GOALS The Samples Why Sample the Population? What is a Probability Sample? Four Most Commonly Used Probability Sampling Methods

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2 GOLS. Epla why a sample s the oly feasble way to lear about a populato.. Descrbe methods to select a sample. 3. Defe ad costruct a samplg dstrbuto of the sample mea. 4. Epla the cetral lmt theorem. 5. Use the cetral lmt theorem to fd probabltes of selectg possble sample meas from a specfed populato. The Samples Why Sample the Populato?. To cotact the whole populato would be tme cosumg.. The cost of studyg all the tems a populato may be prohbtve. 3. The physcal mpossblty of checg all tems the populato. 4. The destructve ature of some tests. 5. The sample results are adequate. What s a Probablty Sample? probablty sample s a sample selected such that each tem or perso the populato beg studed has a ow lelhood of beg cluded the sample. Four Most Commoly Used Probablty Samplg Methods. Smple Radom Sample. Systematc Radom Samplg 3. Stratfed Radom Samplg 4. Cluster Samplg

3 Types of Samples 3

4 ()Smple Radom Sample (SRS): Smple radom sample: sample selected so that each tem or perso the populato has the same chace of beg cluded. Ths ca oly be doe by usg Radom umbers Table, or ay other radomzato devce, e.g. computers, lottery etc. I case that the populato s homogeeous, the best samplg method to follow s the smple radom samplg. SRS s ot a good desg f the populato uder study s very large or heterogeeous. How we used the Radom umbers Table? Loo at the et Eample: We wat selected SRS, f the Populato sze () = 75 & a sample sze () = 5. Step (): ssg umber for each tem the frame from ( to ) or ( to 75). Step (): selected the startg pot the umbers table. Suppose we startg at the tersecto of (row 3 & colum ) ad move horzotally ad oly readg two umbers accordg (go horzotally or vertcally) wthout repetto. Step (3): We selected the tems umber: 7, 57, 8, 37,, 75, 65, 7, 63,,3, 30,9,66,46 ote: We dsregarded ay umber greater tha Part of a table of radom umbers If we movg vertcally the tems a sample umber: 7, 57,8, 6,, 37, 3,6, 40, 47, 46,, 5, 44,

5 ()Systematc Radom Sample Systematc Radom Sample: radom startg pot s selected, ad the every th member of the populato s selected. Here the steps you eed to follow order to acheve a systematc radom sample are: - umber the uts the populato from ( to ) - Determe the sample sze (). 3-Calculate the terval sze, K = /. 4- Radomly select a teger betwee to. 5- The tae every th ut from other tervals. Eample: Selecto of Systematc Sample The frst ut, from the Table s umber 4, the the uts of the sample wll be: 4, 9, 4, 9, 4, Systematc Sample has the advatage to SRS of good coverage of the populato However, t has the same dsadvatages. (3)Stratfed Sample Stratfed Radom Sample: populato s dvded to subgroups, called strata, ad a sample s radomly selected from each stratum. Stratfed radom samplg s used whe we have heterogeeous populato regardg some characterstcs. The populato s sub-dvded to Strata. Each Stratum s homogeeous terally. 5

6 sub-sample s draw from Each Stratum ad the totalty of the sub-samples costtutes the stratfed sample sze. How draw? How may uts from each stratum? The sub-sample ca be draw usg: SRS or Systematc RS. The sze of the sub-sample from each stratum ca be determed through : Proportoal llocato: Sub-sample = Sze of stratum * sample sze Sze of populato = ( h / ) * The sub-sample s tae Proportoal to the sze of the stratum. (Larger strata wll have larger sub-sample). Eample: We wat selected radom sample form the followg table Sample sze = 40 Stratum (Studets) Freshma Sophomore Juor Seor Sze h The Sub-Samples: (Proportoal llocato): = (00/00) *40 = 40 = (600/00) *40 = 0 3 = (00/00) *40 = 0 4 = (300/00) *40 = 60 = = 40 (4) Mult- Stage or Cluster Sample: Cluster sample: populato s dvded to clusters usg aturally occurrg geographc or other boudares. The, clusters are radomly selected ad a sample s collected by radomly selectg from each cluster. The sample s selected to stages; each stage s composed of two steps: - Sub-dvso to smaller Clusters. - Selecto of some clusters. Ths goes o stages tll the ultmate ut whch s dvsble, e.g. household or a dvdual. Eample study umber of traffc accdets doe to all Saud raba: Stage Oe: - Subdvde Saud raba to ctes, (5). - Select some ctes, (say 0). Stage Two: - Sub-dvde each selected cty to streets. - Select some streets from each selected cty. 6

