Sampling Distribution Theory

Transcription

1 Poulatio ad amle: amlig Distributio Theory. A oulatio is a well-defied grou of idividuals whose characteristics are to be studied. Poulatios may be fiite or ifiite. (a) Fiite Poulatio: A oulatio is said to be fiite, if it cosists of fiite or fied umber of elemets (i.e., items, objects, measuremets or observatios). For eamle, all the uiversity studets i Pakista, the heights of all the studets erolled i Karachi Uiversity, etc. (b) Ifiite Poulatio: A oulatio is said to be ifiite, if there is o limit to the umber elemets it ca cotai. For eamle, the role of two dice, all the heights betwee ad 3 meters, etc.. A samle is a art of the whole selected with the object that it will rereset the characteristics of the whole or oulatio or uiverse. The idividuals or objects of a oulatio or a samle may be cocrete thigs like the motor cars roduced i a comay, wheat roduced i a farm, or abstract thigs like the oiio of studets about the eamiatio system. Thus all the studets i schools, colleges ad uiversities form oulatio of studets. The rocess of selectig the samle from a oulatio is called samlig. A samle may be take with relacemet or without relacemet: (a) amlig with Relacemet: If the samle is take with relacemet from a oulatio fiite or ifiite, the elemet draw is retured to the oulatio before drawig the et elemet. (b) amlig without Relacemet: If the samle is take without relacemet from a fiite oulatio, the elemet selected is ot retured to the oulatio. Probability amles ad o-probability amles:. Probability samles are those i which every elemet has a kow robability of beig icluded i the samle. Followig are the robability samlig desigs: (a) imle Radom amlig: refers to a method of selectig a samle of a give size from a give oulatio i such a way that all ossible samles of this size which could be formed from this oulatio have equal robabilities of selectio. It is a method i which a samle of is selected from the oulatio of uits such that each oe of the C distict samles has a equal chace of beig draw. This method sometimes also refers to lottery method. (b) tratified Radom amlig: cosists of the followig two stes:

2 (i) (ii) The material or area to be samled is divided ito grous or classes called strata. Items withi each stratum are homogeous. From each stratum, a simle radom samle is take ad the overall samle is obtaied by combiig the samles for all strata. (c) ystematic amlig: is aother form of samle desig i which the samles are equally saced throughout the area or oulatio to be samled. For e.g., i house-to-house samlig every 0 th or 0 th house may be take. More secifically a systematic samle is obtaied by takig every k th uit i the oulatio after the uits i oulatio have bee umbered or arraged i some way. (d) Cluster amlig: Oe of the mai difficulties i large scale surveys is the etesive area that may have to be covered i gettig a radom or stratified radom samle. It may be very eesive ad legthy task to cover the whole oulatio i order to obtai a reresetative samle. It is ot ossible to take a simle radom or systematic samle of ersos from the etire coutry or from withi strata, sice there is o such list i which all the idividuals are umbered from to. Eve if such a list eisted, it would be too eesive to base the equiry o a simle radom samle of ersos. Uder these circumstaces, it is ecoomical to select grous called clusters of elemets from the oulatio. This is called cluster samlig. The differece betwee a cluster ad a stratum is that a stratum is eected to be homogeous ad a cluster must be heterogeeous as ossible. Clusters are also kow as the rimary samlig uits. Cluster samlig may be cosisted of: (i) (ii) (iii) igle-stage Cluster amlig, ub-samlig or Two-stage amlig, ad Multi-stage amlig.. o-robability samlig desigs cosist of: (a) Judgemet or Purosive amlig: There are may situatios where ivestigators use judgemet samles to gai eeded iformatio. For eamle, it may be coveiet to select a radom samle from a cart-load of melos. The melos selected may be very large or very small. The observer may use his ow judgemet. This method is very useful whe the samle to be draw is small. (b) Quota amlig: is widely used i oiios, market surveys, etc. I such surveys, the iterviewers are simly give quotas to be filled i from differet strata, with ractically o restrictios o how they are to be filled i. Parameters ad tatistic:. A umerical value such as mea, media or stadard deviatio calculated from the oulatio is called a oulatio arameter or simly a arameter. O the

