Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed pared sample s ( X,,,,, If the sample totals are, X X, the fd the covarace betwee X tot ad tot tot tot Soluto: The covarace betwee ote that X tot ad s (, ( E( Xtottot E( Xtot E( tot tot { }{ ( } Cov X E X E X E ( ( tot tot tot tot tot tot E X X X X tot tot E tot tot where X X,, ( E X ( E X X X j ( j j Also, where X X+ X j j j j j X j X X j j X X, The Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page
(, Cov X tot tot E X X ( + E X j X X j j j j ( + ( X X X j j ( + ( ( X X( X ( ( + ( ( ( ( S X where SX ( X X( X X X j Exercse (Smple radom samplg: X X X Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page Uder the smple radom samplg wthout replacemet, fd E( s xy where Soluto: Cosder s X x y ( ( xy x X, y s X x y ( ( xy X xy X x y X X+ X X X j j j j j j
Sce Thus E X X ( E X j X j j j ( j ( ( ( E( sxy X X j j X X j j j X X X ( + X X ( X X X X S X ( ( Exercse 3 (Smple radom samplg: Suppose a populato of sze, ad are two extreme values the sese that deotes the extremely low ad deotes the extremely hgh values amog the sample uts,,, Istead of usg the sample mea y y as a estmator of populato mea such a case, the sample mea estmator s modfed as follows y+ k f sample cotss but ot y y k f sample cotss but ot y for all other samples where k > 0 s kow costat Determe f y s a ubased estmator of populato mea ad fd ts varace Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 3
Soluto: Suppose the populato uts,,, are labeled as,,, So ad ow correspods to labels ad respectvely Sce the sample space correspodg to y has three possble stuatos, so t ca be dvded to three dsjot subsets as follows: Ω Ω Ω { All the uts thesample} { All the uts the sample are such that ut label s preset ad ut label s abset} { All the uts the sample are such that ut labael s preset ad ut label s abset} Ω 3 Ω Ω Ω Whe a sample of sze s draw from a populato of sze, there are the umber of possble subsets of sample draw from populato * * * 3 Uder SRSWOR, ( E y Ω are Ω are Ω are y Ω ( y k ( y k y + + + Ω Ω Ω3 y y y k k + + + Ω Ω Ω 3 y Ω So y s a ubased estmator of populato mea possble subsets Smlarly, Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 4
The varace s Observe that * ar y y Ω ( ( + + + ( y k ( y k ( y Ω Ω Ω3 + + ( y ( y ( y Ω Ω Ω3 + ( ( C C y y Ω Ω3 + Ω Ω Ω3 ( y C C ( y ( y ( ( * Thus Smlarly uts 3 Ω cota the ut label, of them are havg ut labels ( j,3,, ad oe of them cota the ut label ( y y Ω Ω Ω 3 j j + j j + ( y + j Ω j Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 5
Fally, ( ( y ar( y + k k j + ( j + k + j j S y k Example 4 (Smple radom samplg: ( k Let a sample of sze s draw from a populato of sze 3 havg uts, ad 3 The followg estmator s used to estmate the populato mea depedg upo whch samplg uts are chose + y + 3 3 + 3 ( f sample s, ( f sample s, 3 ( f sample s, 3 erfy f y s a ubased estmator of populato mea or ot ad fd ts varace Soluto: The sample space Ω { },, 3 3 3 E( y 3 + + + + + 3 3 ( + + 3 3 whch mples that y s a ubased estmator of Its varace s 3 3 ar( y 3 + + + + + 3 3 4 3 3 + + 3 3 3 + + + + + + + + 4 4 4 4 9 9 4 6 6 3 3 + + 3 9 3 Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 6
Exercse 5 (Smple radom samplg: S The populato coeffcet of varato of a varable s defed as C where S (, s based o populato sze Let the sample mea s based o sample sze by SRSWR whch s usually employed to estmate Assumg y y C to be kow, mprove the sample mea the sese that the estmator has mmum mea squared error Fd out the relatve effcecy of ths estmator relatve to sample mea Soluto: Cosder a scalar multple of y to estmate as yk ky where k s ay scalar The mea squared error of ( M yk E ky E k( y ( k + k ar ( ysrswr + ( k + 0 k S + k ( ( k C + k y k s ow we use the prcple of maxma/mma s follows to fd the mum value of k for whch M ( yk s mmum ( k dm k C ( k dk + dm ( k Substtutg 0 dk k + C k k*, say + C The secod order codto for maxma/mma s satsfed Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 7
Thus the mum estmator s y yk k* y + C ad ts mmum mea squared error s * * ( k + ( M y k C k 4 C C + + C + C C + C + C C + C ar( SRSWR + C M ( k Relatve effcecy + C ar( y Exercse 6 (Samplg for proportos: Suppose we wat to estmate the proporto of me employees (P a orgazato havg 500 total employees I addto, suppose 3 out of 0 employee are me Suppose a sample s draw by smple radom samplg Fd the sample sze to be selected so that the total legth of cofdece terval wth cofdece level 005 s less tha 00 for SRSWR ad SRSWOR Soluto: Frst cosder that a sample of sze s draw from a populato of sze by SRSWR The ( cofdece terval for sample mea ( y or equvaletly the sample proporto ( 00 % p s gve by Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 8 wr
sy s y y z, y+ z, ( ( pwr pwr pwr p wr or pwr z, p+ z The total legth of cofdece terval s It s gve that ( p pwr wr z ( p pwr wr z 00 p ( p wr wr or z 0000 ( z p ( p or wr wr For 005, 3 96 ad 03, we have 0 z p wr 96 03 07 + 8069 0000 The sample sze s greater tha the gve populato sze ths case but sce SRSWR s used, so t s ot mpossble ext we cosder the case whe sample s draw by SRSWOR The related 00( % cofdece terval for y or equvaletly the proporto ( p wor s or y z sy, y+ z sy, pwor ( pwor pwor ( pwor p wor z, pwor + z The total legth of cofdece terval s z p wor ( pwor Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 9
It s gve that pwor ( pwor z 00 z p wor wor or 0000 + z p p 0000 wor ( wor or pwor ( pwor 0000+ z ( p 3 Sce pwor 03, z 96, 500, we have 0 0000 (96 03( 03 + 03( 03 0000 + (96 500 65 Exercse 7: (aryg Probablty scheme: Show that the ates-grudy estmator s o-egatve uder Mdzuo samplg desg Soluto: A suffcet codto for the ates-grudy estmator to be o-egatve s ππ j πj 0, j,, j,,, The expresso of π, π j ad π j are X π + Xtot X j π j + Xtot ( ( ( ( ( ( ( ( X + X j π j + Xtot where ow X tot X s the populato total Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 0
( ( ( ( ( ( ( ( ( ( X X j X + X j ππ j πj + + Xtot Xtot Xtot XX j X + X j + + Xtot Xtot ( ( ( ( ( ( XX j ( ( X + X j ( ( + + Xtot Xtot Each of the term o rght had sde s oegatve ad so that ates-grudy estmator s always oegatve Exercse 8 (Stratfed samplg: Suppose that the populato of sze ca be expressed as populato s dvded to strata where the ( Gh h k+ j, j,,, k, h,,, k where ad k are tegers The th h stratum cotas uts that are labeled as Suppose oly oe ut s radomly selected from every stratum ad a sample of sze s obtaed The values of uts the populato are modeled as + β,,,, where ad β are some costats Fd the varace of populato total Soluto: The estmate of populato total case of stratfed samplg s L h st yh h h where y h L y j hj Compared to the otatos stratfed samplg, we have umber of strata ( L th h stratum sze ( h k sample sze from th h stratum ( h The Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page
k y j st tot h h k h Sce varace of st s ( st ar so we have y h ( ( L h h h Sh, h h h ( 0 ( k k ar k k h j k ( hj h k h j k ( hj h Whe populato values are modeled by the relato + β,,,,, the Thus j k ( hj h ( hj + β h k+ j k h hj k k {( h k j} k + β + j kk ( + β ( h k+ k + hj h β j ( k + k β j + ( k+ j j j 4 β β k k ( + ( + ( + ( + k k k k k k k + ( ( k β ( k ar tot 6 4 Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page
Exercse 9 (Stratfed samplg: Suppose there are two strata of szes ad such that +, (populato sze Let ω ad ω Assume the populato varaces of frst ad secod strata are S ad S ad they are equal, e, S S Cosder a cost fucto C C + C where C ad C are the cost of collectg a observato from frst ad secod strata respectvely Assumg ad to be large, show that arprop ( yst ωc + ω C ar ( yst C C ω + ω where ( ad ( ar y ar y are the varace of stratum mea uder proportoal ad mum prop st st allocatos stratfed samplg Soluto We have ar y ( ω S st K K ω ω S whe s large where K s the umber of strata If cost fucto, geeral, s K C C the uder proportoal allocato Thus or d C K C or d K K or C dc K or C d C Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 3
K C C s the requred sample sze I the set up of gve framework, we have K, S S S, so C,,, C + C Substtutg the expresso for varace ar prop ( y st S S + + S C C C + C C + C C + C + S C C + C S (Usg + C ωc+ ωc S C The sample sze uder mum allocato for fxed cost s L C,,, K S C S The varace of y st uder mum allocato whe s large s Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 4
ar ( y st S S + S C S C S + SC SC C C ( + ( + C C C C C C S + assumg S S S C C S C + C C + C C C S C ω + ω C C From the expresso of ( ( arprop ( yst ωc+ ωc ar ( yst C C ar y ad ar y, we fd prop st st ( ω + ω [ ] Exercses 0 (Stratfed samplg: Suppose there are two strata ad sample of szes ad are draw from these strata Let φ a be the actual rato of ad, e, φ a Let φ be the rato of ad uder mum allocato, e, ( φ Let a ad ( be the varaces of stratum mea uder actual ad mum allocatos respectvely Show that rrespectve of the values,, S, S ad S, the rato 4φ ( +φ φa whe ad are large ad φ φ a s ever less tha Soluto: Whe ad are large, the Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 5
( ar y st S S a + ad varace uder mum allocato s Thus ar yst S S a ( ( + S ( S + S S + S + S S + S Sce uder mum allocato Thus S S + S S S + S S S φa φ φ φ a φ S S + φ a + φ + φ ( + φ Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 6
I ths expresso, replace by + ad ( ( the we have + φ by φ + 4 φ, a ( φ + 4 φ ( φ + ( + φ Usg the result that z + wheever z 0, z + β β over the expresso a, we otce that sce 4φ ( + φ s always true, so we have that a 4φ ( + φ Exercse (Stratfed samplg: Suppose there are two strata ad equal sze of samples are draw from both the strata e, Aother o to draw the samples s mum allocato Let allocato ( ad deotes the varaces uder equal ad mum allocato respectvely the assumg ad are large, show that e e δ where δ, δ + ad are the sample szes draw from frst ad secod strata Soluto: The varace of y st s assumg ( ar y st K S to be large K S Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 7
Whe, say, the varace of y st uder equal allocato s e ar( yst ( S + S Uder mum allocato, ad S S + S S S + S S Sce δ, so S S S ar ( yst + S S + S S S + S S + S ( δ ( δ + ( S + S e S ( + δ S ( + δ S e S S ( + δ ( δ ( + δ Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 8
Exercse (Rato method of estmato Cosder a geeralzed form of rato estmator s some scalar Derve the approxmate bas ad mea squared error of R y X for estmatg the populato mea where x For what value of, the mea squared error s mmum Derve the mmum mea squared error ote that for reduces to usual rato estmator Soluto: Defe x X ε x X + y ε y + E 0, 0 ( ε E( ε ( ε 0 0 ( ε 0 f f f E( ε0 C, E( ε C, E( εε 0 ρc C SX S f, CX, C, X X X X ρ s the populato correlato coeffcet betwee X ad R, R Wrte R y x X ( + ε X ( + ε 0 X ( ε ( ε + + 0 ( + ( + ε ε0 + ε0 + (assumg ε0 < ( + + ε ε0 ε0ε+ ε0 ( + R ε ε0 ε0ε+ ε0 (upto secod order of approxmato The bas R s gve as Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 9
( R ( R Bas E The mea squared error of ( + E ε ε0 ε0ε+ ε0 ( + f f 0 0 ρc C + C fcx + CX ρc ( R ( R MSE E R s gve as ( ε ε0 ε0ε X X E + (upto secod order of approxmato f C + C X ρ C X C To obta the value of for whch the ( R MSE s mmum, we use the prcple of maxma/mma as follows: or dmse d C ( R ρ m CX 0 C ρc C 0, say X X The secod order codto for maxma/mma s satsfed The mmum value of mea squared error s obtaed by substtutg ( R MSE as ( f C C M MSE R C + ρ C X ρ ρcxc CX CX f C ( ρ the expresso for m Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page 0
Exercse 3 (aryg probablty scheme: Suppose there are 5 uts a populato wth values 8 8 y, y, y3, y4, y5 3 3 3 The probabltes of selectg a sample of sze wth dfferet uts as p( y, y, p( y3, y4, p( y3, y5, p( y4, y5 6 6 6 Calculate the frst ad secod order cluso probabltes ad fd the probablty dstrbuto of the π - estmator of the total Suppose the sample total HT s used whch s based o Horvtz-Thompso estmator s used to estmate the square root of populato total ( tot Fd the probablty dstrbuto of HT ad fd f t s ubased for tot or ot Calculate the varace of HT Soluto: The cluso probabltes are π, π, π3, π4, π5, π, π34, π35, π45, π j 0 for all other pars of 3 3 3 6 6 6 (, j The Horvtz Thompso estmator of populato total s π + 4 wth probablty ( / ( / ( 8/3 ( 8/3 + 6 wth probablty + + ( /3 ( /3 6 6 6 Sce the sample sze s, the estmator of varace s y y j j j ππ π ar ( π π π j πj Thus for the sample y y p ( s ar π (,,, 0 Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page
ext sample y 3 y4 p ( s ar π (,,, 0 6 sample y 3 y5 p ( s ar π (,,, 0 6 sample y 4 y5 p ( s ar π (,,, 0 6 wth probablty Thus π 4 wth probablty ( E π + 4 3< 0 tot So π s a based estmator of tot ad t uderestmates t The varace of π s ( ( ar π E π E π 0 9 Samplg Theory Chapter 4 Exercses Samplg Theory Shalabh, IIT Kapur Page