7 7 Revew Eample () : Compute the mea, Varace, Stadard devato of the followg populato values: 6, 3, 5, 4, ad by usg two deferet formulas Soluto: Total Summary Sample Populato The mea S Varace S S Stadard devato

8 Samplg dstrbuto: Defto Let ˆ (read "theta hat") be ay sample statstc; for eample ˆ could be the sample mea ; the sample proporto P ; the sample varace S ad so o. samplg dstrbuto of ˆ s a probablty dstrbuto obtaed by Lstg all possble values that ˆ could have from all possble samples of Sze ad the correspodg probabltes of occurrece How to draw sample from populato umber of samples Wth replacemet Wthout replacemet umber of samples The order of selected objects s ot mportat The order of selected objects s mportat Combato= Permutato= umberof samples umberof samples C P 8

9 Case () the samplg wth replacemet: K umber of samples Case () the samplg wthout replacemet: (-) the order of selected objects s mportat! umberof samples K P ( )( )( 3)...( (-)The order of selected objects s ot mportat! umberof samples K C.!! ( )( )( 3)..!! ( )... ). 9

10 Samplg dstrbuto of the sample mea: probablty dstrbuto of all possble sample meas of a gve sample sze. We study three cases: Case (): The samplg dstrbuto wth replacemet: Eample () Refer to eample () lst all possble samples of eecutves from the populato ad compute ther meas (wth replacemet).fd:. How may dfferet samples of are possble?. The probablty of selectg ay of the possble smple radom samples. 3. The mea of (the mea of samplg dstrbuto of the sample meas), the varace of (the varace of samplg dstrbuto of the sample meas), the stadard devato of (the stadard devato of samplg dstrbuto of the sample meas). Soluto: Wth replacemet. umber of samples= = 5 5. The probablty of selectg ay of the possble smple radom samples P 5 The tems: 6, 3, 5, 4, 0

11 The possble samples: Sample umber() Samples 6,6 6 6, , , , 4 6 3, , , , ,.5 5, , , , , , , , , , 3,6 4, 3.5 3, ,4 3 5, = 5 4 The mea of K 5, I ths case the statstc s a ubased estmator of the parameter.

12 The Varace of Sample umber () Samples 6, , , , , , , , , , , , , , = 5, , , , , , 3 9,6 4 6, , , , The Stadard devato of =

13 Cocluso: For case of samplg wth replacemet, the sample mea s a radom varable wth mea ad varace gve by: Case (): The samplg dstrbuto wthout replacemet: Case (-): The order of selected objects s mportat. Case (-): The order of selected objects s ot mportat. Eample (3) Refer to eample () lst all possble samples of eecutves from the populato ad compute ther meas wthout replacemet f: - The order of selected objects s mportat. - The order of selected objects s ot mportat Fd:. How may dfferet samples of are possble?. The probablty of selectg ay of the possble smple radom samples. 3. The mea of (the mea of samplg dstrbuto of the sample meas), the varace of (the varace of samplg dstrbuto of the sample meas), the stadard devato of (the stadard devato of samplg dstrbuto of the sample meas). Soluto: Case (-): Wthout replacemet ad the order of selected objects s mportat: - umber of samples! K P ( )( )( 3)...( ).! P 5! 5! 0 5!! 5 3. The probablty of selectg ay of the possble smple radom samples P 0 P 3

14 . The possble samples Sample umber() 3 4 Samples 6,3 6,5 6,4 6, Sample umber() 3 4 Samples 3,6 5,6 4,6,6 5 3,5 5 5,3 6 3,4 6 4,3 7 3, 7,3 8 5,4 8 4,5 9 5, 9,5 0 4, 0,4 3. The mea & varace Sample umber() 3 Samples 6,3 6,5 6, Sample umber() Samples 0.5 3, , , , 4 6 4, , , , , , , , , , , , 3 9 0, = The mea of 0, I ths case the statstc s a ubased estmator of the parameter The Varace of = The Stadard devato of =

15 Case (-): Wthout replacemet ad the order of selected objects s ot mportat: - The umber of samples! K C.!! ( )( )( 3).. ( )...! C! 5!! 5. The probablty of selectg ay of the possble smple radom samples P C 0 The possble samples Sample 3 umber() Samples 6,3 6,5 6,4 3. The mea & varace Sample 3 umber() Samples 6,3 6,5 6, , 4 6, ,5 5 3, ,4 6 3, , 7 3, ,4 8 5, , 9 5, , 0 4, = 0 4 The mea of 0, I ths case the statstc s a ubased estmator of the parameter The Varace of = The Stadard devato of =

16 Cocluso: For case of samplg wthout replacemet, the sample mea s a radom varable wth mea ad varace gve by: ad Summary Populato Sample Samplg dstrbuto The mea Varace Stadard devato umber of samples K S S - S K Wth replacemet Wthout replacemet Wth replacemet Wthout replacemet Wth replacemet Wthout replacemet Order mportat P Order ot C mportat 6

17 Eample (4): () ssume havg populato of 6 uts, a radom sample of sze s draw from ths populato wth replacemet. The followg formato s : 34 ad Fd: -The umber of possble samples draw from ths populato equal -The mea of the samplg dstrbuto of the sample mea -The varace of the samplg dstrbuto of the sample mea Soluto: 6 34 K 36, 6. 5, K K 36 () ssume havg populato of 6 uts, a radom sample of sze s draw from ths populato wthout replacemet ad the order s mportat. The followg formato s : 95 ad Fd: -The umber of possble samples draw from ths populato equal -The mea of the samplg dstrbuto of the sample mea -The varace of the samplg dstrbuto of the sample mea Soluto: 6 K P P 30, K 30 5, K 30 (C) ssume havg populato of 6 uts, a radom sample of sze s draw from ths populato wthout replacemet ad the order s ot mportat. The followg formato s: 97.5 ad Fd: -The umber of possble samples draw from ths populato equal -The mea of the samplg dstrbuto of the sample mea -The varace of the samplg dstrbuto of the sample mea 7

18 Soluto: 6 K C C 5, K 5 5, K Eample (5) Radom sample of sze were selected from populatos wth the meas ad varace gve here. Fd the umber of samples, the mea ad stadard devato of the samplg dstrbuto of the sample mea each case: a wth replacemet b wthout replacemet ad theorder s mpor ta c wthout replacemet Soluto: a. The umber of samples or b. The umber of samples P P or c. 876 The umber of samples C 8C ad theorder s ot mpor ta t t 8

19 or Eample (6) Radom sample of sze 3 were selected from populatos. Fd the umber of samples: a. Wth replacemet. b. Wthout replacemet ad the order s mportat. c. Wthout replacemet ad the order s ot mportat. Soluto: 3 a. The umber of samples= b. The umber of samples= P3 c. The umber of samples= C3! 3 6 Eample (7) Suppose that a radom sample of sze 8 from a o-ormal dstrbuto wth mea 50 ad varace 36(wth replacemet) Fd : a. The mea of samplg dstrbuto of the sample mea. b. The stadard error of the samplg dstrbuto of the sample mea. Soluto: a b Eample (8) If 6, Fd a. umber of samples ad the probablty of selectg ay of the possble smple radom samples case of drawg wth replacemet. b. umber of samples ad the probablty of selectg ay of the possble smple radom samples case of drawg wthout replacemet, order are mportat. 9

20 c. umber of samples ad the probablty of selectg ay of the possble smple radom samples case of drawg wthout replacemet, order are ot mportat. Soluto a. P b. P c. P C P ! ( ) 30!! ( )! ( ) P ( ) 6C! ! 6 5 5!!! 6 ote:. The mea of the sample meas s eactly equal to the populato mea : (e. &3) 4. The dsperso (varato, spread) of the samplg dstrbuto of sample meas s less tha the populato dstrbuto.(wth out replacemet) 3. The samplg dstrbuto of sample meas teds to become bell-shape ad to appromate the ormal probablty dstrbuto. (e.3) 0

21 s the "stadard devato of the samplg dstrbuto of the sample mea"; ad always refer to t as the "stadard error of the mea". 4- The factor ; s called the fte populato correcto factor ad ca be gored f : 0.05 (.e. do ot use the fte populato correcto uless the sample s more tha 5 percet of the sze of the populato).

22 Theorem: If the populato follows a ormal probablty dstrbuto, the for ay sample sze the samplg dstrbuto of the sample mea wll be also ormal. Suppose that ~, sze the, ; ~ ; ad let be the mea of a sample of where; case of case of replacmet wthout replacmet Theorem (The Cetral lmt Theorem): If all sample of a partcular sze are selected from ay populato, the samplg dstrbuto of the sample mea s appromately a ormal dstrbuto. Ths appromato mproves wth larger sample ( 30 ). For a populato wth a mea ad a varace the samplg dstrbuto of the meas of all possble samples of sze geerated from the populato wll be appromately ormally dstrbuted wth mea ad stadard devato / as the sample sze becomes larger ( 30 ). Facts: () The epected value of the sample mea equal the populato mea: E Where s the mea of the Samplg dstrbuto of meas. d s the mea of the populato. () If a populato s fte or f samplg s wth replacemet, the the varace of the Samplg dstrbuto of meas s:.

23 (3) If a populato s of sze, f samplg s wthout replacemet, the the varace of the Samplg dstrbuto of meas s:. (4) Samplg dstrbuto of sample meas wll be eactly ormal f the populato s ormally dstrbuted. (5) If a populato from whch samples are tae s ormally dstrbuted wth mea ad varace the the sample mea s ormally dstrbuted wth mea ad varace. (If the populato dstrbuto s ormal, Samplg dstrbuto of the wll be eactly ormal ~ (, ) ). (6) Suppose that s a radom varable havg some dstrbuto fucto wth mea ad varace. The the samplg dstrbuto of the mea approaches a ormal dstrbuto wth mea ad stadard devato / as the sample sze becomes larger, rrespectve of the shape of the orgal dstrbuto (Cetral Lmt Theorem). (If the populato dstrbuto s o-ormal, Samplg dstrbuto of the wll be appromately ormal by Cetral lmt theorem ~ (, ) ). (7) s the sample szes creases, the varablty of each samplg dstrbuto decreases so that they become creasgly more leptourtc Eample (9) Suppose that the radom varable represet the IQ (Itellgece quotet) score for studets at certa uversty, ad that ~ 00, 75. radom sample of sze 5 studets s selected. Fd the followg:. Probablty that the mea IQ score computed from the sample wll be grater tha 5.. Probablty that the mea IQ score computed from the sample wll be less tha Probablty that the mea IQ score computed from the sample wll be betwee 70 ad 30. 3

24 Soluto:. ote that ~ 00,5 00, P P Z P 80 P Z P P Z

25 (ormal populatos) Settg : (Dr. ma l Saleh: Suppose that there are two populatos ad ; ad that s a radom varable defed o the frst populato ad that s aother radom varable defed o the secod populato. radom sample of sze s selected from populato ; let ths radom sample be deoted by: The sample mea of ths sample s deoted by : Smlarly, aother depedet radom sample of sze s selected from populato ; let ths radom sample be deoted by: The sample mea of ths sample s deoted by: We request the samplg dstrbuto of the radom Varable The samplg dstrbuto of the dfferece betwee two depedet sample meas ca be obtaed a smlar way as the samplg dstrbuto of oe sample mea dscussed before. Characterstcs of the samplg dstrbuto of the dfferece betwee two sample meas : 5

26 ~, If ;, Hece, ~ ~, The Z ~ 0, Eample (0) ssume there are two types of plat. The mea heght of type s 3 whle the mea heght of type s. The varaces of the two types are 60 ad 70 respectvely ad the heghts of both types are ormally dstrbuted. Two samples radomly selected from the two populatos, from the populato of type, 0 plats are selected ad from the populato of type, 4 plats are selected. What s the probablty that the mea of the 0 plats of type wll eceed the mea of the 4 plats of type by 5 or more? Soluto: From the above formato we fd that: d; d hece:

27 ~ The we request the followg probablty: P 5 PZ P 5 0 0, Z The samplg dstrbuto s show the followg fgure; the area above 5 s shaded. Eample () The electrc lght bulbs of maufacturer have a mea lfetme of 400 hours wth a stadard devato of 00 hours, whle those of maufacturer have a mea lfetme of 00 hours wth a stadard devato of 00 hours. If radom sample of 5 bulbs of each brad are tested (If the populato dstrbuto s ormal), Fd:. The probablty that the brad bulbs wll have a mea lfetme whch s at most 350 hours.. The probablty that the brad bulbs wll have a mea lfetme whch s at most0 hours. 3. The probablty that the brad bulbs wll have a mea lfetme whch s at least 60 hours more tha the brad bulbs. 4. The probablty that the brad bulbs wll have a mea lfetme whch s at least 50 hours more tha the brad bulbs Soluto:. The probablty that the brad bulbs wll have a mea lfetme whch s at most 350 hours , hr

