covetry Uiversity Press Kis, L. 19 Survey samplig. ew ork, Wiley Seti, V.K. (193), A ote o optimum ratificatio of populatios for eimatig te populatio meas. Vol. pg Sukatme P.V (19), Major Developmets i samplig Teory & Practice Jo Wiley & Sos Lodo. ew ork Sydey. Tompso S.K. (01) Samplig. 3 rd Editio. Jo Wiley ad Sos. Holboke, ew Jersey. Prit ISS: 03-9, Olie ISS: ISS 03-10