Statistics: Part 2 Hypothesis Testing

Transcription

1 Hey Stk d Joh W. Woods obbty Sttsts d Rdom Vbes o Egees 4th ed. eso Eduto I. 0. ISBN: Chte 7 Sttsts: t Hyothess Testg Setos 7. Byes Deso Theoy kehood Rto Test Comoste Hyotheses 40 Geezed kehood Rto Test (GRT) 403 How Do We Test o the Equty o Mes o Two outos? 408 Testg o the Equty o Ves o Nom outos: The F-test 4 Testg Whethe the Ve o Nom outo Hs edetemed Vue: Goodess o Ft Odeg eetes d Rk 43 How Odeg s Useu Estmtg eetes d the Med 45 Codee Itev o the Med Whe Is ge 48 Dstbuto-ee Hyothess Testg: Testg I Two outo e the Sme Usg Rus 49 Rkg Test o Smeess o Two outos 43 Summy 433 obems 433 Reeees 439 B.J. Bzu F 06 o 0 ECE 3800

2 7. Hyothess Testg Now tht we hve oets o etbe tevs bsed o oeted sttsts we ete ths to whethe sttemet bsed o the sttsts s etbe o ot Sttst Deso Mkg. A sttemet s mde tems o Hyothess. The go o teetg the mesued vues s to et o ejet the Hyothess. Emes: the o beg ed s two dom ose sgs (oesses) hve the sme me the etme stted o ght bub s ote desto o the me the etme stted o the ght bub s ote desto o mmum the sme me mesued o set o omoets s wth the 95% odee tev o the desed me vue (% esstos e wth % o vue wth 95% odee) Whe thee s oy oe Hyothess t s eeed to s the u Hyothess (H 0 ). Thee e otety mute Hyotheses we geete te to et oe ove othe (estbshg theshods o deso mkg). The tet w oh ths sttg wth the stutued Byes Deso theoy whee bsed o -o d kowedge deso theshod betwee two ossbtes be deed. B.J. Bzu F 06 o 0 ECE 3800

3 7. Byes Deso Theoy I the bsee o dve gude the Byes oh to mkg desos dom (stohst) evomet s guby the most to oedue devsed by hums Demm d es Eme 7. A 40 ye od tet hs odue e s o the et ug. The tet s o-smoke. The doto hs some ossbtes: Is the odue beg o eous? Do you eom sugey o ot? Useu dt: odues ey-mdde ge o-smokes e beg 70% o the tme Moe tos: Reduto e eety ude deet odtos I the stte o tue be desbed tems o ds o ommo s we mght hve Fgue 7.- Thee s vue o (to be detemed) tht w mmze the eeted sk. A dtum ot the ego ζ ( ] s moe key to be ssoted wth beg odto d w ed to to (do t oete) whe dtum ot Γ [ ) s moe key to be ssoted wth e d w ed to to (oete). B.J. Bzu F 06 3 o 0 ECE 3800

4 The we dee obbtes d d whee the st s the eo obbty tht the evdee suggests thee s e tht eques sugey whe t thee s o e d the seod dees the eo obbty tht the evdee suggests thee s o e d theeoe o to s ed o eve though thee ey s e. A sk ue deed s R d o e o beg bsed o odto eettos s Rd R d The tot eeted sk the ootes obbty s B d R d R d We ow seek to mmze the sk uto wth eset to d. ettg d Bd Rd Rd Edg the jot obbty tems B d d B d Fom evous detos d d d d d d d d B.J. Bzu F 06 4 o 0 ECE 3800

5 B.J. Bzu F 06 5 o 0 ECE 3800 The d d d d B d d d B d Usg to mmze the eeted sk by tkg the devtve d d d d d d d d d d B d d d Usg d d d 0 0 B d d d A deso theshod be deed bsed o the etve es o the egos s b k The vue k b estbshes theshod o deso ue. I k b the beeve t s e d oete I b k the beeve t s beg d do othg

