SIMPLE RANDOM SAMPLING

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1 UIT IMPL RADOM AMPLIG mple Radom amplg tructure. Itroducto Obectves. Methods of electo of a ample Lottery Method Radom umber Method Computer Radom umber Geerato Method.3 Propertes of mple Radom amplg Merts ad Demerts of mple Radom amplg.4 mple Radom amplg of Attrbutes.5 ample ze for pecfc Precso.6 ummary.7 olutos/aswers. ITRODUCTIO mple radom samplg refers to the samplg techque whch each ad every tem of the populato s havg a equal chace of beg cluded the sample. The selecto s thus free from ay persoal bas because the vestgator does ot make ay preferece the choce of tems. ce selecto of tems the sample, depeds etrely o chace, ths method s also kow as the method of probablty samplg. Radom samplg s sometmes referred to a represetatve samplg. If the sample s chose at radom ad the sze of the sample s suffcetly large, t wll represet all groups the populato. A elemet of radomess s ecessary to be troduced the fal selecto of the tem. If that s ot troduced, bas s lkely to eter ad make the sample urepresetatve. Methods of selecto of a smple radom sample are eplaed ecto.. I ecto.3 the propertes of smple radom samplg are descrbed. mple radom samplg of attrbutes s troduced ecto.4 whereas ecto.5 the sample sze determato for specfc precso s descrbed brefly. Obectves After studyg ths ut, you would be able to descrbe the smple radom samplg; epla the method of RWR ad RWOR; epla ad derve the propertes of smple radom samplg; calculate the varace of the estmate of the sample mea; descrbe the R for attrbute ad ts propertes; ad descrbe ad draw a approprate sample sze for specfc precso. 5

2 tatstcal Techques. MTHOD OF LCTIO OF A AMPL The radom sample obtaed by a method of selecto whch every tem has a equal chace to be selected the sample. The radom sample depeds ot oly o method of selecto but also o the sample sze ad ature of populato. mple Radom amplg If a sample of uts s selected radomly from a populato of sze, ths method s kow as smple radom samplg. As the ame suggest, smple radom samplg s a method whch the requred umber of elemets /uts are selected smply by radom method from the target populato. Oe ca select a smple radom sample by ether of these two methods wth replacemet method ad wthout replacemet method. mple Radom amplg wth Replacemet (RWR) Whe smple radom samples are selected the way that uts whch has bee selected as sample ut s remed or replaced the populato before the selecto of the et ut the sample the the method s kow as smple radom samplg wth replacemet. mple Radom amplg wthout Replacemet (RWOR) Whe smple radom sample are selected the way that a ut s selected as sample ut s ot med or replaced the populato before the selecto of the et ut. Ths method s kow as smple radom samplg wthout replacemet.e. oce a ut s selected the sample wll ever be selected aga the sample. ome procedures are smple for small populato ad s ot so for the large populato. Proper care has to be take to esure that selected sample s radom. Radom sample ca be obtaed by ay of the followg methods: 6. By Lottery method;. By Mechacal Radomzato or Radom umbers method; ad 3. By Computer Radom umber Geerato method... Lottery Method Ths s very popular method of selectg a radom sample uder whch all tems of the populato are umbered or amed o separate slps of paper. These slps of paper should be of detcal sze, color ad shape. These slps are the folded ad med up a cotaer or bo or drum. A bld fold selecto s the made of the umber of slps requred to costtute the desred sze of sample. The selecto of tems thus depeds etrely o chace. For eample f we wat to select caddates out of. We assg the umbers to. Oe umber to each caddate ad wrte these umbers ( to ) o slps whch are made as homogeeous as possble. These slps are the put bag ad thoroughly shuffled ad the slps are draw oe by oe. The the caddates correspodg to umbers o the slp draw wll costtute a radom sample. If we draw a slp ad ote dow the umber wrtte o the slp ad the aga replace the slp the bag ad the draw the et. These umber of slps costtute a sample of requred sze s called a sample of RWR. If we do ot replace the frst slp whch has already bee draw the bag for the subsequet draws the t s called RWOR.