3 other had, a umerical value such as mea, media or D calculated from the samle is called a samle statistic or simly a statistic.. Parameters are fied umbers, i.e., they are costats. tatistics very from samle to samle from the same oulatio. 3. I geeral, corresodig to each oulatio arameter there will be a statistic to be comuted from the samle. 4. The urose of samlig is to gather iformatio that will be used as a basis for makig geeralisatio about the ukow oulatio arameters. 5. A arameter is usually deoted by a Greek letter ad a statistic is usually deoted by a Roma letter. For e.g., the oulatio mea is deoted by μ while the samle mea is deoted by. imilarly, the D of a oulatio is deoted by σ while the samle D is deoted by. amlig ad o-amlig Errors: (a) amlig Errors:. The samle data deals with oly a ortio of the oulatio uder cosideratio rather tha the whole oulatio. Because of this artial iformatio about the oulatio, there is always a chace of errors or discreacies to eist. This discreacy or error is simly kow as samlig error. It is also kow as samlig variatios ad chace variatios.. amlig error is reset wheever a samle is draw. Mathematically, the samlig error is defied as the differece betwee the samle statistic ad oulatio arameter. The covetioal rocedure cosists of subtractig the value of arameter, θ, from that of the statistic t; that is, the samlig error, E, is: E = t θ 3. The samlig errors are egative if the arameter is uder estimated, ad ositive if it is over-estimated. 4. The chace of samlig error ca be reduced by icreasig the size of the samle. (b) o-amlig Errors:. uch errors eter ito ay kid of ivestigatio whether it is a samle or a comlete cesus.. o-samlig errors arise from the followig reasos: Faulty iterviews ad questioaires, Icomlete ad iaccurate resoses, Mistakes i recordig or codig the data, Errors made i rocessig the results, etc. 3. These errors ca be cotrolled if the volume of data rocessed is small. 4. o-samlig errors are less sigificat i a samle.

4 Bias:. It is refer to the overall or log-ru tedecy of the samle results to differ from the arameter i the articular way.. Bias should be ot be cofused with samlig errors. Mathematically, it is defied as below: B = m μ Where μ is the true oulatio value ad m is the mea of the samle statistics of a ifiity of samles. 3. The bias may be ositive or egative accordig to as m is greater or less tha μ. Precisio ad Accuracy:. Accuracy refers to the size of deviatios from the true mea μ, whereas, the recisio refers to the size of deviatio from the overall mea m obtaied by reeated alicatio of the samlig rocedure.. Precisio is a measure of the closeess of the samle estimates to the cesus cout take uder idetical coditios ad is judged i samlig theory by the variace of the estimates cocered. amlig Distributio:. The value of a statistic varies from oe samle to aother eve if the samles are selected from the same oulatio. Thus, statistic is a radom variable.. The distributio or robability distributio of a statistic is called a samlig distributio. For e.g., the distributio of samle mea is a samlig distributio of mea ad the distributio of the samle roortio is a samlig distributio of roortio. The D of the samlig distributio of a statistic is called the stadard error of the statistic. amlig Distributio of Mea: From a fiite oulatio of uits with mea μ ad D σ, draw all ossible radom samles of size. Fid the mea of every samle. tatistic is ow a radom variable. Form a robability distributio of, kow as samlig distributio of mea. The samlig distributio of mea is oe of the most fudametal cocets of statistical iferece ad it has the followig roerties:. The mea of the samlig distributio of mea is equal to the oulatio mea: or E( )

5 . If the samlig is doe without relacemet from a fiite oulatio, the stadard error of mea is give by: Where is Fiite Poulatio Correctio (f..c.) is samlig fractio 3. Whe f..c. aroaches oe, the stadard error of mea is simlified as: with relacemet fiite The f..c. aroaches oe i each of the followig cases: (i) (ii) (iii) whe the oulatio is ifiite, whe samlig fractio is less tha 0.05, ad whe the samlig is with relacemet. Wheever, the samlig is with relacemet, the oulatio is cosidered ifiite. For e.g., a bo cotais 5 balls, whe a samle is draw with relacemet, the samle size ca be eteded from = to = 00 or whatever size is desired. Hece, the oulatio is cosidered to be ifiite. Mea ad tadard Deviatio of amlig Distributio: Like other distributio, the samlig distributio of has a mea ad stadard deviatio: f Mea of samlig distributio The stadard deviatio of samlig distributio of is kow as stadard error ( ). The stadard error of mea is always less tha the D of oulatio, i.e., σ. It deeds o the size of the samle draw. If the samle size icreases, the stadard error of mea decreases ad cosequetly the value of samle mea will be closer to the value of oulatio mea. f D of samlig distributio

6 or alteratively f ( ) D of samlig distributio o. of Possible amles: The umber of ossible samles ca be calculated as below: (i) (ii) Whe samlig is doe without relacemet, all ossible samles = C Whe samlig is doe with relacemet, all ossible samles = Eamle: A oulatio cosists of followig data:,, 3, 4 uose that a samle of size is draw with relacemet. You are required to calculate the followig: (a) Poulatio mea, (b) Poulatio stadard deviatio, (c) Mea of each samle, (d) amlig distributio table of samle mea with relacemet, ad (e) Mea ad stadard deviatio of samlig distributio. olutio: = 4 = o. of samles (whe samlig is with relacemet) = = 4 = 6 (a) Poulatio Mea (μ): (b) Poulatio tadard Deviatio (σ):