6 Usg vues evousy estbshed wth =0.3 d =0.7 (eo tetbook. 394) d d B B d d 3.5 The mmum w ou by mmzg the eemets m 9 d 3.5 d kb d d Theeoe I I the beeve t s e d oete the beeve t s beg d do othg As ote o mute eemets ths deso theshod tke the om o kehood to whee k b Evey Byes sttegy eds to kehood to test howeve ot evey RT s the esut o Byes deso sttegy. You w ote the evous emes we dd ot hve the udeg d utos but we oud estbsh obbtes d sk (o ost) uto. B.J. Bzu F 06 6 o 0 ECE 3800

7 The oowg otes e bsed o the gdute ouse tetbook moe smy stted Byes deto. Abeto eo-g obbty Sttsts d Rdom oesses Fo Eet Egeeg 3d ed. eso ete H Ue Sdde Rve NJ 008 ISBN: Byes Deso Methods I ths seto we eoe methods tht ssume dom vbe d tht we hve o kowedge o ts dstbuto. Ths ew ssumto eds to ew methods o ddessg estmto d hyothess testg obems Byes Hyothess Testg Cosde sme by hyothess obem whee we e to dede betwee two hyotheses : bsed o dom sme d we ssume tht we kow tht H 0 ous wth obbty 0 d H wth obbty 0. Thee e ou ossbe outomes o the hyothess test d we ssg ost to eh outome s mesue o ts etve mote:. H 0 tue d dede H 0 Cost = C 00. H 0 tue d dede H (Tye I eo) Cost = C 0 3. Htue d dede H 0 (Tye II eo) Cost = C 0 4. H tue d dede H Cost = C It s esobe to ssume tht the ost o oet deso s ess th the ost o eoeous deso tht s C00 C0 d C C0. Ou objetve s to d the deso ue tht mmzes the vege ost C: C C00 H 0 H0 C0 H H0 0 C H H C0 H 0 H Note: deteto d estmto otm s tyy deed bsed o ost uto. The ost utos e ote edy eted to the oeto o mete (e.g. mmum mesque-eo mmum kehood et.) but they ude bty tems s og s they be edy deed bsed o the mesued vues deees sttsts o obbty. B.J. Bzu F 06 7 o 0 ECE 3800

8 The oowg theoem detes the deso ue tht mmzes the vege ost. B.J. Bzu F 06 8 o 0 ECE 3800

9 B.J. Bzu F 06 9 o 0 ECE kehood Rto Test Whe we do t hve udeyg obbtes o ost uto but do hve the odto obbtes we dee kehood to test s k et s hoe k ejet s hoe whee the theshod k s detemed om te tht oud be bsed o y deso theoy desed. Eme 7. Heth ood weght eduto Test gou: 98 bs. st-dev 5 bs Coto gou: 0 bs. st-dev 5 bs Guss ds o the u hyothess (othg hged sk ot heu) 0 e 5 H Guss ds o the tete hyothess (hged sk heu) 98 e 5 H A det to test woud estbsh H H 98 0 e H H 0 98 e H H e

10 H H H H e The deso uto beomes 4 e e e e K e ˆ I I K K 4 e ˆ k the et the H hyothess d ejet H 5 4 e ˆ k the et the H hyothess d ejet H 5 The ds o ths deso e show the et gue og kehood test. As og s the og uto s mooto we oud use 4 K ˆ e k 5 4 ˆ 5 5 ˆ 4 k K k K o ˆ B.J. Bzu F 06 0 o 0 ECE 3800

11 Fo ths equto the eo obbtes be deed s dedeh H dedeh H The st eo s ed tye I eo dedg gst the u hyothess whe t s tue. Ths s so deed s the sge o the test. The seod eo s ed tye II eo dedg o the u hyothess whe t s ot tue. The vue (-ths obbty) s deed s the owe o the test. We dee the tye I eo o ths obem tems o the me vues d 0 ˆ H e 5 I we set h s ˆ H d The theshod test vue o vous sme szes s show the oowg gue. 0 0 Theshod eve h= smes () B.J. Bzu F 06 o 0 ECE 3800

