Chapter 3: Inference about Population Proportions

Transcription

1 Chpter 3: Iferece bout Populto Proportos We re ofte cocered wth mg fereces bout populto proportos For exmple: - Accordg to recet Gllup poll, 60% of Amercs re dsstsfed wth the wy thgs re gog the Uted Sttes - I 990, the proporto of femle US s 53% Secto : Ifereces bout sgle populto proporto (Revso) Recll: Norml Approxmto to the Boml The boml rdom vrble X hs me µ p d stdrd devto σ p( p) If for boml dstrbuto s lrge ( p>5 d (-p)>5)we my use orml pproxmto to the boml X p - We stdrdzed X to obt Z N(0,) p( p) Remr: The smple proporto (, ( ) / ) Norml p p p where p>5 d (-p)>5 x umber of successes pˆ s pproxmtely totl umber of observto Hece, ) If pˆ > 5 d ( pˆ) > 5, 00 ( α)% CI for the populto proporto, p, s gve by pˆ ± z( α / ) s( pˆ) where s( pˆ) pˆ( pˆ) b) To test the hypothess H : p p 0 0 H : p p 0, use the test sttstc * pˆ p0 p0( p0) z where s( pˆ ) s( pˆ ) Reject H 0 f z * > z( α / ) The P-vlue s PZ ( > z * ) Pge of 9

2 Exmple: It hs bee reported tht pproxmtely 60% of US households hve two or more televso sets d tht t lest hlf of Amercs sometmes wtch televso loe Suppose tht 75 US households re smpled, d of those smpled, 49 hd two or more televso sets d 35 respodets sometmes wtch televso loe ) Two clms c be tested usg the smple formto Wht re the two clms? ) Costruct 95% cofdece tervl for the proporto of the Amercs tht hve two or more televso sets? Do the dt preset suffcet evdece to show tht the 60% fgure clmed the mgze rtcle s correct? 3) Do the dt preset suffcet evdece to cotrdct the clm tht t lest hlf of Amercs sometmes wtch televso loe? Pge of 9

3 Secto : Ifereces bout two populto proportos (Revso) x x Let pˆ d pˆ d ssume pˆ > 5, ( pˆ ˆ ˆ ) > 5, p > 5, d ( p) > 5, the ) A00( α)% CI for dfferece betwee the populto proportos, D p p, s gve by ˆ ˆ ˆ pˆ ( pˆ ) pˆ ( pˆ ) D± z( α / ) sd ( ) where sd ( ) + where D ˆ p ˆ p ˆ A c) To test the hypothess H : p p 0 0 H : p p 0, use the test sttstc z * Dˆ 0 ˆ pˆ ˆ ˆ ˆ ( p) p( p) where sd ( ) + sd ( ˆ ) Reject H 0 f z * > z( α / ) The P-vlue s PZ ( > z * ) Exmple: A expermet ws coducted to test the effect of ew drug o vrl fecto The fecto ws duced 00 mce, d the mce were rdomly splt to two groups of 50 The frst group, the cotrol group, receved o tretmet for the fecto The secod group, the expermetl group, receved the drug After 30-dy perod, the proportos of survvors the two groups were foud to be 036 d 060, respectvely ) Is there suffcet evdece to dcte tht the drug s effectve tretg the vrl fecto? ) Use 95% cofdece tervl to estmte the ctul dfferece the cure rtes for the expermetl versus the cotrol groups Pge 3 of 9

4 Sec 3: Ifereces bout Severl Proportos If rdom vrble X follows the gmm dstrbuto the the probblty desty fucto s gve by f( x) x e α β Γ( α) α x/ β Ch-Squre dstrbuto wth r degree of freedom s specl cse of gmm dstrbuto where α r / d β Recll tht f X hs the boml(,p) dstrbuto the x! PX ( x) p( p) where p p p!( p)! Proportos re relly just specl cses of mes To see ths, let x be or 0 f the th US ctze s mle or femle, respectvely, d let p represet the ctul proporto of mle ctzes The f N s the populto sze, N p x N So p s relly the verge of the N s d 0s Pge 4 of 9

5 Ths mes we c te smple of sze d estmte p wth the ubsed pot estmtor pˆ x, where X s the umber for the th rdomly chose perso Sce the populto of the US s rther lrge, we c vew the perso should be mle wth probblty p The X s s depedet Beroull(p) rdom vrbles Tht s, the th X ~ boml( p, ), whch mes tht for {0,,, } P pˆ P X P X p ( p) Ad hece the me d stdrd devto of ˆp re p d p( p) / respectvely (these results we hve used sectos d ths chpter, but we dd ot see why But ow we ow why ) The multoml dstrbuto s jot dstrbuto geerlzto of the boml dstrbuto Tht s, suppose tht depedet expermets re to be performed, ech of whch results outcome wth probblty p, outcome wth probblty p,, outcome wth probblty p Let X deote the umber of the expermets resultg outcome, the for d p, we hve! p PX ( x, X x,, X x) p p p!!!!! Π The expresso bove s the jot probblty mss fucto for the multoml dstrbuto We c verfy ths probblty expresso by otg tht the probblty of y prtculr sequece of the outcomes where evet occurs exctly tmes for,, s exctly Π p Also, the umber of these sequeces s exctly!!!! Pge 5 of 9

6 Exmple: Cutthrot s three-plyer gme of pool tht Joh, George, d Rgo le to ply (Pul s ded) Joh s very good, d ws 60% of the tme, George ws 30% of the tme, d Rgo ws 0% of the tme Suppose they ply te gmes Wht s the probblty tht Joh ws t lest fve gmes d George ws t lest four? Bc to sttstcs The ch-squre goodess-of-ft test s hypothess test for determg whether cert probbltes (or proportos) te o prtculr vlues To do ths, we perform multoml expermet, d record the observed umbers, of trls resultg ech outcome type,, We the compre these umbers to the expected vlues of the umbers of outcomes of ech type uder the ull hypothess Ths depeds o the test sttstc ( E ) X where E p E (expected cell frequecy) The Ch-Squre Goodess-of-Ft Test: H : p p, p p,, p p H : t lest oe of the proportos dffers from ts hypotheszed vlue, Reject H o f X > χ ( α, ) Pge 6 of 9

7 Exmple: The uts populto cosst of oe of fve types A rdom smple of 300 uts s clssfed s follows: Ctegory Observed Cout, Totl 300 It s hypotheszed tht H0 : p 00, p 05, p3 040, p4 05, p5 00 At α 005 level, do the 300 uts pper to be from populto wth these vlues for p Pge 7 of 9

8 Ch-Squre Test of Homogeety The followg tble s r ccotgecy tble where the r rows correspod to the r popultos d the c colums correspod to the c ctegores of clssfcto Ctegores of clssfcto Populto c Totl c c r r r rc r Totl c To test the hypothess H p p p H 0 : r : Not ll the populto proportos re equl Or o the other word H 0 H : The populto re homogeeous : The popultos re ot homogeeous We use the test sttstc ( ) r c j Ej where Ej (expected cell frequecy) χ E j Reject H o f X > χ ( α, ( r )( c )) Pge 8 of 9

9 Exmple: A resercher studed the chrcterstcs of subjects ttedg fve-dy hum sexulty progrm The results re show the followg tble: Mrtl Sttus Group sgle Mrred or Totl dvorced Medcl Studets Nursg Studets 5 37 Other studets Group leders Totl Test whether or ot the four popultos represeted the study re homogeous wth respect to mrtl sttus Pge 9 of 9