ISyE 512 Review for Midterm 1

Size: px

Start display at page:

Download "ISyE 512 Review for Midterm 1"

Chester Cummings
5 years ago
Views:

1 ISyE 5 Review for Midterm Istructor: Prof. Kaibo Liu Departmet of Idustrial ad Systems Egieerig UW-Madiso kliu8@wisc.edu Office: Room 307 (Mechaical Egieerig Buildig) ISyE 5 Istructor: Kaibo Liu

2 Outlie About the eam Checklist Problems review ISyE 5 Istructor: Kaibo Liu

3 Whe Whe & Where /3 (Thursday) :30PM 3:45PM Where 63 MECH ENGR Buildig (last ame s iitial letter from A to J ) 540 Egieerig Hall Buildig (last ame s iitial letter from K to Z ) Please take a seat where the paper is distributed o ISyE 5 Istructor: Kaibo Liu 3

4 Eam Eam covers all the materials i the lecture util (icludig) Chapter 0 SPC with Autocorrelated Process Data. All homework problems ad eamples i lecture otes are importat. Thigs Allowed Oe-page (double-sided) cheat sheet Calculator Tables eeded i calculatio will be provided ISyE 5 Istructor: Kaibo Liu 4

5 5 problems totally Eam Problems st: Multiple choice questio d: Hypothesis testig, cofidece iterval, p value 3rd: Cotrol chart for variables 4th: Cotrol chart for attributes 5th: Type I ad type II calculatio uder differet decisio rules ISyE 5 Istructor: Kaibo Liu 5

6 Outlie About the eam Checklist Problems review ISyE 5 Istructor: Kaibo Liu 6

7 Distributios. Discrete Probability Distributio Hypergeometric distributio Biomial distributio Poisso Distributio. Cotiuous Probability Distributio Normal distributio Chi-Square distributio Studet t distributio ISyE 5 Istructor: Kaibo Liu 7

8 Useful Results o Mea ad Variace If is a radom variable ad a is a costat, the E(a+)=a+E() E(a*)=aE() V(a+)=V() V(a*)=a V() If,,, are radom variables, E( + + )=E( )+ +E( ) If they are mutually idepedet, ad a,,a are costats V(a + + a )=a V( )+ +a V( ) ISyE 5 Istructor: Kaibo Liu 8

9 INTERRELATIONSHIPS BETWEEN DISTRIBUTIONS Hypergeometric, Biomial, Poisso, Normal Samplig without replacemet i fiite populatio Hypergeometric fiite populatio if /N0. N: populatio size :sample size The sum of a sequece of Beroulli trials i ifiite populatio with probability of success p Number of defects per uit Poisso if 5 p=d/n, Biomial if large, small p <0., or large, large p > 0.9, p =-p Normal =, = If p>0 ad 0. p 0.9 =p, =p(-p) a 0.5 p a 0.5 p Pr( a) p( p) p( p) b 0.5 p a 0.5 p Pr( a b) p( p) p( p) Pr( pˆ ) p p( p) / p p( p) / ISyE 5 Istructor: Kaibo Liu 9

10 Iferece About Process Quality Estimatio poit estimatio iterval estimatio Hypothesis Testig Defiitio Testig o mea kow ad Ukow variace Testig o Variace ISyE 5 Istructor: Kaibo Liu 0

11 The Use of P-Values i Hypothesis Testig P-Value: the smallest level of sigificace that would lead to rejectio of the ull hypothesis if the predefied >P= mi, reject the ull hypothesis ISyE 5 Istructor: Kaibo Liu

12 Iferece o the MEAN of a Normal Populatio Variace Kow Assumig Kow Populatio Variace / ~ N(0,) Z Z / / pvalue H : 0 0 H Reject H 0 : 0 if Reject H 0 : < 0 if 0 > Z(/) / 0 < -Z() / 0 ( ) / 0 ( ) / Reject H 0 : > 0 if 0 / > Z() ( ) / 0 ISyE 5 Istructor: Kaibo Liu

13 Iferece o the MEAN of a Normal Populatio Variace Ukow Assumig Ukow Populatio Variace s/ H : 0 ~ t (-) H 0 Reject H 0 : 0 if Reject H 0 : < 0 if s t/ t/ 0 s/ 0 s/ > t(/, -) <- t(-) s pvalue CDF 0 t( ) ( ) s/ t ( 0 s / ) Reject H 0 : > 0 if 0 s/ > t(-) ( ) s / 0 t ISyE 5 Istructor: Kaibo Liu 3