12 ookg t bet 98 ˆ H e 5 d 98 d ˆ H d Q 5 d 5 98 Note tht oe we dee the theshod bsed o h 98 vue o bet om the questos. Q we oud deteme the 0 0 h bet@ 0 - obbty o eo smes () I we wted bet d h to be equ the theshod must be h wy betwee the two mes. Theeoe o 4. 0 B.J. Bzu F 06 o 0 ECE 3800

13 4-5 Hyothess Testg Bsed o : obbst Methods o Sg d System Ayss (3d ed.) by Geoge R. Cooe d Ce D. MGem Ood ess Nu Hyothess Testg A sge test bsed o deso ue must be detemed. The sge test estbshes eve otety the odee eve o odee tev to deteme whethe to et o ejet the hyothess. Ths s stted s deso ue whee Aet H 0 : Rejet H 0 : the omuted vue sses the sge test. the omuted vue s the sge test. I gee ths voves sge test tht dees d equto o uto tht be omuted bsed o the mesued dt (sttsts). A eome theshod s the deed tht dees Aet/Rejet o ss/ boudy. Isde o outsde the odee tev. Aetby meet desed te o ot. B.J. Bzu F 06 3 o 0 ECE 3800

14 Eme: (. 74) A to mutue ms tht the tos hve me bekdow votge o 300V o gete. We estbsh sge test o 99% odee eve. (I ths se we e ookg o vues bove the mmum odee eve s etbe Ths w be oe sded test.) I testg 00 tos e tested (ote ths s destutve test) d me vue o 90V wth ubsed sme stdd devto o 40 V. Is the Hyothess eted o ejeted? The sge test t the 99% odee eve eques ~ t S ˆ Usg v=99 d F=0.99 ( sded test) A. G-4 t=.358 (usg 0). Theeoe ~ 40 t S ˆ o ˆ The mesuemet esut o 90V uses us to ejet the Hyothess! Theeoe we woud sy tht the me med s ot vd. F ote: 99% odee eve ws seeted o the sge test ths eme 99.5% eve wee seeted the Hyothess woud hve bee eted! ( =89.53) To evte (?) ths ouso eve o sge be deed tht s eve o sge =.0 - odee eve. Ths woud sy tht the bove test ws to % eve o sge. Theeoe % eve o sge s ejeted but 0.5% eve o sge woud be eted. B.J. Bzu F 06 4 o 0 ECE 3800

15 Eme: Testg Co The bom dom vbe ows us to deveo test to see o s whe ed. We eed to out the umbe o heds tht ou d test t eve o sge o 5% o 95% odee eve. Aet H 0 : Rejet H 0 : the umbe o heds sde 95% odee eve t s. the umbe o heds outsde 95% odee eve t s ot. We ssume tht umbe o ts hs ed to Guss sttsts eet o - o kowedge o the me d ve eeted o o (=0.5) usg the bom dom vbe. Fo ths dstbuto the me d ve e E V o Assumg 00 ts d tht the sttsts hve beome Guss. k k ˆ We use two-sded test theeoe k=.96 d we hve k Theeoe the test ego o the Hyothess s 4.9 ˆ 4.9 o 45. ˆ Now you hve te to estbsh o s. Fom Mtb: (46<=<=54) = B.J. Bzu F 06 5 o 0 ECE 3800