3 The above method s very popular lottery draws where a decso about przes s to be made. However, whle adoptg lottery method, t s absolutely essetal to see that the slps are as homogeeous as possble terms of sze, shape ad color, otherwse there s a lot of possblty of persoal preudce ad bas affects the result. mple Radom amplg.. Radom umber Method The lottery method, dscussed above, become qute cumbersome to use f the sze of the populato s very large. A alteratve method of radom selecto s that of usg the table of radom umbers. The most practcal ad epesve method of selectg a radom sample cossts the use of Radom umber tables whch have bee so costructed that each of the dgts 0,,,, 9 appears wth appromately the same frequecy ad depedetly. If we have to select a sample from a populato of sze ( 99) the the umbers ca be combed two by two to gve pars from 00 to 99. The method of drawg the radom sample cossts the followg steps:. Idetfy the uts the populato wth the umbers from to ;. elect at radom, ay page of the radom umber tables ad pck up the umbers ay row or colum or dagoal at radom ad dscart the umber whch s greater tha ; ad 3. The populato uts correspodg to the umbers selected step-, costtute the radom sample. The sample may be selected wth replacemet or wthout replacemet. I the case of samplg wth replacemet, a umber occurrg more tha oce s accepted. A ut s repeated as may tmes as a radom umber occurs. But the case of samplg wthout replacemet, f a umber radom umber table or remader occurs more tha oce s omtted at ay sequet stage. I the above selecto procedure umberg of uts from 00 owards ad makg use of remaders has a advatage, as o radom umbers s beg wasted durg the selecto procedure. Ths saves tme ad labor. For eample a populato cossts of 0 uts ad a sample of sze 5 s to be selected from ths populato. ce 0 s a two dgt fgure, ut are umbered as 00, 0 9. Fve radom umbers are obtaed from a two dgt radom umber table. They are gve as 85, 63, 5, 34, ad 46. O dvdg 85 by 0 the remader s 5, hece select the ut o seral o. 5. mlarly, dvdg 63, 5, 34 ad 46 by 0, the respectvely remader are 3,, 4 ad 6. Hece selected uts are at seral umbers 05, 03,, 4 ad 06. These selected uts costtute the sample. A umber of radom umber tables are avalable such as:. Tppet s Radom umber Tables These tables cosst of 0,400 four dgts umbers, gvg all 0,400 4.e. 4,600 dgts.. Fsher ad Yate s Radom umber Tables These tables cosst of 5000 dgts arraged two dgts umbers. 3. Kedall ad Babgto mth s Radom Tables These tables cosst of,00,000 dgts grouped to 5,000 sets of 4 dgts radom umbers. 7

4 tatstcal Techques 4. Rad Corporato Radom umber Tables These tables cosst of oe mllo radom dgts cosstg of,00,000 radom umbers of 5 dgts each...3 Computer Radom umber Geerato Method The ma dsadvatage of radom umber method s that f we wat to draw a sample of large sze from the target populato the t wll take a log tme to draw radom umbers from the radom umber table. Thus to save tme ad eergy geerato of large umbers of a radom umbers oe ca opt computer radom umbers geerato method. May kds of methods have bee used for geerato of radom umber from computer but we shall ot dscuss all of them here ad descrbe oly lear cogruetal geerato method. Lear Cogruetal Geerato Method Lear cogruetal method ca take may dfferet forms but the most commoly used form s defed by z = (az + c) mod m for =,, where, a, c ad z are to be the rage (0,,,, m-) ad tegers a - Multpler Iteger c - hft or Icremet Iteger m - Modulus Here, () mod m meas remader term whe s dvded by m. uppose we wat to draw a sample of sze 4 from the populato of sze 8. Take m = 8, a = 5, c =7 ad z 0 = 4 the resultg sequece of radom umbers s calculated as z = ( ) mod 8 = 3 z = ( ) mod 8 = 6 z 3 = ( ) mod 8 = 5 z 4 = ( ) mod 8 = 0 For above calculato we may also make a computer program whch s beyod ths course. We are, therefore, ot gog to dscuss t here..3 PROPRTI OF IMPL RADOM AMPLIG Termologes = Populato sze = ample sze 8 = Value of the character uder study for the th ut the populato

