SydU STAT3014 (2015) Second semester Dr. J. Chan 1

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1 Refereces STAT3014/3914 Appied Statistics-Sampig Preimiary 1. Cochra, W.G. 1963) Sampig Techiques, Wiey, ew York.. Kish, L. 1995) Survey Sampig, Wiey Iter. Sciece. 3. Lohr, S.L. 1999) Sampig: Desig ad Aaysis, Duxbury Press. 4. McLea, W. 1999) A Itroductio to Sampe Surveys, A.B.S. Pubicatios, Caberra. Sectio outie 1. Simpe radom sampes ad stratificatio. Fiite popuatio correctio factor. Sampe size determiatio. Iferece over subpopuatios.. Stratified sampig. Optima aocatio. 3. Ratio ad regressio estimators. Ratio estimators. Hartey-Ross estimator. Ratio estimator for stratified sampes. Regressio estimator. 4. Systematic sampig ad custer sampig. 5. Sampig with uequa probabiities. Probabiity proportioa to sizepps) sampig. The Horvitz-Thompso estimator. SydU STAT ) Secod semester Dr. J. Cha 1

2 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1 Simpe Radom Sampes SRS) 1.1 The Popuatio We have a fiite umber of eemets, where is assumed kow. The popuatio is Y 1... Y, where Y i is a umerica vaue associated with i-th eemet. We adopt the otatio where capita etters refer to characteristics of the popuatio; sma etters are used for the correspodig characteristics of a sampe. Popuatio Tota: Y = Y i, Popuatio Mea: µ = Ȳ = Y = 1 Y i, Popuatio Variace : σ = 1 Y i Ȳ ) ad S = 1 σ = 1 1 Y i Ȳ ). These are fixed popuatio) quatities, to be estimated. If we have to cosider two umerica vaues: Y i, X i ), i = 1,, a additioa popuatio quatity of iterest is R = Y i X i = Y X = Ȳ X the ratio of totas. SydU STAT ) Secod semester Dr. J. Cha

3 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1. Simpe Radom Sampig Focus o the umerica vaues Y i, i = 1,,. A radom sampe of size is take without repacemet : the observed vaues y 1,, y are radom variabes ad are stochasticay depedet. The sampig frame is a ist of the vaues Y i, i = 1,,. The atura estimator for Ȳ is Ȳ = 1 y i = ȳ ad hece for Y = µ is Ŷ = ȳ. Distributioa properties of ȳ are compicated by the depedece of the y i s. Sampe variace: s = 1 1 y i ȳ) Fudameta Resuts Eȳ) = µ. Var ȳ) = 1 )S = 1 f)s = ) σ 1 var ȳ) = 1 )s Es ) = S where f is the sampig fractio ad the fiite popuatio correctio f.p.c.) is 1 f. SydU STAT ) Secod semester Dr. J. Cha 3

4 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe Proof: Let y = 1 y i = 1 idicator First, we have I i = i S y i I i where the sampe membership { 1 if eemet i is i the sampe, 0 if otherwise. EI i ) = 0 PrI i = 0) + 1 PrI i = 1) = π i =, EI i ) = 0 PrI i = 0) + 1 PrI i = 1) = π i =, EI i I j ) = 0 0 PrI i = 0&I j = 0) PrI i = 0&I j = 1) PrI i = 1&I j = 0) PrI i = 1&I j = 1) 1) = π ij = 1) VarI i ) = EI i ) E I i ) = π i 1 π i ) = 1 ), CovI i, I j ) = EI i I j ) EI i )EI j ) = π ij π i π j = The Ey) = 1 y i EI i ) = 1 [ ) ] def. s y E[vary)] = E y i = 1 P f. = 1) 1) y i = Y. ) S y ). Ubiased P f.1 = Vary) Ubiased. Proof 1: show that Vary) = ) S y. SydU STAT ) Secod semester Dr. J. Cha 4

