How To Write An Acctac Samlig

Transcription

1 REVISA INVESIGACION OPERACIONAL VOL. NO. -9 A FRAMEWORK FOR HE DERIVAION AND VERIFICAION OF VARIABLES ACCEPANCE SAMPLING PLANS K. Prsto Whit Jr.* ad Kth L. Johso** *Dartmt of Sstms ad Iformatio Egirig P.O. Bo 777 Uivrsit of Virgiia Charlottsvill Virgiia 9 USA **Statistics ad rdig NASA/NESC Sstms Egirig Offic. Mail Sto E/NESC Bldg. 7 Rm. NASA Marshall Sac Flight Ctr Hutsvill Alaama 58 USA ABSRAC Acctac samlig is a mthod for vrifig qualit or rformac rquirmts usig saml data. Varials acctac samlig is a altrativ to attriuts acctac samlig which i ma istacs rquirs sigificatl smallr samls. I this ar w cosolidat th litratur o varials acctac samlig rovidig a uifid ositio of th aroach usd to dvlo such las. From withi this framwork w rviw th drivatio of las for otial ormal gamma Wiull ad Poisso radom varials. W vrifid ths drivatios o a st of tst rolms. KEYWORDS: qualit girig rliailit isctio rquirmts vrificatio MSC: 6P RESUMEN El mustro d actació s u método ara vrificar la calidad o l rqurimito dl comortamito usado datos mustrals. El mustro d actació or varials s ua altrativa ara l mustro d actació or atriutos l cual muchos casos rquir d mustras sigificativamt quñas. E st traajo cosolidamos la litratura sor l mustro d actació or varials rstado ua osició uificada dl foqu usado ara dsarrollar tals las. Dsd l itrior dl marco d traajo rvisamos la drivació d las ara varials alatorias co distriució ocial ormal gamma Wiull Poisso. Vrificamos stas drivacios sor u cojuto d rolmas d rua.. INRODUCION O of th oldst rolms i qualit girig is to assss th acctailit of itms that a customr rcivs from a roducr. Acctac samlig is a altrativ to % isctio alid wh isctio is dstructiv or wh th tim ad/or cost of % isctio ar uwarratd or rohiitiv. h customr dcids th disositio of a icog lot asd o a stadard scifig th maimum roortio of ocoforg itms i th saml. h dcisio ca to acct or rjct th tir lot or to cotiu samlig. Acctac samlig also has adatd to rolms ot idtifid with rocurmt. Smith t al. [] advocatd th alicatio of acctac samlig i th cott of watr qualit assssmt. Baard t al. [] alid th udrlig iomial modl to th assssmt of futur aircraft ruwa-icursio cotrols. Whit t al. [] showd that attriuts acctac samlig ca alid to a samlig rimt icludig thos commol mlod to vrif dsig rquirmts usig simulatio ad Mot Carlo mthods. Acctac samlig attriuts ASA stcs a lot asd a cout of th umr of ocoforg itms rlativ to th saml si. h isctio varial is iar ass/fail ad thrfor th cout is cssaril iomial. ASA ca usd with catgorical oututs or with oututs masurd o a cotiuous or discrt scal rfrc to a rquird limitig valu. Coctuall siml asil alid ad uivrsall alical ASA is th first choic for samlig isctio. Acctac varials ASV is a altrat aroach which i ma istacs rscris sigificatl smallr samls tha ASA. ASV rquirs that isctio varial is masurd o a cotiuous or discrt scal that th kwhit@virgiia.du kth.l.johso@asa.gov