5 = Value of the character uder study for the th ut the sample mple Radom amplg Populato mea amplemea Populato mea square s σ ample mea square Populato varace Theorem : Prove that the probablty of selectg a specfed ut of the populato at ay gve draw s equal to the probablty of ts beg selected at the frst draw. Proof: I smple radom samplg method a equal probablty of selecto s assged to each ut of the populato at the frst draw. Thus, R from a populato of uts, the probablty of drawg ay ut at the frst draw s, the probablty of drawg ay ut the secod draw from amog the avalable s ad so o. Let, r be the evet that ay specfed ut s selected at the r th draw. P ( r ) = Prob.{A specfc ut s ot selected at ay oe of prevous P ( r ) = P ( r ) = That meas r r r P ( It s (r-) draws ad the selected at the r th draw} ot selected at th draw) P (It s selected at r th draw that s ot selected at the prevous (r-) draw) r r P ( r ) = P ( r ) = P ( r ) = = P( ) 3... r r r 9

6 tatstcal Techques Theorem : The probablty that a specfed ut s selected the sample of sze s Proof: ce a specfed ut ca be selected the sample of sze mutually eclusve ways, vz. t ca be selected the sample at the r th draw (r =,,, ) ad sce the probablty that t s selected at r th draw s P ( r ) = ; r,, 3,..., Therefore, the probablty that a specfed ut s cluded the sample would be the sum of the probabltes of cluso the sample at st draw, d draw,, th draw. Thus, by addto theorem of probablty, we get P ( r r ) = r Theorem 3: The possble umbers of sample of sze from a populato of sze f samplg s doe wth replacemet s. Proof: The frst ut ca be draw from uts ways. mlarly, secod ut ca also be draw ways because the frst selected ut aga med wth the populato. o o up to the selecto of th ut. Thus, the total umber of ways are C. C. C... C ( tmes) C Theorem 4: I RWOR the sample mea s a ubased estmator of populato mea. Proof: We have a where, a = f th ut s cluded the sample ce, a takes oly two values ad 0 0 f th ut s ot cluded the sample a =.P a 0.P(a 0) =.P ( th ut s cluded a sample of sze ) + 0.P ( th ut s ot cluded the sample) Hece, = ()

7 Theorem 5: I RWR, the sample mea s a ubased estmator of populato mea. mple Radom amplg Proof: We have ( ). = Theorem 6: I RWOR, the sample mea square s a ubased estmate of the populato mea square,.e. Proof: We have s s = ( ) s We have a a where, a = f th ut s cluded the sample () () 0 f th ut s ot cluded the sample (3) 3

8 3 tatstcal Techques Therefore, (4) ad a a a a (5) where, a ad a are defed equato (3) Therefore, a a.p a a 0.P a a 0 P a a a P a.p a ) ( ) ( (6) Because th a P ut s cluded the sample ad sample the cluded s ut that gve sample the cluded s ut P a a P th th ubsttutg equato (5), we get (7) ubsttutg from equatos (4) ad (7) equato (), we get ) ( s

9 33 mple Radom amplg (s ) = Theorem 7: I RWOR, the varace of the sample mea s gve by Var Proof: We have Var (8) ow (9) From equato (4), we have But Therefore, (0) Also from equato (7), we have ( ) ()

10 tatstcal Techques ubsttutg from equatos (0) ad () equato (9), we get... () ubsttutg from equato () equato (8), we get Var Var Theorem 8: I RWR, varace of sample mea s gve by Var Proof: We have Aga, Var Var Var Var ( ) ce case of RWR each observato s depedet, therefore Var σ. (3) But (4) ubsttutg from equato (4) equato (3) we get Var Theorem 9: The varace of the sample mea s more RWR comparso to ts varace RWOR,.e. RWR RWOR Var Var 34 Proof: We have VarRWR