7 (c) Mea ( ) of Each amle: amles (with relacemet): (,) (,) (3,) (4,) (,) (,) (3,) (4,) (,3) (,3) (3,3) (4,3) (,4) (,4) (3,4) (4,4) Mea ( ): (d) amlig Distributio: amlig Distributio of amle Mea ( ) with Relacemet Frequecy Distributio of Probability Distributio of Tally Marks f = f Total 6 (e) Mea ad stadard deviatio of samlig distributio: f f () f ( ) f ( ) Total f ().5

8 or alteratively f ( ) f ( ) (.5) Eamle: Take the data of revious eamle ad assume samlig without relacemet, ad comute: (a) Poulatio mea, (b) Poulatio stadard deviatio, (c) Mea of each samle, (d) amlig distributio table of samle mea w/o relacemet, ad (e) Mea ad stadard deviatio of samlig distributio. olutio: (a) ad (b) Poulatio mea ad D: As calculated above (c) Mea of each samle: o. of ossible samles = C = 4 C = 6 samles amles (without relacemet): (,) (,3) (,4) (,3) (,4) (3,4) Mea: (d) amlig Distributio: amlig Distributio of amle Mea ( ) without relacemet f( ) f f ( ) f ( ).5 / / / / / Total (e) Mea ad D of amlig Distributio: f ().5

9 or alteratively f ( ) f ( ) (.5) amlig Distributio of the Differeces of Meas:. uose we have two ifiite oulatios I ad II with meas μ ad μ, ad D σ ad σ resectively.. is the samle mea of from oulatio I ad of from oulatio II with Ds ad resectively. 3. From the two fiite oulatios, we ca obtai a distributio of differeces of meas. is called amlig Distributio of Differeces of the Meas : Var f Var Var f f Provided that ad = 0.05 The distributio of is ormal if: (i) the samles are draw from ormal (or ymmetrical) oulatios, or (ii) ad both are at least 30. The distributio of z will be stadard ormal: z

10 Eamle: Poulatio I = {,, 3, 4} Poulatio II = {3,4,5} amles draw from each oulatio with relacemet: = = Comute meas of each samles, ossible differeces betwee ad, samlig distributio of, ad mea ad D of samlig distributio of. olutio: o. of ossible samles from Poulatio I = = 4 = 6 samles amles I:,,,3,4,,,3,4 3, 3, 3,3 3,4 4, 4, 4,3 4,4 : o. of ossible samles from Poulatio II = = 3 = 9 samles amles II: 3,3 3,4 3,5 4,3 4,4 4,5 5,3 5,4 5,5 :

11 Differeces of Ideedet amle Meas ) ( amlig Distributio of with Relacemet Tally Marks f f f f Total f ( f.5)

12 hae of the amlig Distributio of : The Cetral Limit Theorem describes the shae of the samlig distributio of mea. The theorem states that the samlig distributio of mea is ormal distributio either if the oulatio is ormal or if the samle size is more tha 30. Cetral limit theorem also secifies the relatioshi betwee μ ad relatioshi betwee σ ad. ad the If the samlig distributio of mea is ormal, we would eect 68.7%, 95.45% ad 99.73% of the samle meas to lie withi the itervals, ad 3 resectively. amlig Distributio of Proortio:. The samlig distributio of roortio is defied as: Where is the umber of successes (values with a secified characteristic) i a samle of size.. If the samlig rocedure is simle radom, with relacemet, is recogised as Biomial Radom Variable with arameters ad π, π is the robability of success. π ca also be iterreted as the oulatio roortio, sice: P(success) E ( ) V ( ) 3. To determie the mea ad variace of : o.of successesi the oulatio o.of items i the oulatio Ifiite Poulatio with Relacemet: P P or alteratively P

13 Fiite Poulatio without Relacemet: Eamle: A coordiatio team cosists of seve members. The educatio of each member as follows: (G = Graduate, PG = Post Graduate) Members Educatio G PG PG PG PG G G (i) (ii) (iii) Determie the roortio of ost-graduates i the oulatio. elect all ossible samles of two members from the oulatio without relacemet, ad comute the roortio of ost-graduate members i each samle. Comute the mea (μ ) ad the D (σ ) of the samle roortio comuted i (ii). olutio: (i) Proortio of PG i the oulatio: = 7 o. of PG = 4 π = 4/7 = 0.57 (ii) o. of ossible samles (without relacemet) = C = 7 C = samles.,,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 The corresodig samlig roortios are:

14 amlig Distributio of Proortio Tally Marks f P() 0 3 3/ = /7 = / = 4/7 = / = /7 = 0.86 Total P().P() P.P() Total (iii) Mea ( ) ad D ( ) of samle roortio distributio: P or alteratively P P (0.575 ) The results are verified as below: ( 0.57) hae of the amlig Distributio of Proortio : The cetral limit theorem also holds for the radom variable, which states that: (i) (ii) The samlig distributio of roortio aroaches a ormal distributio with mea ad D (with relacemet) If the radom samlig is without relacemet ad the samlig fractio 0.05, the f..c. must be used as below i the formula of D: (iii) Whe 50 ad both.π ad ( π) are greater tha 5, the samlig distributio ca be cosidered ormal.