2 distriutio of this varial is kow a riori ad stal ad that a la ists for this articular distriutio. Whil far mor rstrictiv i its assumtios ASV should cosidrd wh th largr samls rquird ASA ar uavailal. h ojctiv of this ar is to cosolidat th scholarl litratur o ASV ad rovid a radal tutorial o th coct ad gral aroach usd to dvlo ASV las for altrativ art distriutios. Whil ASV i fact mlos a cosistt rocdur for dtrig samlig las this commoalit is oscurd diffrt rstatio stls ad th somtims vastl diffrt otatio adotd rsarchrs i th fild. Hr w offr a uifid rstatio ad a cosistt otatio. I additio ach of th las dscrid hr was imlmtd i a calculator ad vrifid o a rag of saml rolms.. PROBABILISIC REQUIREMENS AND LIMI SANDARDS I ucrtai viromts rquirmts vrificatio sks to dtr whthr a masural quatit is coforg or ocoforg i.. to dtr whthr or ot th art oulatio from which a saml is draw achivs a scifid lvl of qualit or rformac. h rquirmt ca statd roailisticall as a I limit stadard [] whr: I is th rformac idicator. h masurd quatit ma ihrtl catgorical or qualitativ i atur ad rformac idicatd occurrc or ooccurrc of som vt. Altratl th masurd quatit ma ihrtl quatitativ ad rformac idicatd succss or failur i achivig a limit or tolrac. is th imum rliailit for th oulatio. his is th imum acctal log-ru roortio of osrvatios o which th oulatio achivs th dsird rformac. If is th failur roailit for th oulatio th rquirmt is < -. is th maimum acctal cosumr s risk. his is th roailit of icorrctl acctig a ocoforg oulatio as th rsult of samlig rror. Formall w rrst a masural quatit as th radom varial X with roailit distriutio F; ad dsit f; whr is a ukow distriutio aramtr. Lt {X i ; i= } a radom saml with osrvd valus { i ; i= }. h vrificatio rolm is to dtr whthr or ot th oulatio as a whol coforms to a scifid limit stadard asd o th statistics of th saml osrvd.. SAMPLING PLANS AND OPERAING CHARACERISICS A samlig la is th air whr is th imum saml si i.. th imum umr of osrvatios rquird to vrif statisticall th rquirmt imosd stadard. is a costat factor which is usd to assss whthr or ot th oulatio is coforg. h itrrtatio of dds o whthr th charactristic is cotiuous or discrt as discussd i Sctio. For a giv distriutio vr samlig la has a uiqu oratig charactristic OC. h OC is a fuctio which dfis th roailit of acctig a oulatio P a for vr valu of th failur roailit i.. th OC is th fuctio P a whr []. A samlig la is drivd first dfiig two oratig oits - ad whr < ad ad ar small roailitis. W th rquir that th OC must imall satisf th followig iqualitis P a > - P a < Altratl it is somtims mor covit to rss ths i trms of th rjctio roailitis as th owr coditios P r =-P a < ad P r =-P a > -. Not that wh =- ad is th cosumrs risk iqualit forcs th limit stadard. Udr this la w will acct a oulatio with failur roailit as coforg with small roailit. Iqualit caturs th comtig good udr this la w will acct a oulatio with roailit as coforg with high roailit-. h roailit of icorrctl rjctig a coforg sstm is calld th roducr s risk. Wh giv as a rctag %- o is calld th acctal qualit lvl AQL ad %- is calld th lot tolrac rct dfctiv LPD. Not that th limit stadard dos ot scif th o - oratig oit. It is commo to

3 dvlo a rag of las with th rquird cosumr s risk ad diffrig Plas with largr saml sis th hav smallr corrsodig.. HYPOHESIS ESING AND ACCEPANCE LIMIS h udrlig rolm ca framd as a hothsis tst for which w itd to forc oth sigificac ad owr rquirmts. h ull ad altrat hothss ar H : = ad H : = > Udr H fail to rjct th oulatio as ocoforg ad udr H w rjct th oulatio as ocoforg. Iqualit stalishs th sigificac of tst as ad iqualit stalishs th owr of th tst as - whr ad ar th roducr s ad cosumr s risk rsctivl. W us th saml data to choos tw th ull ad altrat hothss. o accomlish this w d to dtr th critical valu of a aroriat tst statistic. Dot th tst statistic as th acctac limit A whr th argumts ar th aramtrs of th samlig la. For a cotiuous quatit =k ad th acctac limit ad ticall has th form whr th lus is usd with a ur oud ad th us with a lowr oud. It follows that A ˆ k ˆ ˆ whr ad ar stimators for th ukow aramtr ad th stadard dviatio of th oulatio. For I a lowr scificatio limit with valu th dsird rformac o th i th osrvatio is achivd if ad ol if i >. For th saml as a whol w rjct th ull hothsis if A k. hat is th acctac limit is smallr tha th scifid limit. For I a ur scificatio limit with valu ma th dsird rformac o th i th osrvatio is achivd if ad ol if i < ma. For th saml as a whol w rjct th ull hothsis if ˆ A k ma A k ˆ k ˆ. hat is th acctac limit is largr tha scifid limit. h ffct i ithr cas is to mov th critical oit of th acctac limit awa th stimat of th ukow quatit i th dirctio of th scifid limit k stadard dviatios. Ituitivl this hdg is itdd to comsat for rror i th stimat for a saml of osrvatios i a statisticall act wa. Not that i this ar w will ot addrss th third cas whr two limits ar scifid. I this cas I is a tolrac itrval ma > i >. h drivatio of samlig las for tolrac is modstl mor comlicatd tha that dscrid i Sctio 5. h itrstd radr is rfrrd to th cosidral litratur o tolrac itrvals. For a discrt oulatio charactristic th acctac limit has th form A c Y c whr Y is som fuctio of X with ogativ itgr valus. It follows that c Y A c For a lowr scificatio limit o Y th acctac umr c is th imum acctal valu of Y ad th rjctio critria is Ac<. For a ur scificatio limit o Y c is th maimum acctal valu of Y ad th rjctio critria is Ac>. 5. GENERAL PROCEDURE FOR DEVELOPING VARIABLES ACCEPANCE SAMPLING PLANS For a cotiuous distriutio with lowr scificatio limit th rocdur comriss four sts: Dtr th valu of ticall th ma for th ull roailit distriutio with failur roailit = Pr[X< ]=F ; ; dtr th valu of for th altrat distriutio with failur roailit =Pr[X<.]=F ;. his is accomlishd usig th ivrs distriutio. As show i Figur th rquirmt that < imlis that >. Dtr th imum valu of from th samlig distriutios for ad such that oth iqualitis ad ar satisfid.

4 Dtr th maimum acctac limit A from th ull samlig distriutio with Pr[ ˆ A ] F ˆ; ; dtr th imum acctac limit A from th altrat samlig Pr[ ˆ A ] F ˆ; distriutio with ; as illustratd i Figur. Dtr th factors k ad k corrsodig to A ad A rsctivl from Equatio. I gral ths factors ar ot qual. A cosrvativ choic is to us th largr of th two valus as th factor k for th samlig la. Altratl th k is somtims tak as th avrag of ths two valus. For I a ur scificatio limit th failur roailit of th oulatio is =Pr[X> ma ]=-F;. h rocdur is idtical with this sigl ctio. For a discrt distriutio istad of th factors k ad k at St th costat c is dtrd. F; F; Figur. Distriutio fuctios ad critical valus for X udr th ull ad altrat hothss. - k k Figur. Samlig distriutios for th stimator mi ˆ A A udr th ull ad altrat hothss.

5 If th radom varial X ad th ukow aramtr ca stadardid it is grall mor covit to us th stadardid distriutio ad stadardid samlig distriutio to driv samlig las. his covic is illustratd i Sctio 6 for otial ad ormal radom varials. 6. MEHODS IMPLEMENED A litratur sarch foud drivatios of k samlig las for otial ormal gamma ad Wiull distriutios. hs drivatios follow th gral rocdur giv i Sctio 5. A drivatio also was foud for th c samlig la for th Poisso distriutio. I this sctio w outli ths drivatios with rfrc to lowr scificatio limits. h corrsodig drivatios for ur scificatio limits ar asil dducd ad ar ot covrd hr. 6. Eotial Guthr [] cosidrs th otial radom varial X with ukow ma suort [ ad distriutio fuctio Not that otial ad chi-squard distriutios ar oth scial cass of th gamma distriutio with otial=chi-squar=gamma. Dot a radom varial distriutd chi-squard with dgrs of frdom as Y. X ca stadardid as Y X so that ; ad whr is ithr or. Similarl th stimator for th ma is saml ma X which ca stadardid as Y X 6 Sustitutig Equatio 5 ito Equatio 6 ilds whr is ithr or. Iqualitis ad iml ogthr ths i tur iml that or Pr X Pr Yv ; F ; h imum rquird saml si is th smallst valu of th itgr satisfig iqualit 7 which ca dtrd tal looku. / ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; 7 5