5 Vary) = Var STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1 ) y i I i [ = 1 y i VarI i ) + ] y i y j CovI i, I j ) i j,i<j { = 1 [ y i 1 )] + [ 1) ) ] } y i y j 1) i j,i<j { = 1 1 ) y i ) } y i y j i j,i<j = 1 1 ) { } 1 yi ) 1 1) 1) y i y j 1) i j,i<j = 1 1 ) { } 1 yi + 1 y i y j 1) i j,i<j { } = 1 1 ) 1 1 yi 1 y i y j 1 i j,i<j { = 1 1 ) 1 yi 1 yi + )} y i y j 1 i j,i<j { = 1 1 ) 1 yi 1 ) )} y i y i 1 = 1 ) S y = σ ) y σ 1 = y 1 Proof : show that Es y) = S y where S y = 1 1 [ Es 1 y) = E 1 ] y i y) y i Y ) = 1 σ y. { } = 1 1 E [y i Y ) y Y )] SydU STAT ) Secod semester Dr. J. Cha 5

6 = = = = = = STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe [ ] 1 1 E y i Y ) y Y ) [ ] 1 Ey i Y ) Ey Y ) 1 [ ] 1 Vary i ) Vary) 1 [ ) ] 1 σ σy y 1 1 ) σ y = 1 1 σ y = S y ) σ y 1 Cetra Limit Property For arge sampe size > 30 say), ad sma to moderate f, we have the approximatio ȳ µ)/ Varȳ) 0, 1). Cofidece Iterva for µ = Ȳ Repacig S by s, a approximate 95% C.I. for µ ad Y are respectivey ȳ ± 1.96 s 1 f, Read Tutoria 10 Q1. ȳ ± 1.96 s 1 f) SydU STAT ) Secod semester Dr. J. Cha 6

7 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe Exampe: Idustria firm) A idustria firm is cocered about the time spet each week by staff o certai tasks. The time-og sheets of a SRS of = 50 empoyees show the average amout of time spet o these tasks is hours, with a sampe variace s =.5. The compay empoys = 750 staff. Estimate the tota umber of ma-hours used each week o the tasks ad costruct a 95% CI for the estimate. Soutio: From = 750 time-og sheets, a SRS of = 50 sheets was obtaied. The average amout of time used i the sampe is y = hours/week. Sice = 50 is arge, we take z 0.05 = Hece Ŷ = y = = hours varŷ ) = 1 ) s = ).5 = 3, seŷ ) = 3, 65 = % CI for Y = y 1.96 seŷ ), y seŷ )) = , ) = 7431., ) SydU STAT ) Secod semester Dr. J. Cha 7

8 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1.3 Simpe Radom Sampig for Attributes. The method for SRS ca be appied to estimate the tota umber, or proportio or %) of uits which possess some quaitative attribute. Let this subset of the popuatio be C. Let ad simiary for y i s. Customary otatio: µ = Y = Y i = 1 if i C = 0 if i / C Ȳ = P is the popuatio proportio, Y i = P is the popuatio tota cout, ȳ = p is the sampe proportio. Let y = y i = p, Q = 1 P ad q = 1 p. Thus Sice Y i Ȳ ) = we have ad simiary, Ŷ = p = y/. Y i Ȳ = Ȳ Ȳ = P 1 P ), S = P 1 P )/ 1) P 1 P ) s = p1 p)/ 1) p1 p) ad s = p1 p) 1) = p1 p) 1 p1 p) SydU STAT ) Secod semester Dr. J. Cha 8

9 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1.4 Sampe Size Cacuatios To cacuate the sampe size eeded for sampig yet to be carried out, we wat to be at east 1001 α)% sure the estimate ȳ of µ is withi 100δ% of the actua vaue of µ e.g. 1 α = 0.95, z α/ = 1.96). That is Pr{ ȳ µ δ µ } 1 α ) ȳ µ δ Pr µ Var ȳ) Var ȳ) 1 α δ µ / Var ȳ) z α/ 1 ) S δ µ) zα/ S S δ µ) zα/ S δ µ) zα/ + S δ µ ) z α/ S + S S δ µ ) /z α/ + S orma Stadard orma µ δ µ δ µ SEȳ) Area 1 α z α/ = 1.96 µ 0 ȳ z µ + δ µ δ µ SEȳ) Igorig f.p.c. i.e. takig f = 0 or Var ȳ) = S / whe is ukow) z α/ S z α/ s δ µ ) δȳ) where s ad ȳ are estimates from a piot survey ad s p1 p) for attributes. SydU STAT ) Secod semester Dr. J. Cha 9