11 Var ad RWOR Therefore, Var RWR Var RWOR ) Var 0 That mples VarRWR RWOR That meas varace of the sample mea s more RWR as compared wth ts varace the case of RWOR. I other words RWOR provdes a more effcet estmate of sample mea relatve to RWR..3. Merts ad Demerts of mple Radom amplg Merts mple radom samplg has the followg merts:. I smple radom samplg each ut of the populato has equal chace to be cluded the sample; ad. ffcecy of the estmates ca be foud out smple radom samplg because all the estmates are calculated by usg the probablty theory. mple Radom amplg Demerts Despte merts, smple radom samplg has some demerts too vz.. A up-to-date frame of populato s requred smple radom samplg;. ome admstratve coveece arses smple radom samplg f some of the uts are spreaded a wde area. o collectg formato from these related uts may be problem; ad 3. R requred larger sample sze tha ay other samplg for a f level of precso. ample : A populato have 7 uts,, 3, 4, 5, 6, 7. Wrte dow all possble samples of sze (wthout replacemet) whch ca be draw from the gve populato ad verfy that sample mea s a ubased estmate of the populato mea. Also calculate ts sample varace ad verfy that Var RWR Var RWOR oluto: We have =,, 3, 4, 5, 6,

12 tatstcal Techques All possble samples of sze are as follows: ample o. ample values ample Mea (,) (,3) (,4) (,5) (,6) (,7) (,3) (,4) (,5) (,6) (,7) (3,4) (3,5) (3,6) (3,7) (4,5) (4,6) (4,7) (5,6) (5,7) (6,7) Total From the table, we have k 84.0 ad C C 84 4 c Var C

13 Verfcato: I RWOR the varace of sample mea s gve by mple Radom amplg Var I RWR the varace of sample mea s gve by Var Var Var Hece, RWR RWOR ) Draw all possble samples of sze from the populato {, 3, 4} ad verfy that also fd varace. ) How may radom samples of sze 5 ca be draw from a populato of sze 0 f samplg s doe wth replacemet? 3) From a populato of 50 uts, a radom sample of sze 0 s draw wthout replacemet. From the sample followg result are obtaed. 48, 36 Calculate the sample mea ad ts varace. 4) Draw all possble samples of sze from the populato {8,, 6} ad verfy that ad fd varace of estmate of the populato mea. 5) From a populato of sze =00, a radom sample of sze 0 s draw wthout replacemet. From the sample followg results are obtaed. 45 Calculate the varace of sample mea..4 IMPL RADOM AMPLIG OF ATTRIBUT A qualtatve characterstc whch caot be measured umercally s kow as a attrbute.e. hoesty, tellgece, beauty, etc. I may stuatos, t s ot possble to measure the characterstc uder study but possble to classfy the populato to varous classes accordg to the attrbutes uder study. For eample, we ca dvde a populato of a coloy to two classes oly say lterate ad llterate wth respect to attrbute lteracy. Hece the uts the populato ca be dstrbuted these two classes accordgly as t possesses or does ot possess the gve attrbute. After takg a sample of sze, we may be terested estmatg the total umber of proporto of the defed attrbute. 37

14 tatstcal Techques otatos ad Termologes Let us suppose that a populato havg uts,,, s classfed to k mutually dsot ad ehaustve classes. The = The proporto of uts processg the gve attrbute populato A/ = The proporto of uts ot processg the gve attrbute populato A / = - Let us cosder RWOR sample of sze. From ths populato f a s the umber of uts a sample possessg the gve attrbute the p = proporto of sampled uts possessg the gve attrbute = a / q = proporto of sampled uts ot possessg the gve attrbute = a / Let be the th ut of the populato, where =,,,. The, = f th ut possesses the gve attrbute = 0 f t does ot possess the gve attrbute mlarly, deote th ut the sample The, =, f th sampled ut possesses the gve attrbute = 0, f th sampled ut does ot possess the gve attrbute The = A, the umber of uts the populato possessg the gve attrbute. ad = a, the umber of sampled uts possessg the gve attrbutes. Thus, A ad mlarly, A π a p ad a p 38 ( )