6 With kow th k-factor ca comutd. h stimator for oth th ma ad stadard dviatio is th saml ma X thrfor th acctac limit is It is customar to lt k=-k ad us th k-factors ad 6. Normal -kow h drivatio of samlig las for masural quatitis distriutd N is widl ulishd [] [5] [6] [7] ad [8] amog othrs ad th asis for stadards MIL-SD- ad ANSI/ASQC Z.9 ad ISO 859. Cosidr th ormal radom varial X with ukow ma kow stadard dviatio suort - ad dsit fuctio Dot a radom varial distriutd a stadard ormal N as Z. X ca stadardid as Z X such that whr Pr X A h ctral limit thorm tlls us that th ma stimator as X Z whr Z also is a stadard ormal N dviat. h acctac critrio is A k k Iqualitis ad iml ˆ ˆ A k k' k X k' Y Pr Z X Pr k k f ; X is distriutd N / which ca stadardid so that w will rjct th oulatio if h acctac-limit critria rssd as owr rquirmts from iqualitis ad th iml ; ; ; ; Pr Z / k X k Pr X Pr Pr Z k k k k k 8 9 5

7 k ogthr ths iml that k k ad thrfor h rquird saml si is th smallst itgr valu of satisfig iqualit. With kow th k-factors ca comutd as from quatios. 6. Normal -ukow h mor usual cas is whr ukow ad must stimatd from th saml data. Whil a act aroach is availal usig a o-ctral t-distriutio s [] amog othrs for modst saml sis cllt aroimatios for ad k ar achivd alig a rsult Cramér [9]. Lt X ad S saml ma ad stadard dviatio rsctivl. For sufficitl larg th distriutio of th acctac limit is asmtoticall ormall distriutd N A A with ma A k ad variac A k h limitig variac / is wightd th asio factor k []. For a o-normal radom varial X with ma stadard dviatio skw ad kurtosis k k k whr th lus is usd with a ur oud ad th us with a lowr oud. For a Normal radom varial X th skw is = ad th kurtosis is =. h asio factor thrfor rducs to [] k k h drivatio of ad k th rocds as i th kow cas rsultig i ad k 6. Gamma k k A k X ks 6

8 7 akagi [] cosidrs th gamma radom varial X with ukow locatio << ad stimatd sha> ad scal > scal aramtrs suort [ ad dsit fuctio h momts for this distriutio ar With th sam ormal aroimatio N A A for Ak th asio factor giv i Equatio is comutd usig th gamma momts i Equatio. h drivatio is sstiall th sam as that for th ormal distriutio rsultig i ad 5 Hr is a stadard gamma dviat with dsit for which = ad =. h asio factor i quatio is comutd usig th Wiull momts i Equatios. h rssios for ad k ar agai giv Equatios ad 5 whr it is udrstood that t is th Wiull dviat. 6.5 Wiull akagi [] also cosidrs th Wiull radom varial X with ukow locatio << ad stimatd sha> ad scal > scal aramtrs suort [ ad dsit fuctio h momts for this distriutio ar 6 Hr is a stadard Wiull radom varial with dsit / / ; f 6 / t t k t t k X t t t f ; f ; 6 9