15 (iv) Whe the distributio of is ormal, the followig statistic will be stadard ormal variable: z z o o o or amlig Distributio of Differece betwee Two Proortios:. If two radom samles of size ad are draw ideedetly from two oulatios with roortios π ad π the samlig distributio of ( ) the differece betwee two samle roortios, aroaches ormal distributio with: ) ( ) ( ) ( ) ( ad, as ad icrease. Moreover: z will be stadard ormal variable.. For ukow π ad π, samle estimates ad are used thus: 3. Whe the two ukow oulatio roortios ca be assumed equal, a estimated ˆ is obtaied as below: ˆ ad the estimated stadard error as below: ˆ ˆ

16 amlig Distributio of t:. If a radom samle of size is draw from a kow ormal Poulatio with mea μ ad D σ, the samlig distributio of the samle mea is a ormal distributio with mea ad stadard error be a stadard ormal variable:, ad hece z would. But whe the oulatio is ukow with ukow D σ, the value of σ is relaced the samle D, as give below: Therefore, the stadard error is equal to : 3. Accordig to W.. Gossett, the followig statistics is deoted by t istead of z, which follows aother distributio kow as studets t-distributio or simly t-distributio. 4. The samle stadard deviatio is give by: I the above equatio the ( ) is called Degree of Freedom or simly d.f., through which we ca obtai t-value from t-table. 5. The t-distributio aroaches stadard ormal distributio as icreases. Tyically whe > 30, the t-distributio is cosidered aroimately stadard ormal.

17 Proerties of t-distributio:. The t-distributio, like the stadard ormal, is bell shaed, uimodal ad symmetrical about the mea,. There is a differet t-distributio for every ossible samle size, 3. The eact shae of t-distributio, deeds o the arameter, the umber of degrees of freedom, deoted by ν. 4. As the samle size icreases, the shae of t-distributio becomes aroimately equal to the stadard ormal distributio: z-distributio t-distributio ( = 8) t-distributio ( = 5) 5. The mea ad stadard error of t-distributio are: t 0 t for amlig Distributio of Variaces: Poulatio Variace: or alteratively Mea of samlig distributio of ( ): f f

18 Eamle: A oulatio cosists of the followig umbers:,3,5,7. Fid the oulatio variace (σ ) ad the mea of samlig distributio of variaces ( ), if all samles are draw with relacemet of size from the oulatio. olutio: o. of ossible samles (with relacemet) = = 4 = 6 samles amles: Meas of samles: Variaces of samles:,,3,5,7 3, 3,3 3,5 3,7 5, 5,3 5,5 5,7 7, 7,3 7,5 7, amlig Distributio of : Tally Marks f f Total 6 40 f f s

19 Pooled Estimate of Variace:. If radom samles of size ad are draw ideedetly from two ormal oulatios with meas μ ad μ ad variaces σ ad σ, the samlig distributio of the differece betwee the samle meas follows a ormal distributio with mea ad stadard error give as below: ad Thus, the π will be equal to: z ad it will be a stadard ormal variable.. But if σ ad σ are ukow ad equal, their estimators ad are defied as: ad Whe the σ ad σ are relaced by the estimators ad the distributio of ca be stadardised rovided that the samles are large ( ad > 30). 3. But whe samles are small, i.e., less tha 30 ( ad 30), σ ad σ are relaced by a sigle estimator kow as ooled variace deoted by : Weighted Average of ad : Where ( + ) is the degree of freedom.

20 4. With same size of samles ad, the estimator is the simle average of ad : for 5. The ooled variace assumes that the oulatio variace is ukow ad equal. However, the same is used to relace σ ad σ for slightly uequal oulatio variaces rovided that the samles are of equal size, i.e., =. 6. I both of the above situatios, i.e., equal oulatio variace ad slightly uequal oulatio variace with equal samles (i.e., = ), the statistic t is calculated as below: t Where is ooled D. 7. ow cosider the situatio where σ ad σ are cosiderably differet (both ukow) ad it is imossible to draw samles of equal size, the statistics used i this case would be: t Where the degree of freedom ν is as follows: