Z = = = = X np n. n n. npq. npq pq

Transcription

1 Stt 4, secto 4 Goodess of Ft Ctegory Probbltes Specfed otes by Tm Plchowsk Recll bck to Lectures 6c, 84 (83 the 8 th edto d 94 whe we delt wth populto proportos Vocbulry from 6c: The pot estmte for populto proporto ws smple proporto umber of successes X p ˆ totl umber tested ypothess test for sgle smple, from secto 84 (83 the 8 th edto: We dd lttle mthemtcl fglg wth the the z-score formul for boml dstrbutos to get z-score formul for proportos X p X p X pˆ Z pq pq pq pq For the populto proporto p the umertor, d for the stdrd error ( the deomtor, we used the vlue from the ull hypothess, p Lkewse, for the populto prmeter q, we used the ull vlue q p Note tht we could express the umertor s observed vlue mus expected vlue For the hypothess test volvg proportos, we used the sme rule of thumb s we dd for usg orml dstrbuto to pproxmte boml dstrbuto If both p d lso q, the the hypothess test s cosdered lrge-smple test α α α/ α/ p ( q q > p p ( q q < p We tested to verfy lrge-smple stuto, the computed the vlue of the test sttstc: p ( q q p pˆ z ypothess test for comprg two smples, from 94: We oly cosdered cses where m p, mq, p, q, tht s, lrge smple cse for whch both smplg dstrbutos, p ˆ X Y m d p ˆ, re pproxmtely orml Addtolly, we focused solely o hypothess test for whch (whch mtches prctce for the vst mjorty of ctul stutos p q > α α α/ α/ pˆ ˆ Verfy lrge-smple stuto, d compute the vlue of the test sttstc: z pq ˆ ˆ + < ( ( Note tht, oce g, we could express the umertor s observed vlue mus expected vlue m

2 I ll of the prevous ecouters wth populto proportos, we were cosderg boml expermet cosstg of sequece of depedet trls whch ech trl could result oe of two possble outcomes: S (for success d F (for flure The probblty of success, deoted by p, ws ssumed to be costt from trl to trl, d the umber of trls,, ws fxed t the outset of the expermet A multoml expermet geerlzes boml expermet by llowg ech trl to result oe of k possble outcomes, where k > Exmple A bckgroud um begs hve the followg blood types I ddto to the A d B tges, there s thrd tge clled the Rh fctor, whch c be ether preset (+ or bset ( I geerl, Rh egtve blood s gve to Rh-egtve ptets Rh postve blood or Rh egtve blood my be gve to Rh postve ptets So, there re eght ctegores of blood type (k 8: The uversl red cell door hs blood type O The uversl plsm d pltelet door hs blood type AB + Sde ote: Whle whole blood hs shelf lfe of 4 dys, doted pltelets must be used wth 5 dys of collecto There s costt eed for pltelets, whch re essetl prt of the tretmet for y low-pltelet codto, cludg some types of ccer You c dote pltelets every seve dys, up to 4 tmes yer Accordg to the Amerc Red Cross, proportos of the dfferet blood types the US populto re: Cucs Afrc Amerc spc As O + 37% 47% 53% 39% O 8% 4% 4% % A + 33% 4% 9% 7% A 7% % % 5% B + 9% 8% 9% 5% B % % % 4% AB + 3% 4% % 7% AB % 3% % % Why would ths formto be mportt? As oe exmple, some geetc dsorders re much more effectvely treted by usg blood from door who comes from the sme ethc group Notto: I geerl, we wll refer to the k possble outcomes o y gve trl s ctegores, d p wll deote the probblty tht trl results ctegory If the expermet cossts of selectg dvduls or objects from populto d ctegorzg ech oe, the p s the proporto of the populto fllg the th ctegory Ths type of expermet wll be pproxmtely multoml provded tht s much smller th the populto sze Why would we eed much smller th the populto sze?

3 The ull hypothess of terest wll specfy the vlue of ech Exmple A If we were to test the proportos of blood types mog ethc spcs the US populto gve the tble bove, wht would the ull hypothess be? p For multoml lyss, the ltertve hypothess wll stte tht s ot true tht s, tht t lest oe of the p s hs vlue dfferet from tht sserted by (mplyg tht t lest two must be dfferet, sce ll of the proportos dded together must equl [Ths s smlr to the sttemet of the lterte hypothess ANOVA] Notto: The otto p, p sub ought, wll represet the vlue of p clmed by the ull hypothess [Ths s the sme s the otto for the ull vlue ler regresso lyss, β ] I the blood types mog ethc spcs Exmple A bove, p 53, p 4, p 9 3 Theory: Before multoml expermet s performed, the umber of trls tht wll result ctegory (,,, k s rdom vrble just s the umber of successes d the umber of flures boml expermet re rdom vrbles Ths rdom vrble wll be deoted by N, d ts observed vlue by Sce ech trl results exctly oe of the k ctegores, (tht s, ech observto plced exctly oe ctegory, t wll lwys be true tht k N, etc, where s the totl umber of trls Lkewse, the sum of the observed s wll ecessrly be For exmple, expermet wth 5 totl trls d k 4, rdom vrble N mght tke o vlue, N mght tke o vlue 5, d N 3 mght tke o vlue 3 5 The N 4 must tke o vlue N 4 I other words, there s uderlyg ssumpto tht the k ctegores re comprehesve, d every observto wll ft to oe of those ctegores The sum of the s wll equl the smple sze, d terms of proportos, p For multoml lyss, the hypotheses wll be stted terms of reltve frequecy, wll be clculted terms of frequecy, p, but the test sttstc I boml expermet, the expected umber of successes d expected umber of flures re p d q, respectvely (Recll the formul for the me of boml probblty dstrbuto, Lecture 34 Whe p ( q q s true, the expected umbers of successes d flures re p d q, respectvely Smlrly, multoml expermet the expected umber of trls resultg ctegory s E N p (,,, k ( Whe p, p p, K, p k pk s true, these expected vlues become E N p E N p, K E N p (, (, ( k k Exmple A revsted A resercher wts to test the proportos of blood types mog ethc Ass the US populto wth 8 Wht would the hypotheses be? The expected frequeces whe s true re

4 The s d correspodg expected frequeces re ofte dsplyed tbulr formt I ths blood type Exmple, gve As ethcty, for 8 (d k 8, ths tble mght look lke the oe below Ctegory O+ ( O ( A+ ( 3 Observed 3 Expected A ( 4 B+ ( B ( 6 AB+ ( Note tht the Expected vlues re those tht re clculted ssumg the ull hypothess s true AB Row totl ( 8 8 The Observed s re usully referred to s observed cell couts (or observed cell frequeces, d Expected vlues E ( N p re clled the correspodg expected cell couts uder Whe s true, ech E N If, however, severl of the observed couts should be resobly close to ts correspodg ( p dffer substtlly from ther expected couts, we my hve suffcet evdece to coclude tht the ctul vlues of the p s dffer mrkedly from wht the ull hypothess sserts The test procedure volves ssessg the dscrepcy betwee ech Observed d ts ssocted Expected vlue E ( N, wth beg rejected whe t lest two dscrepces re suffcetly lrge p We mght, method smlr to wht we dd whe comprg multple populto mes Chpter, bse our mesure of dscrepcy o the squred devtos ( ( ( p, p, K, k pk, d clculte sum of squres, ( p owever, sce we re testg proportos, equl umerc dffereces mght trslte to very dfferet proportos I the text s exmple, the uthors suppose 95, p, 5 d p I both cses the squred umerc dfferece s 5 But, 95 s oly 5% less th ts expected vlue p, whle 5 s 5% less th ts expected vlue p To tke reltve mgtudes of the devtos to ccout, we wll tke ech squred devto d dvde t by ts correspodg expected cout: Note tht ths vlue wll lwys be postve ( observed expected ( expected The probblty dstrbuto of ths sttstc s ether Z or T dstrbuto, but s rther oe clled χ dstrbuto (See text secto 44 I my Stt 4 Lectures I bypssed ths dstrbuto ch-squred ( p f ν Γ ( ν ( x ν x x ; e ν, x

5 The ch-squred dstrbuto hs sgle prmeter, ν k degrees of freedom, wth possble vlues,, 3, (There re oly k freely determed cell couts: oce y k re kow, the remg oe s uquely determed sce the sum must equl Alogous to the crtcl vlue tα,ν for the t dstrbuto, χ α,ν s the vlue such tht α of the re uder the χ curve les to the rght of χ α,ν Selected vlues of χ α,ν re gve Appedx Tble A7 We ll terpret the χ test sttstc the sme wy tht we terpreted the F sttstc Chpter A vlue of the test sttstc whch s greter th the vlue of the crtcl vlue, clculted χ, wll mply P-vlue α, d we wll reject the ull hypothess χ α, k Just s wth other hypothess tests, we hve some uderlyg ssumptos We must hve smple rdom smple of depedet observtos, for whch there re k > ctegores All observtos must be used Dt c be expressed ether s frequecy, or s reltve frequecy tht s the coverted to frequecy The rule of thumb for beg ble to use χ test s tht for ll cells, we must hve 5 for ll ctegores p Exmple B: Ech yer, DuPot Automotve releses ts Color Populrty Report, study lyzg d predctg color treds throughout the world

6 The ew-cr-sles mger of delershp, kowg tht color prefereces c chge from oe yer to the ext, polls recet customers d gets the followg results: Whte, 3%; Blck, %; Slver, 8%; Gry, 4%; Red, 8%; Blue, 6%; Brow/Bege, 6%; Gree, %; Yellow/Gold, %; Others, % Do the poll results dcte tht the delershp should djust the proportos of colors tht they keep ther vetory (α 5? Ctegory Observed ˆ Whte Blck Gry Slver Blue Brow Red Gree Yellow p Expected p ( O E ( O E ( p ( O E ( p ( E p p Other χ ( O E ( p E p IMPORTANT: The symbol χ s otto! Do ot squre the sum the lst colum hypotheses: clcultos for Observed pˆ : clcultos for Expected p :

7 ( O E ( p clcultos for ( ( ( ( O E p, O E p, : E p crtcl vlue: cocluso: We hve ppled the ch-squred test to stuto whch k > owever, t c lso be used whe k Not surprsgly, the the ch-squred test for k s mthemtclly equvlet to the comprso of two populto proportos test we used Lecture 84 (83 the 8 th edto It c be show tht ( Z χ d ( χ z α / α, so tht χ > χ α, f d oly f Z z α / IMPORTANT: As s the cse wth ll sttstcl test procedures, oe must be creful ot to cofuse sttstcl sgfcce wth prctcl sgfcce A clculted χ tht s greter th crtcl vlue χ my be result of very lrge smple sze rther th y prctcl dffereces betwee the hypotheszed p s d true p s Before rejectg, the pˆ s should be exmed to see whether they suggest model dfferet from tht of from prctcl pot of vew Good ews: We re ot gog to cosder the use of Tble A to fd P-vlues for the χ test sttstc For ths clss, we ll ether rely o softwre to clculte the P-vlue or we ll use the rejecto rego method used Exmple B bove (You mght recogze ths s the sme process used to evlute the F-dstrbuto test sttstc chpter We ll lso be skppg χ Whe the P s Are Fuctos of Other Prmeters α, k

8 owever, we re gog to tke look t χ Whe the Uderlyg Dstrbuto Is Cotuous The uderlyg cocept s frly strghtforwrd Let X deote the vrble beg smpled d suppose the x As the costructo of frequecy dstrbuto hypotheszed probblty desty fucto of X s ( f [, [,, K, [, Chpter, subdvde the mesuremet scle of X to k tervls, k k (Note tht the left-sde boudry s closed d the rght-sde boudry s ope The cell probbltes specfed by re the p P ( X < f ( x dx The cell tervls should be chose so tht 5 for,,, k to meet our rule-of-thumb crter I p prctce, the cells re ofte selected so tht the p s, d therefore the p s, re equl We wo t hve tme to do Exmple, so I ll refer you to the text s Exmple 44 The followg otes mght help: For the 9 th percetle ( exctly 9% of ll studets wll fsh, the crtcl vlue s z 8 The uthors determed σ 563 by replcg µ µ + 8σ wth ts crtero vlue µ d solvg for σ 3 The eght z-tervls were selected so tht ech hs probblty of /8 Tht s, the probblty s uform for ech tervl, esurg tht the Expected vlue for ech tervl wll be uform /8 5 You should do the clcultos for yourself for prctce, checkg your work gst the text s results

9 Appedx Tble A7 Crtcl Vlues for Ch-Squred Dstrbutos α ν