15 mlarly, mple Radom amplg s p p pq Theorem 0: ample proporto p s a ubased estmate of populato proporto π,.e. (p) = Proof: We have a p A π We kow that smple radom samplg the sample mea provdes o ubased estmate of the populato mea Therefore (p) = Theorem : I RWOR, show that Var (p) = Proof: We have, Var (p) = Var =. ( ).. ( ) =. ( )..5 AMPL IZ FOR PCIFIC PRCIIO A very frst problem faced by a statstca ay sample survey s to determe the sample sze so that the populato parameters may be estmated wth a specfed precso. The degree of precso ca be determed terms of. The level of sgfcace the estmate; ad. The cofdece terval wth whch ths estmate le wth respect to gve level of sgfcace. Let us cosder the parameter the populato mea of the populato of sze. We kow that sample mea based o uts s ubased estmate of. Let the dfferece betwee estmate value ad the populato mea s d ad level of cofdece s (-), the the sample sze s determed by the equato. 39

16 tatstcal Techques P d α or P d (5)... (6) where, s very small preassged probablty ad s ow as the level of sgfcace.. If s suffcetly large ad we cosder RWOR, the the statstc Var (7) where, Z s a stadard ormal varate. Accordgly, f we take = 0.05, the we have P Z P P Comparg wth equato (6), we get d.96 d.96 d d (8) Ths formula gves the sample sze RWOR for estmatg populato mea wth cofdece level 95 % ad marg of error d, provded s large. mlarly, f s small the statstcs z follows the studet s t dstrbuto wth (-) degree of freedom s gve by t 40

17 If t s the crtcal value of t for ( ) df ad at level of sgfcace the s gve by the equato mple Radom amplg P. t (9) Comparg wth equato (6) we get d. t d t.6 UMMARY t t d t t d I ths ut, we have dscussed:. The smple radom samplg;. The method of RWR ad RWOR; 3. The propertes of smple radom samplg; 4. Method of fdg the varace of the estmate of the sample mea; 5. The smple radom samplg for attrbute ad ts propertes; ad 6. The sample sze determato for specfc precso..7 OLUTIO / AWR ) I RWOR the umber of sample s 3 C C 3 The samples wth ther meas are as follows: r. o ample Mea,3.5, ,

18 tatstcal Techques Therefore, C C Aga V = Therefore, Var ) The frst ut ca be draw from 0 uts 0 C = 0 ways. ce samplg s doe wth replacemet so the secod ut ca be draw 0 C = 0 ways so o upto the selecto of 5 th Ut. Thus the total ways are = 0 5 ways. 4 3) We have o s s whch s the estmate value of.

19 Therefore, Varace mple Radom amplg 4) I RWOR the umber of samples s C = 3 C = 3 ad samples wth ther meas are Therefore, Aga Therefore,. o. ample Mea (8, ) 0 (8, 6) 3 (, 6) 4 Total C, C Aga estmator of populato mea s sample mea ad so ts varace Var where, Therefore, Var

20 tatstcal Techques 5) We have, ad Var s ( ) Whch s the estmate value of. Therefore, Var

Sixth Edition. Chapter 7 Point Estimation of Parameters and Sampling Distributions Mean Squared Error of an 7-2 Sampling Distributions and

Sixth Edition. Chapter 7 Point Estimation of Parameters and Sampling Distributions Mean Squared Error of an 7-2 Sampling Distributions and 3//06 Appled Statstcs ad Probablty for Egeers Sth Edto Douglas C. Motgomery George C. Ruger Chapter 7 Pot Estmato of Parameters ad Samplg Dstrbutos Copyrght 04 Joh Wley & Sos, Ic. All rghts reserved. 7