Department of Mathematics, IST Probability and Statistics Unit

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1 Depatmet of Mathematics, IST Pobability ad Statistics Uit Reliability ad Quality Cotol d. Test ( Recuso) st. Semeste / Duatio: hm // 9:AM, Room V. Please justify you aswes. This test has two pages ad fou uestios. The total of poits is... Coside the followig multiple choice uestios, ad select ad justify the best possible aswe. (i) If the cotol limits based o thee stadad deviatios of the cotol statistic ae eplaced with (.) those based o two stadad deviatios: A. the esultig cotol chat is moe sesitive to the eal demads of the pocess; B. adjustmets to the pocess based o the two sigma limits may icease pocess vaiatio. Best possible aswe B. Usig two sigma limits i place of thee sigma limits esults i espodig to adom vaiatio as if it wee due to assigable causes ad leads to uecessay adjustmets ad tampeig with the pocess; this i tu leads to a icease i pocess vaiatio. (ii) Whe a X chat is used to cotol the pocess mea usig a sample size of : (.) A. the S chat should be used to cotol the pocess vaiace; B. the R chat (age chat) should be used to cotol the pocess stadad deviatio. Best possible aswe A. R loses its e ciecy as a estimato of, as the sample size iceases, theefoe the age chat should ot be used whe the sample size is as lage as. [Kelle (, p. ) metios that the age chat should ot be used fo subgoups lage tha.]. A maiteace goup impoves the e ectiveess of its epai wok by moitoig the factio p of maiteace euests that euie a secod call to complete the epai. Te weeks of data led to: i 9 Total umbe of euests ( i ) Reuests euiig a d. call (y i ) 9 (a) A appoach to dealig with vaiable sample size ( i ) is to use a stadadized cotol chat, whee (.) the poits ae plotted i stadad deviatio uits. Such a cotol chat has the cete lie at zeo, ad uppe ad lowe cotol limits of + ad, espectively. Idetify its cotol statistic whe you ae tyig to moito p ad the maiteace goup cosides a taget value p.. Does the pocess appea to be i statistical cotol? Relevat.v. ad its distibutio Y i umbe of euests euiig a d. call i the i th sample of size i, i N Y i Biomial( i,p) p P (euest euiig a d. call) Kelle, P. (). Statistical Pocess Cotol DeMystified (Had stu made easy). New Yok: MacGaw Hill. Cotol statistic of the stadadized chat fo p Judgig by the desciptio above ad the fact that estimato of p is Yi i at sample i ad that i cotol E(Y i / i ) p V (Y i / i ) p ( p ), i the cotol statistic of the stadadized chat fo p should be Y i Z i i p. p ( p ) i Checkig whethe the pocess is i statistical cotol Fo istace, takig the lagest value of y i coicidetally associated with the smallest sample size we obtai y z p p ( p ).. (.).99 [LCL, UCL] [, ]. Hece we ca add that the pocess does ot appea to be i statistical cotol. (b) Now, admit a much smalle sample size fixed ad eual to. Obtai values of the i-cotol (.) ARL of a p chat with sigma limits ad also its out-of-cotol ARL whe the factio of ocofomig items suddely shifts fom its taget value p. to. (esp..). Commet. Cotol statistic ad distibutio X N YN,NIN X N Y N Biomial(, p p + ), whee, p. ad ( < < p ) epesets the magitude of the shift i p. Cotol limits of the p chat ( ) p ( p ) LCL max,p ( ). (.) max,. max{,.} p ( p ) UCL p +. (.). +. Pobabilities of tiggeig a sigal Sice X N ad LCL, we get:

2 ( ) P (X N [LCL, UCL] ) P ( X N > UCL ) F Biomial(,pp+ )( UCL) F Biomial(,p.+ ) (.) F Biomial(,p.+ ) () >< >:.99.9, p p. ( ).99., p. (.).99., p. (.). Ru legth ad euested ARL We ae dealig with a Shewhat chat, thus, the umbe of samples collected util the chat tigges a sigal give, RL( ), has the followig distibutio: Thus, RL( ) Geometic ( ( )). ARL( ) ( ) ><.9 '., p p. (, i-cotol). >: ' 9., p. (., out-of-cotol, upwad shift). '., p. (., out-of-cotol, dowwad shift). Commet I this case we have ARL(.) > ARL(), that is, we deal with a[ uppe oe-sided] p chat, whose ARL i the pesece of a dowwad shift is ueasoably lage tha the i-cotol ARL. (c) What is the miimum sample size that guaatees a positive lowe cotol limit? List oe (.) coseuece of such a limit i the pefomace of the p chat. Obtaiig a positive lowe cotol limit Give that p., : LCL > p ( p ) p > > ( p ) p (.) >. >9, ad the miimum sample size that guaatees a positive lowe cotol limit is 9. Impotace of a positive lowe cotol limit Whe dealig with a p chat it is essetial to have a positive lowe cotol limit i ode to detect i a faily uick fashio a decease i the factio of ocofomig items (i.e., uality impovemet), amely with ARL( < ) < ARL().. I 9, A.A. Michelso measued the speed of light i ai usig a modificatio of a method poposed by the Fech physicist J.B.L. Foucault. (a) Te of these idividual measuemets, epoted i kilometes pe secod ad with 99 km/s (.) subtacted fom it, ae: Measuemet A statisticia pefomed a Adeso Dalig goodess-of-fit test usig Mathematica ad got a p value of.. Is thee ay evidece that the measuemets above ae omally distibuted? Result of the goodess-of-fit test Recall that the p value is the lagest sigificace level leadig to the o ejectio of the ull hypothesis. Thus, fo these paticula data set ad ull hypothesis: we should ot eject H : T Nomal(, ),, > fo ay sigificace levels apple p value., amely the usual sigificace levels (%, %, %); we should eject H fo ay sigificace levels >p value.. The family of omal distibutios, {Nomal(, ):, > }, seems to be vey easoable i light of the data set. (b) Suppose you decide to collect samples of speed measuemets ad to opeate a stadad X chat ad a uppe oe-sided S chat to detect shifts fom (, ) to( + / p, ), R ad. (i) Detemie the pobability that a sigal is tiggeed by the stadad X chat with (.) ARL (, ) samples, whe. Recalculate this pobability fo the uppe oe-sided S. chat with the same i-cotol ARL. Commet. Quality chaacteistic X speed measuemet X Nomal(, ) Cotol statistics X N mea of the N th adom sample of size S N vaiace of the N th adom sample of size Distibutios X N Nomal + p, ( ),whee epesets the magitude of a shift i (esp. a upwad shift i ). ( )S N ( ) ( ). Cotol limits LCL p UCL + p LCL UCL / p apple (esp. The Adeso-Dalig test is a statistical test of whethe a give sample of data is daw fom a give pobability distibutio; whe applied to testig if a omal distibutio adeuately descibes a set of data, it is oe of the most poweful statistical tools fo detectig most depatues fom omality ( test). )

3 Pobability of tiggeig a sigal Takig ito accout the distibutio of the cotol statistics, the X sigal with pobability (, ) P LCL apple X N apple UCL,... apple ad the uppe oe-sided S chat with pobability () P S N [LCL,UCL ], R,, chat tigges a apple ( ) UCL F ( ) F ( ),. Ru legths ad ARL We ae dealig with Shewhat chats theefoe the umbe of samples collected util the X ad S chats tigge a sigal, give ad, ae such that: RL (, ) Geometic( (, )); ARL (, ) (, ) ; RL () Geometic ( ()) ; ARL () (). Obtaiig ad The costat is such that the i-cotol ARL, ARL (, ), is eual to samples, that is, : (, ) ARL (, ) Similaly, : [ ( ) ( )] apple apple (.99).. ARL (, ) ARL (, ) ARL (, ) ARL () () F ( ) ( ) ARL () F apple ( ) ARL () F ( ) F (.99) (). Pobabilities of tiggeig a valid sigal whe ad apple (, ) apple h A ' [ (.)] (.9). () F ( ). F ( ) ' F () (.) Commet Fo ad., the pobabilities coicide, that is, the RL of the X ad S chats have the same geometic distibutio with paamete.. [Let us emid the eade that the distibutio of X chat also depeds o, theefoe this chat is also sesitive to shifts i spead. Moeove, ou umeical ivestigatios lead us to coclude that, fo, RL (, ) RL (, ),. Ivestigate!] (ii) Compute the out-of-cotol ARL ad SDRL of you joit scheme fo ad i the pesece (.) of the shift metioed i (b)(i). RL of the joit scheme Accodig to Sectio. of the lectue otes: st RL, (, ) mi{rl (, ),RL ()} Geometic (, (, )), (, ) (, )+ () (, ) (). Reuested out-of-cotol ARL ad SDRL Fo ad.,, (, ) (b)(i) ARL, (, ), (, )

4 SDRL, (, ).9 '. Table 9. p, (, ), (, ) p.9.9 ' 9.. (c) Assume a shift fom (, ) to( + / p, ) occued. I this case the (.) pobability that a abitay X chat (esp. S chat) detects such a shift is epeseted by (, ) (esp. ()). Pove that P [RL (, ) RL ()] (,) () (,)+ () (,) (). Itepet this esult. Ru legths [Fo R ad,] RL (, ) Geometic( (, )) RL () Geometic( ()). To pove P [RL (, ) RL ()] (,) () (,)+ () (,) () Poof If we apply the total pobability law ad capitalize o the idepedece betwee X ad S, thus, of those two RL, we successively get P [RL (, ) RL ()] +X i +X i +X i P [RL (, ) RL () RL () i] P [RL () i] P [RL (, ) i] P [RL () i] [ (, )] i (, ) [ ()] i () (, ) () Itepetatio Sice the joit scheme fo ad the esult, (, ) (, )+ () P [RL (, ) RL ()] This is temed the pobability of a simultaeous sigal. +X i {[ (, )] [ ()]} i (, (). (, ) () [ (, )] [ ()] (, ) () )+ () (, ) QED tigges a sigal with pobability (, ) (), (, ) () (, )+ () (, ) (), ca be obviously itepeted as a coditioal pobability it coespods to the pobability that the chat fo ad both tigge a sigal, give that the joit scheme was esposible fo a alam.. The desity of a plastic pat used i a cellula telephoe is euied to be at most.g/cm. (a) Admit a samplig pla by vaiables is adopted with kow stadad deviatio. Set such a (.) pla with isk poits (p, ) (%,.9) ad (p, ) (%,.). Samplig pla by vaiables with kow stadad deviatio (sample size) k (acceptace costat) (kow stadad deviatio) U (uppe specificatio limit) Poduce s ad cosume s isk poits (p, ) (%,.9) (p, ) (%,.) Obtaiig ad k Accodig to (.), h i < ( ) ( ) (,k ) : (p ) (p ) : k (p ) ( ) (p ) ( ) <. ( ) ( ) i h (.9) (.) (.) (.) : k (.) (.9) (.) (.). (.) (.9) h <. (.9) (.) (.)i. : k (.). (.) (.9) (.9)... We should take d.e ad k.. I fact p P a (p ) k (p ) p [. (.)] P a (p ) ' (.).99.9 p k (p ) p [. (.)] ' (.).9. apple.. (b) Admit a statisticia suggested the adoptio of a samplig pla by vaiables with ukow (.) stadad deviatio. Use the appopiate fomulae to obtai s ad k s ad veify that whe

5 s 9 ad k s. the appoximate values of P a (p )(esp.p a (p )) is lage (esp. smalle) tha o eual to (esp. ). Commet o these values of s ad k s i light of (a). Note: I case you did ot solve (a), coside ad k.. Sigle samplig pla by vaiables with ukow stadad deviatio s (sample size) k s (acceptace costat) Obtaiig s ad k s Capitalizig o (.) ad o the fact that we obtai u (k ) + (. ) + '.9 v k. ' 9. s + u + p u + v p '.9 s k s s k.9 '.9. ' apple.. Commet Admittig that the stadad deviatio is ukow is moe ealistic: it does ot chage the acceptace costat sigificatly (i this execise k s ' k dow to the fist decimal place); but it euies the collectio of a much lage sample (i this case s. ). (c) Suppose that a sample of the appopiate size was take fom a lot, ad x. ad s (.).. Should the lot be accepted o ejected? Checkig whethe o ot the lot should be accepted The lot should be accepted i Q U x s k s, whee Q is the uality idex, U is the uppe specificatio limit, x ad s epeset the mea of a sample with size s, ad k s the acceptace costat. Fo this sample, we have Q....., theefoe we should eject the lot. If we coside s 9 ad k s. the P a (p ) (.9) ' ( p ) (.) ' P a (p ) ' (.9).9 ( p ) k s s s + sk s s s ' (.)