10 Exampe: bood group) STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1. What size sampe must be draw from a popuatio of size = 800 i order to estimate the proportio with a give bood group to withi 0.04 i.e. a absoute error of 4%) with probabiity 0.95?. What sampe size is eeded if we kow that the bood group is preset i o more tha 30% of the popuatio? Soutio: 1. We have = 800, S ˆp1 ˆp) = 0.5 = 0.5, δ µ = 0.04 ad z 0.05 = ote that S = p1 p) is max at p = 0.5 ad this gives the most coservative. p 1 p) 0.5 Take = 343. p S δµ/z α/ + = ) S )/ = ow S = p1 p) = ) = 0.1. Take = 310. Read Tutoria 10 Q. S δµ/z α/ + = ) S )/ = SydU STAT ) Secod semester Dr. J. Cha 10

11 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe 1.5 Iferece over Subpopuatios-Poststratificatio Motivatig exampe: detist) There are 00 chidre i a viage. Oe detist takes a simpe radom sampe of 0 ad fids 1 chidre with at east oe decayed tooth ad a tota of 4 decayed teeth. Aother detist quicky checks a 00 chidre ad fids 60 with o decayed teeth. Estimate the tota umber of decayed teeth. C 1 : 1 decayed teeth C : o decayed teeth Tota 1 = 140 = 60 = 00 1 = 1 = 8 = 0 y i = 4 = y i i C 1 C 1 C 1 = 140 = 60 1 = 1 = 8 y i i C 1 = 4 From a popuatio of idividuas, oe simpe radom sampe of idividuas y i, i = 1,, is draw. Separate estimates might be wated for oe of a umber of subcasses {C 1, C, } which are subsets of the popuatio sampig frame) usig post-stratificatio. Reasos: 1. Uavaiabiity of a suitabe sampig frame for each stratum eve though the stratum sizes 1,..., L are ofte obtaiabe from officia statistics.. Iabiity to cassify popuatio eemets ito a appropriate stratum without actua cotact, e.g. persoa characteristics such as educatioa eve ad poitica preferece ad househod characteristics such as owed/reted accommodatio, icome eve ad househod size are ukow 3. Muti-variate ad muti-purpose ature of most surveys. 4. Post-stratificatio is to correct the distorted sampe proportio due to o-respose. SydU STAT ) Secod semester Dr. J. Cha 11

12 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe Exampe: POPULATIO SAMPLIG FRAME) Austraia popuatio retaiers the empoyed SUBPOPULATIO uempoyed Queesaders supermarkets the empoyed workig overtime Soutio: The estimate for the overa average o. of decay teeth usig overa sampe mea is Ȳ = ȳ = 1 y i = 4 0 =.1. i C 1 The estimate for the overa or coditioa tota umber of decayed teeth, Y is Ŷ pst,1m1 = ȳ = 00.1 = 40, igorig the iformatio of 1 = 1 ad 1 = 140 from the secod detist. Usig these iformatio ad coditio o those with at east oe decayed teeth, the average o. of decay teeth is Ȳ pst,1m1 = ȳ = 00.1) = 3 Aterative estimate for the average o. of decayed teeth usig the coditioa sampe mea is Ȳ pst,1m = ȳ 1 = ȳ 1 = 4 1 = 3.5 Hece the estimate for the tota o. of decayed teeth is Read Tutoria 10 Q3. Ŷ pst,1m = 1 ȳ 1 = ) = 490 SydU STAT ) Secod semester Dr. J. Cha 1

13 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe Formuae: Deote the tota umber of items i cass C that is geeray ukow but we ca estimate by by. ote = / where w = / is the sampe proportio of uits faig ito C ad the correspodig popuatio proportio is W = /. The same techique is used to estimate popuatio mea & tota i C : Ȳ = i C y i /, ad Y = i C y i usig The y i = ȳ = { yi if i C 0 if i / C y i / estimates the mea Ȳ Data C 1 C Mea y i : y 1 y... y 1 +1 y ȳ y i : y 1 y ȳ 1 y i : y 1 y = y i /. The ubiased estimator for the tota Y = Ȳ = variace are Ŷ pst,m1 = ȳ = Y i ȳ 1 i C ad its y i / ad varŷpst,m1) = 1 ) s 1. Whe is kow, the ubiased estimator for mea Ȳ = Y / i C ad its variace estimate are Ȳ pst,m1 = ȳ / = ȳ /W ad var Ȳ pst,m1 ) = 1 W 1 ) s SydU STAT ) Secod semester Dr. J. Cha 13