14 Iterrelatioships betwee statistical ifereces Z Assumig Kow Populatio / Variace Z / Assumig Kow Populatio Variace ISyE 5 Istructor: Kaibo Liu / ~ N(0,) Reject H 0 : 0 if 0 Reject H 0 : < 0 Reject H 0 : > 0 / ~ N(0,) > Z(/) Reject / H 0 : 0 if if 0 / 0 < -Z() if > Z() Reject / H 0 : < 0 if P=[-( Z 0 0 )] with 0 > Z(/) / 0 < -Z() / two-sided H, i.e., H: 4

15 ISyE 5 Istructor: Kaibo Liu Iferece o the Differece i Meas of Two Populatios Variace Kow / / - - Z Z Assume Kow Populatio Variaces (0,) ~ - N Reject 0 : H if / - Z Reject 0 : H if Z - Reject 0 : H if Z - 0 : H H pvalue ) - ( ) - ( - [ ( )] Observatios i TWO samples are all i.i.d. 5

16 ISyE 5 Istructor: Kaibo Liu a) Assume Homogeeity ( ) ) ( ~ - t S p where ) ( ) ( s s S p Reject 0 : H if ), / ( - t S p Reject 0 : H if ), ( - t S p Reject 0 : H if ), ( - t S p, /, / - - p p S t S t ; 0 : H H Iferece o the Differece i Meas of Two Populatios Variace Ukow ( ) [ ( )] p t S pvalue ) ( ) ( S t p ) ( ) ( S t p i.i.d. Equal variace 6

17 ISyE 5 Istructor: Kaibo Liu b) Assume Heterogeeity ( ) ) ( ~ - v t s s where ) / ) / ) / / ( ( ( v s s s s Reject 0 : H if ), / ( - v t s s Reject 0 : H if ), ( - v t s s Reject 0 : H if ), ( - v t s s Iferece o the Differece i Meas of Two Populatios Variace UNKow 0 : H H 7

18 Iferece o the Variace of a Normal Distributio ( )s ~ ( ) ( ) s /, /, H0 : 0 H Reject H : if 0 o ( - ) s o > (/, - ) or ( - ) s o < ( - /, - ) Reject H 0 : < o if ( - ) s o < (-, - ) Reject H 0 : > o if ( - ) s o > (, - ) C.I. ( ) S ( ) S, Pr{ /, /, /, } / ISyE 5 Istructor: Kaibo Liu 8

19 S S / / ~ F, Iferece o the Variaces of Two Normal Distributios With H 0 : = for H : Reject H 0 for H : < Reject H 0 for H : > Reject H 0 C.I. S S ISyE 5 Istructor: Kaibo Liu if if if s s s s s s > F (/,, ) or > F (,, ) > F (,, ) S s s F /,, /,, /,, /,, /, F F F S < F ( /,, ) The two d.f. are echaged 9

20 Type I error ( producer s risk): Two Types of Hypothesis Test Errors = P{type I error} = P{reject H 0 H 0 is true} =P{coclude bad although actually good}= P{coclude statistically out of cotrol although the process is truly i cotrol} Type II error (cosumer s risk): = P{type II error} = P{fail to reject H 0 H 0 is false} =P{coclude good although actually bad} = P{coclude statistically i cotrol although the process is truly out of cotrol} Power of the test: Power = - = P{reject H 0 H 0 is false} Reality : You Coclude " H " 0 is True " H " is True " H 0 is True " C o f i d e c e P r o d u c e r E r r o r, " H is True " C o s u m e r E r r o r, P o w e r / LCL f() 0 UCL / ISyE 5 Istructor: Kaibo Liu 0

21 Geeral Model for a Cotrol Chart Let w be a sample statistic that measures some quality characteristics of iterest, ad suppose that the mea of w is w ad the stadard deviatio of w is w. The the ceter lie, the upper cotrol limit, ad the lower cotrol limit become UCL = w + L w Ceter lie = w LCL = w - L w 3 sigma cotrol limits: Actio limits: L=3 (p=0.007) Warig limits: L= (p=0.0455) where L is the "distace" of the cotrol limits from the ceter lie, epressed i stadard deviatio uits Probability limits (Wester Europe): Actio limits: 0.00 limits (p=0.00) Warig limits: 0.05 limits (p=0.050) ISyE 5 Istructor: Kaibo Liu