28 ~ 400, P 350 P Z P Z P Z P Z PZ The probablty that the brad bulbs wll have a mea lfetme whch s at most0 hours. 0000, hr 00 hr 5 ~ 00, P 0 P Z P Z P Z P Z PZ The probablty that the brad bulbs wll have a mea lfetme whch s at least 60 hours more tha the brad bulbs hr ~ 00, P 60 P Z PZ PZ 0 0 P Z The probablty that the brad bulbs wll have a mea lfetme whch s at least 50 hours more tha the brad bulbs hr ~ 00,

29 P 50 P Z PZ PZ 0 0 P Z

30 (Large samples) If a radom sample of observatos s selected form a bomal populato wth parameter p, the the samplg dstrbuto of the sample proporto (P = /) Wll have a mea ad a varace Whe the sample sze s large, the samplg dstrbuto of p ca be appromated by a ormal dstrbuto. The appromato wll be adequate f 00, 0.05 < < 0.95, ( 5 ) If ~ (, ), 5 ad large, where, ~ P ~ Where,, where, p themea of Var( ) p p samllg dstrbuto of proporto E the var aceof samllg dstrbuto of proporto Var( ) ( ) p ( ) ( ) p p Var P E E Eample () I a survey, 500 parets the US were ased about the mportat of sports for boys ad grls.of the parets tervewed, 300 agreed that the geders are equal ad should have equal opportutes to partcpate sports. Fd:. The sample proporto of parets who agree that the geders are equal ad should have equal opportutes.. The true proporto the populato s equal to some uow value that call,estmate the true proporto P 30

31 Soluto: 300. P p p p P ~ , The mea of the sample dstrbuto of proporto of parets who agree that the geders are equal ad should have equal opportutes. 4. The varace of the sample dstrbuto of proporto, case of samplg wth replacemet, Eample (3) Suppose that ~ I00.0.3). Usg the ormal appromato for the bomal probabltes fd the followg probablty:. P P 0.5,. P P 0.5 Soluto: p 0.3, p 0.00, p 00 P ~ 0.3, PP 0.5 P Z P Z P Z PP 0.5 P Z P Z P Z

32 Eample(4) The "cademc Ceter of Huma Studes" has a small medcal ceter whch has lmted facltes ad serous cases the patet should trasfer to the earest hosptal for treatmet, however ths s cosder as a rare case. Ffty-four percet of all studets are favor of mprovg the medcal ceter. radom sample of 000 studets s selected, ad ased f they favor of mprovg the medcal ceter. Fd the followg:. The probablty that the percetage of studets favor of mprovg the medcal ceter s less tha 50%. Soluto: For the populato of all studets, the proporto favor of mprovg the medcal ceter ; s equal to For a radom sample of sze 000, we ca verfy that both ad ( ) are greater tha 5. The, the sample proporto of studets favor of mprovg the medcal ceter follows appromately a ormal dstrbuto wth mea ad varace gve by: 0.54 ; P P hece; ~ P 0.54, Frst: Fdg the probablty that the percetage of studets favor of mprovg the medcal ceter s less tha 50%, usg the correcto for cotuty, ths probablty s computed as follows: P ( P 0.5) P Z P Z Remar :. The samplg dstrbuto of P s completely descrbed by both ad.. I case of samplg wthout replacemet, the varace of P becomes: P 3. The samplg dstrbuto of P s appromately ormally dstrbuted f s farly large( 00 ) ad ot close to 0 or (0.05 < < 0.95). rule of thumb s that the appromato s good f both ad ( ) are greater tha or equal 5 4. s sample sze a creases, the stadard error of P s decreases.

33 (Large ad depedet samples) Theorem: Gve that P ad P are sample proportos from two depedet ( ) ( ) P P ~,, , 5 Provded that 00 ( ) 5 for,. P P Eample (5) ssume that 0.8 of uversty graduates are able to pass ICDL test, whle oly 0.5 of secodary school graduates are able to pass the test. Suppose that 0 persos are sampled from the populato of uversty graduates ad that 5 persos are sampled from the populato of secodary school graduates, ad the ICDL test s held to all of them. What s the probablty that P P 0.? Soluto: fd that: P P d P P p p ( ) ( ) (0.8)(0.) 0 ad (0.5) (0.5) 5 The; P P P 0. P Z P Z Z