14 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe where S ca be estimated by s : S = 1 1 y i Ȳ ) ad s = 1 1 y i ȳ ). ote: the radom variabe are ot icuded i these cacuatios.. Whe is ukow, we ca estimate the tota Y by ȳ but we caot estimate Ȳ by ȳ /W. A atura way is to estimate W = by w = : Ȳ pst,m = ȳ / = ȳ ad var Ȳ pst,m ) 1 W 1 Simiary the tota estimator ad its variace estimate are ) s Ŷ pst,m = ȳ = ȳ / ad varŷpst,m) W 1 ) s where s = 1 y i ȳ ) is the sampe variace i C ad 1 i C W i = W. Theorem: For the estimator Ȳ = ȳ = / )ȳ, Eȳ ) = Ȳ Varȳ ) 1 1 ) S W provided we defie Eȳ ) = Ȳ whe = 0. SydU STAT ) Secod semester Dr. J. Cha 14

15 STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe Proof: Usig coditioa expectatio, Eȳ ) = E {Eȳ )} = Eȳ = 0) Pr = 0) + Eȳ = i) Pr = i) i 1 = Ȳ Pr = 0) + i 1 Ȳ Pr = i) = Ȳ where for each fixed, a simpe radom sampe of size is draw from C. Further usig coditioa variace, sice Varȳ ) = VarEȳ )) }{{} = E [ 1 = 1 W 1 ) S ) 1 E +EVar ȳ )) = 0 + EVarȳ )) Ȳ ) ] ) S S = E S S S W W W W W extra var. = 1 W W S 1 W where the igored term: is the extra variabiity due to the radom sampe size. ) 1 Theorem: E W W W. 1 Proof: = W + W ) 1 = ) 1 W W W [ = 1 ) ) ] W W W W W SydU STAT ) Secod semester Dr. J. Cha 15

16 Hece ) 1 E STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe = 1 W 1 W + 1 W [ 1 E W ) + E W ) W W ) W 1 W ) W sice E ) = W ad V ar ) = W 1 W ). ]... = W W W Coroary: The estimator Ŷpst,m = ȳ is ubiased for Y Var Ŷ) = Varȳ ). ad has Whe is kow, the estimator Ŷpst,m = ȳ uses more iformatio both & ) tha ȳ ad so is a better ubiased estimator. Usig this method, the overa tota ad mea estimates are: ad Ŷ pst = Ȳ pst = =1 =1 W ȳ ad var Ŷpst) 1 ) 1 W ȳ ad var Ȳ pst ) 1 ) 1 W s =1 W s =1 respectivey sice Ȳ pst = 1 Ŷ pst,m = 1 var Ȳ pst ) = =1 =1 W varȳ ) Compare Ȳ pst to Ȳ = ȳ = ȳ = =1 W ȳ =1 W 1 1 ) s 1 W ) 1 =1 w ȳ ad var Ȳ ) = 1 ) s =1 for oe SRS without post-stratificatio, we have W s =1 SydU STAT ) Secod semester Dr. J. Cha 16

17 W repaces w ad STAT3014/3914 Appied Stat.-Sampig C1-Simpe radom sampe L =1 Read Tutoria 11 Q1,c,d. W s repaces s to correct sampe proportios. Post-stratified estimator based o 1 SRS Par. Estimator Variace Ȳ kow Ȳ pst,m1 = ȳ / var Ȳ pst,m1 ) = 1 W 1 ) s Y ukow Ŷpst,m1 = ȳ varŷpst,m1) = 1 ) s Ȳ ukow Ȳ pst,m = ȳ var Ȳ pst,m ) 1 1 ) s W Y kow Ŷ pst,m = ȳ varŷpst,m) W 1 ) s Y kow Y pst = W y =1 varŷ pst) 1 ) s W =1 Y kow Ŷ pst = y varŷpst) 1 ) s W =1 =1 where y i = y i if i C & 0 otherwise, ȳ = 1 y i, ȳ = 1 y i, i C i C s = 1 1 y i ȳ ), ad s = 1 y i ȳ ). 1 i C SydU STAT ) Secod semester Dr. J. Cha 17