22 Average Ru Legth (ARL)-i cotrol ARL: The average umber of poits that must be plotted util a poit idicates a out-of-cotrol coditio. If the process is i-cotrol, The followig table illustrates the possible sequeces leadig to a "out of cotrol" sigal: Ru legth Probability ( ) ARL0 ARLicotrol 3 ( ) : : : k ( ) k Eample: ARL i-cotrol = /= /0.007 = 370. Eve the process is i cotrol, a out-of-cotrol sigal will be geerated every 370 samples o the average. If the process i actually i-cotrol, we wish to see as may icotrol samples as possible => less false alarm ISyE 5 Istructor: Kaibo Liu

23 Average Ru Legth (ARL) Out of Cotrol If the process is actually out-of-cotrol, ad the probability that the shift will be detected o the first sample is - the secod sample is (-) the rth sample is r- (-) The epected umber of samples take util the shift is detected is ARL ( ) ARLout of cotrol r r r Power: p(correctly detect o.o.c) power Remark: we wat to be large. Thus, the "out of cotrol" coditio ca be quickly detected. If the process i actually out-of-cotrol, we wish to see out-of-cotrol samples ASAP=>fewer false detectio ISyE 5 Istructor: Kaibo Liu 3

24 Sample Size ad Samplig Frequecy Larger sample size easier to detect small mea shift Average ru legth ad average time to sigal (ATS=ARL*samplig iterval h) are cosidered i desig ad the check the detectio power. ISyE 5 Istructor: Kaibo Liu 4

25 Cotrol Chart for X ad R Kow, Statistical Basis of the Charts suppose { ij, i=,,m, j=,,} are ormally distributed with ij,~n(, ), thus, ~ N(,( / ) ) X i X bar chart moitors betwee-sample variability (variability over time) ad R chart measures withi-sample variability (istataeous variability at a give time) If ad are kow, X bar chart is (if k=3) 3 3 A LCL A CL ULC A A 3 ISyE 5 Istructor: Kaibo Liu 5

26 Cotrol Chart for X ad R Kow, (Cot s) Rage R i =ma( ij )-mi( ij ) for j=,.. If ad are kow, the statistical basis of R charts is as follows: Defie the relative rage W=R/. The parameters of the distributio of W are a fuctio of the sample size. Deote W =E(W)=d, W =d 3, (d ad d 3, are give i Appedi Table VI of Tetbook P75) R =d, R =d 3, which are obtaied based o R=W R chart cotrol limits d 3d (d 3d ) R 3 R 3 3 Chart Statistic LCL D CL d ULC D D D d d 3d 3d 3 3 ISyE 5 Istructor: Kaibo Liu 6

27 ISyE 5 Istructor: Kaibo Liu Need to estimate ad X bar chart Cotrol Chart for ad R Ukow ad 7 X m R R ; d R ˆ ; m m X ˆ m i i m i j ij m i i R A d R / 3 ˆ 3 3ˆ ˆ R A ULC CL R A LCL d 3 A A is determied by, Appedi Table VI

28 ISyE 5 Istructor: Kaibo Liu Cotrol Chart for ad R Ukow ad (cot s) X 3 3 R m i i R d R d ˆ d ˆ ; m R R ˆ Need to estimate based o R=W, W =d 3, R chart (if k=3) R, R d R ˆ )R d d 3 ( d R d 3 R 3ˆ ˆ 3 3 R R R D ULC R CL R D LCL d 3d D d 3d D D 3 ad D 4 are determied by, Appedi Table VI 8

29 Summary of Cotrol Charts (if k=3) Process Parameters X bar chart R chart S chart kow LCL A CL ULC A LCL D CL d ULC D LCL B CL c 4 5 ULC B 6 X bar & R chart ˆ X R ˆ d LCL A CL ULC A R R LCL D R CL R 3 ULC D R 4 X bar & S chart ˆ X ˆ S c 4 ; LCL A CL ULC A 3 3 S S LCL B CL S 3 ULC B 4 S S ISyE 5 Istructor: Kaibo Liu 9

30 OC Curve for bar ad R Chart X bar chart 0 UCL k / ; LCL k / ; 0 0 ( k) i cotrol d 0, P{ LCL UCL d} 0 Pr{ UCL } Pr{ LCL } k d k d out of cotrol The epected umber of samples take before the shift is detected (the process is o.o.c) ARL outofcotrol r r r ( ) If process is i cotrol: ARL is the epected umber of samples util a false alarm occurs ARL icotrol i k k ISyE 5 Istructor: Kaibo Liu