9 ad ma ad variac whr =/. h asio factor i Equatio is comutd usig th Wiull momts i quatios 6. h rssios for ad k ar agai giv Equatios ad 5 whr it is udrstood that t is th Wiull dviat. 6.6 Poisso Guthr [] [] cosidrs th Poiso radom varial X with ukow ma suort [ ad distriutio fuctio h mas ad ca dtrd form th ivrs distriutio. Altratl Guthr loits th rlatioshi tw Poisso ad chi-squard distriutios ad calculats th mas usig th rssio W ca thik of X as a discrt radom varial rrstig th umr of usal itms i a lot ad scificatio limit as th scifid imum umr of usal itms i ach lot. If w otai a saml of lots th total umr of usal itms otaid is th sum which is distriutd Poisso with a ma of i.. F;. Bcaus w ar workig with discrt radom varials as is th cas i attriuts acctac samlig th rjctio critrio is asd o th imum total umr of usal itms i all of th lots > c+=d istad of th distac k. Iqualitis ad ar P a > =- F d ; > - P a > =- Fd; < or mor siml th owr rquirmts Fd; >- ad Fd; <. Agai loitig rlatioshi tw Poisso ad chi-squard distriutios th owr rquirmts iml his rssio ca solvd for a d samlig la umratio. h valu of itgr valus of d is icrasd util a itgr valu of is foud which satisfis iqualitis ad. 7. CONCLUSIONS f t; t h work rortd i this ar is th first has of a rojct itdd to mak ASV a ractical altrativ to ASA wh aroriat varials las ar availal. W iitiall rviwd th litratur discovrig ulishd las for otial ormal gamma ad Wiull radom varials. With th ctio of ormal las our sarch for offth-shlf ASV la calculators was fruitlss. o rfct our ow udrstadig ad to facilitat th futur dvlomt of varial las for a widr slctio of distriutios w itroducd a cosistt otatio ad itrrtd th rocdur for dvloig las withi th commo framwork of hothsis tstig [] []. h rsult rstd hr is a cosolidatio of th istig litratur. I rviwig th mthods rstd w also vrifid th mathmatical drivatios. h comltio of this foudatioal rsarch allowd us to imlmt la calculators ad tst th accurac ths las miricall. stig is cssar sic as w hav show hr th majorit of varials las ar asd o aroimatig th stadard dviatio th mloig a asio factor to th saml stadard dviatio. I th F ; Y i ; X j j t d; i i! d; 8

10 fial has of this rojct w will addrss th ractical alicatio of ASV wh distriutioal forms ar thmslvs ucrtai. RECEIVED OCOBER REVISED FEBRUARY Ackowldgmts: his work was suortd udr cotract th NASA Egirig Saft Ctr. REFERENCES [] SMIH E. P. ZAHRAN A. MAHMOUD M. AND YE K. : Evaluatio of watr qualit usig acctac samlig varials. Eviromtrics [] BAYARD R. CVIJEIC M. GRAHAM J. MARA MORALES M. AND NAFF B. 7: A Sstms Aroach to Ruwa Icursio Prvtio uulishd rort Dartmt of Sstms ad Iformatio Egirig Uivrsit of Virgiia Charlottsvill VA. hird-lac ri i th 7 FAA Airort Dsig Comtitio for Uivrsitis. [] WHIE K.P. JR. JOHNSON K.L. AND CREASEY R.R. JR. 9: Attriut acctac samlig as a tool for vrifig rquirmts usig Mot Carlo simulatio. Qualit Egirig -. [] GUENHER W. C. 977: Samlig Isctio i Statistical Qualit Cotrol. Macmilla Nw York. [5] BOWKER A. H. AND GOODE H.P. 95: Samlig Isctio Varials. McGraw-Hill: Nw York. [6] KAO J. H. K. 97: MIL-SD-: Samlig rocdurs ad tals for isctio varials for rct dfctiv. Joural of Qualit cholog 8 7. [7] LIEBERMAN G. J. AND RESNIKOff G.J. 955: Samlig las for isctio varials. Joural of th Amrica Statistical Associatio [8] MONGOMERY D. C. 5: Itroductio to Statistical Qualit Cotrol. 5th d. Wil Nw York. [9] CRAMÉR H. 95: Mathmatical Modls of Statistics. Pricto Uivrsit Prss Pricto NJ. [] AKAGI K. 97: O dsigig ukow-sigma samlig las o a wid class of o-ormal distriutios chomtrics [] WALLIS W.A. 97: Us of varials i acctac isctio rct dfctiv i Slctd chiqus of Statistical Aalsis. McGraw-Hill Nw York. [] GUENHER W. C. 97: Varials samlig las for th Poisso ad iomial. Statistica Nrladia