16 Eme: Hyothess testg ommutos Sg us ose ut to the eeve. Dgt symbo eeve oututs vue oesodg to symbo us ose. Hyothess testg estbshes the ues to seet oe symbo s omed to othe. Ioet seetos esuts symbo eos d dgt bt eos oe the symbos e tsted to bts. t s t t o T t T Bed Sk Dgt Commutos Fudmets d Atos ete H TR Seod Edto 00. Aed B. Fo eh symbo deteto sttst s geeted o the symbo eod T. T T T z 0 Hyothess testg the detemes the estmted symbo vue om the deteto sttst. The umbe o Hyothess s equvet to the umbe o ossbe symbos tsmtted. B.J. Bzu F 06 6 o 0 ECE 3800

17 7.3 Comoste Hyotheses 40 Geezed kehood Rto Test (GRT) A Geezed kehood Rto Test be used whe mute Hyothess est. I ths se we oud thk o eh hyothess vovg deet subset o subse o the ossbe outomes. Ag test sttst oud be geeted to dee uso o euso suh s M * z GM The umeto tem voves o mmzto o the mete thet whe the deomto voves gob mmzto tems o the u ge o thet. Mute vues e omuted bsed o the seh estted egos desed o eh o the o mm. Ths s moe osey ssoted wth detetg mute sg eves om mut-eve tsmsso ovdg deteto sttsts wth eh eeted egos o ossbe symbo detto. Not beg etued o you e teested: How Do We Test o the Equty o Mes o Two outos? 408 Testg o the Equty o Ves o Nom outos: The F-test 4 Testg Whethe the Ve o Nom outo Hs edetemed Vue: 46 B.J. Bzu F 06 7 o 0 ECE 3800

18 B.J. Bzu F 06 8 o 0 ECE Goodess o Ft 47 eso test o vd sttst mesues g os og de A t set o smutos s tht the dt tuy hs the desed htests deed o desed. Ths s tuy mott etg dom vbes o mthemty deed destes d dstbutos. I the smuto dt s eo the esuts w be questobe d y desos o desgs bsed o the esut oud. The gee mode s tht o sotg the dt to bs d omg the estmted obbtes wth the seed obbtes. The o the estmtes e e the desed esuts t s key tht outo e oet. I they de two o moe bs t s key tht t s ot the oet desty uto. I eomg ths test the umbe d wdths o the bs s mott. Fo sme dsete tests bs be osdeed. Fo otuous tests the umbe d sze dve omety the mout o dt eeded to be tke d the ove uy. The R.V. s deed s ese j 0 b o obsevto jth The obbty s j The we hve the sum R.V. the umbe o outomes the b. j Y j Ths beomes mutom obbty uto d s bsed o Y Y Y!!!! The omto s Y Y Y e!!!!

19 Skg the mmum kehood dsusso the tet Ude the ge sme ssumto we omte usg the Guss om wth N whe the U Y o o The sttst test kow s eso Test Sttsts beomes V Y o o The eso s test sttsts hs the om o h-squed RV wth - degees o eedom. (Note h-squed s bsed o the sum o Guss R.V. squed). Theeoe ssgg egos o ovegee o obbty the h-squed tbe must be used (G-5 Tbe 3). Eme 7.4 Co ess Assumg heds d ts e obbty 0.. Assume 00 s t eve o sge o h =0.05. I we obseve 6 heds V Y o o V Usg the tbe -= degee o eedom 0.95 esuts z=3.84 Theeoe The teo s ot ssed. I thee wee 4 d 59 heds d ts F V V The teo s ssed. B.J. Bzu F 06 9 o 0 ECE 3800

20 Eme 7.4 F De Assume 000 os wth the oowg sequet esuts: V Y o o V Usg the tbe -=5 degee o eedom 0.95 esuts z=. Theeoe The teo s ssed. F V Mte ot dsussed o equed 7.5 Odeg eetes d Rk 43 How Odeg s Useu Estmtg eetes d the Med 45 Codee Itev o the Med Whe Is ge 48 Dstbuto-ee Hyothess Testg: Testg I Two outo e the Sme Usg Rus 49 Rkg Test o Smeess o Two outos 43 Summy 433 obems 433 Reeees 439 B.J. Bzu F 06 0 o 0 ECE 3800