31 Wester Electric Rules (Zoe Rules for Cotrol Charts) Ehace the sesitivity of cotrol charts for detectig a small shift or other oradom patters. Oe or more poits outside 3-sigma limits.. Two of three cosecutive poits outside -sigma limits 3. Four of five cosecutive poits beyod the -sigma limits 4. A ru of eight cosecutive poits o oe side of the ceter lie. The rules above apply to oe side of the ceter lie at a time. Other sesitizig rules for Shewhart cotrol chart; Table 5-, P05 The overall type I error: the process is declared out of cotrol if ay oe of the rules is applied: k ( ) i is the type I error of usig oe rule i aloe i if all k rules are idepedet. i ISyE 5 Istructor: Kaibo Liu 3

32 Procedures for Establishmet of Cotrol Limits Ukow ad If ad are ukow, we eed to estimate ad based o the prelimiary i-cotrol data (ormally m=0~5). The cotrol limits established usig the prelimiary data are called trial cotrol limits, which are used to check whether the prelimiary data are i cotrol. First check R or S chart to esure all data i-cotrol, ad the check X-bar chart Collect Prelimiary Data X Estimate R or S Establish Trial Cotrol Limits Check Prelimiary Data I-cotrol Future Moitorig Update Estimatio Elimiate the Outliers due to Assigable Causes Out-of-cotrol ISyE 5 Istructor: Kaibo Liu 3

33 Estimatio of the Process Capability Get process specificatio limits (USL, LSL) Estimate based o ˆ R / (R chart) or ˆ S / c (S chart) d 4 Estimate the fractio of ocoformig products p (or p0 6 PPM) LSL USL pˆ Pr{ LSL} Pr{ USL} ( ) ( ) ˆ ˆ Process-Capability Ratio C p USL - LSL ; ˆ USL - LSL = C = ; p 6s 6ˆ s Estimated process stdev ISyE 5 Istructor: Kaibo Liu 33

34 Differeces amog NTL, CL ad SL ad Impact o Process Capability There is o relatioship betwee cotrol limits ad specificatio limits. C p is a ide relatig atural tolerace limits to specificatio limits. Eterally determied NTL: atural tolerace limits 3 Ceter lie o bar Distributio of bar values Distributio of idividual process measuremet LSL LNTL 3 LNTL 3 LSL UNTL 3 UNTL 3 USL USL C p >, P<00% C p =, P=00% ISyE 5 Istructor: Kaibo Liu LNTL LSL 3 3 USL Eterally determied Width defied by NTL s is larger tha that defied by CL, why? C p <, P>00% UNTL 34

35 How to Establish a p-chart? m=0-5 samples for costructig trial cotrol limits If p ukow, coduct a test ad trial cotrol limits with pˆ p m D i p m E(p) p i m i m pˆ i UCL p 3 Ceterlie LCL p 3 p( p) p p( p) Trial Cotrol Limits Is there a assigable cause for out-of-cotrol poits or a oradom patter? If so, fid the root causes ad delete these poits, ad the update cotrol limits. Whe a poit is ON a cotrol limit, it is cosidered as either out-ofcotrol or i-cotrol depedig o how the problem asks ISyE 5 Istructor: Kaibo Liu

36 p Cotrol Chart (The umber of ocoformig items) Rather tha plottig the fractio ocoformig, we plot the umber of ocoformig items with a p Chart : UCL X = p + 3 p( p) Ceter lie = p LCL X = p 3 p( p) If LCL X <0, set LCL X =0 Np ad p cotrol charts ca be trasferred to each other. From p to p, multiple to the UCL, CL, ad LCL; From p to p, divide to the UCL, CL, ad LCL. ISyE 5 Istructor: Kaibo Liu 36

37 Cotrol Charts for Nocoformities - c Chart Cotrol limits for the c chart with a kow c (kow mea ad variace) UCL c 3 c CL c If LCL<0, set LCL=0 LCL c 3 c If ukow c, c is estimated from prelimiary samples of ispectio uits for costructig trial cotrol limits m Nocoformities UCL c 3 c ci i total # of defects i all samples CL c ĉ c m umber of samples LCL c 3 c The prelimiary samples are eamied by the cotrol chart usig the trial cotrol limits for checkig out-of-cotrol poits Whe a poit is ON a cotrol limit, it is cosidered as O.O.C ISyE 5 Istructor: Kaibo Liu 37

38 Cotrol Charts for Nocoformities Per Uit - u Chart c: total ocoformities i a sample of ispectio uits ( is ot ecessary be iteger) u: average # of ocoformities per ispectio uit i a sample m u u LCL u 3 c i i i u ; u CL u i i m u UCL u 3 If ukow u, is estimated from prelimiary samples of ispectio uits for costructig trial cotrol limits u ISyE 5 Istructor: Kaibo Liu 38

39 Outlie About the eam Checklist Problems review ISyE 5 Istructor: Kaibo Liu 39

40 Hypothesis testig Problem 4 i HW The iside diameters of bearigs used i a aircraft ladig gear assembly are kow to have a stadard deviatio of 0.00 cm. A radom sample of 5 bearigs has a average iside diameter of cm. (a) Test the hypothesis that the mea iside bearig diameter is 8.5 cm. Use a two-sided alterative ad α=0.05. (b) Calculate the p-value of the test i (a). (c) Costruct a 95% two-sided cofidece iterval o mea bearig diameter. ISyE 5 Istructor: Kaibo Liu

41 Hypothesis testig Solutio: (a) H 0 : u = 8.5 vs. H : u 8.5. Reject H 0 if Z 0 > Z α/ (b) Z value 6.78 is out of the rage i the table, therefore, P-value is close to 0, < (c) ISyE 5 Istructor: Kaibo Liu

42 Problem 5 i HW Cotrol chart for variables ISyE 5 Istructor: Kaibo Liu 4

43 Cotrol chart for variables ISyE 5 Istructor: Kaibo Liu 43

44 Cotrol chart for attributes Problem i HW 3 Samples o the cotrol limits are regarded out-of-cotrol. ISyE 5 Istructor: Kaibo Liu

45 Cotrol chart for attributes 00, p 0.08, UCL 0.6, LCL 0 (a) p 00(0.08) 8 UCL p 3 LCL p 3 p( p( p) 8 3 8( 0.08) 6.4 p) 8 3 8( 0.08) (b) p = 0.08 < 0. ad = 00 is large, so use Poisso approimatio to the biomial. Pr (type I error) Pr{ pˆ LCL p} Pr{ pˆ UCL p} Pr{ D LCL } Pr{ D UCL } Pr{ D 0 8} Pr{ D 6.4 8} POISSON(0,8, true) POISSON(6,8, true) (c) p ew = 00(0.) = 0 > 0 ad 0. p 0. 9, so use the ormal approimatio to the biomial. ew If the studets use p-chart cotrol limits: [Pr{ pˆ UCL p} Pr{ pˆ LCL p}] ( 0.) 00 0.( 0.) 00 ( 0.975) ( 5) UCL p p( p) LCL p p( p) (d) Pr(detect shift by at most 4 th sample) = Pr(ot detect by first four samples) = (0.648) 4 ISyE 5 Istructor: Kaibo Liu =

46 Problem 4 i HW Type I ad Type II error calculatio uder differet decisio rules What is the Type-I error probability for each of these rules. If the mea of the quality characteristic shifts oe process stadard deviatio ( ), ad remais there durig the collectio of the et seve samples, what is the Type-II error probability associated with each decisio rule? ISyE 5 Istructor: Kaibo Liu

47 Type I ad Type II error calculatio uder differet decisio rules (a) Rule : Pr{out of cotroli cotrol} Pr{out of 7 beyod} Pr{ out of 7 beyod} Pr{7 out of 7 beyod} Pr{0 out of 7 beyod} (0.007) ( 0.007) Rule : Pr{out of cotroli cotrol} Pr{all 7 o oe side} (0.5) ISyE 5 Istructor: Kaibo Liu

48 Type I ad Type II error calculatio uder differet decisio rules (b) Rule : Sice ormal distributio is symmetric, the error should be the same o matter the mea shifts up or dow. Discussio o is eough. 3 / 3 / 0 Pr{sample withi out of cotrol} ( ) ( ) / / (3 5) ( 3 5) (0.76) (If, we have the same 0.) Pr{i cotrol out of cotrol} Pr{7 withi} ( ) Rule : Agai we oly discuss. Pr{fall i upper side} 0 ( ) ( 5) / (.3) Pr{fall i lower side} ( ) ( 5) / (.3) Pr{i cotrol out of cotrol} Pr{all 7 o oe side} ( ) ISyE 5 Istructor: Kaibo Liu

49 ISyE 5 Istructor: Kaibo Liu 49

Statistical Fundamentals and Control Charts

Statistical Fundamentals and Control Charts Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,