Acceptance Sampling Powerpoint

Acceptance Sampling Webinar20101129*

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Acceptance Sampling Webinar20101129

Acceptance Sampling Webinar20101129*Acceptance Sampling I


Acceptance Sampling Webinar20101129*What you will learn The purpose of SamplingHow to draw a statistically valid SampleHow to Develop a Sampling PlanHow to construct an O-C curve for your sampling planHow to use (and understand) ANSI/ASQ Z1.4How to use ANSI/ASQ Z1.9Assessing Inspection Economics


Acceptance Sampling Webinar20101129*What is SamplingSampling refers to the practice of evaluating (inspecting) a portion -the sample - of a lot the population for the purpose of inferring information about the lot. Statistically speaking, the properties of the sample distribution are used to infer the properties of the population (lot) distribution. An accept/reject decision is normally made based on the results of the sampleSampling is an Audit practice


Acceptance Sampling Webinar20101129*Why Sample?EconomyLess inspection laborLess timeLess handling damageProvides check on process controlFewer errors ???i.e. inspection accuracy


Acceptance Sampling Webinar20101129*What does Sampling not do?Does not provide detailed information of lot qualityDoes not provide judgment of fitness for use (of rejected items)Does not guarantee elimination of defectives any AQL permits defectives


Acceptance Sampling Webinar20101129*Sampling CaveatsSize of sample is more important than percentage of lotOnly random samples are statistically validAccess to samples does not guarantee randomnessAcceptance sampling can place focus on wrong place Supplier should provide evidence of qualityFocus should be on process controlMisuse of sampling plans can be costly and misleading.No such thing as a single representative sample


Acceptance Sampling Webinar20101129*Representative Sample?There is no such thing as a single representative sampleWhy?Draw repeated samples of 5 from a normally distributed population.Record the X-bar (mean) and s (std.dev) for each sampleWhat is the result?


Acceptance Sampling Webinar20101129*Distribution of MeansThe Distribution of Means obeys normal distribution regardless of distribution of parent population.


Acceptance Sampling Webinar20101129*Standard Error of the MeanCentral Limit TheoremThe relationship of the standard deviation of sample means to the standard deviation of the population Note: For a uniform distribution, Underestimates error by 25% with n=2, but only by 5% with n=6


Acceptance Sampling Webinar20101129*The Random SampleAt any one time, each of the remaining items in the population has an equal chance of being the next item selected One method is to use a table of Random Numbers (handout from Grant & Leavenworth)Enter the table Randomly ( like pin-the-tail-on-the-donkey)Proceed in a predetermined direction up, down, acrossDiscard numbers which cannot be applied to the sample


Acceptance Sampling Webinar20101129*Random Number TableSource: Statistical Quality Control by Grant & Leavenworth


Acceptance Sampling Webinar20101129*Stratified SamplingRandom samples are selected from a homogeneous lot. Often, the parts may not be homogeneous because they were produced on different machines, by different operators, in different plants, etc. With stratified sampling, random samples are drawn from each group of processes that are different from other groups.


Acceptance Sampling Webinar20101129*Selecting the SampleWrong way to select sampleJudgement: often leads to BiasConvenienceRight ways to select sampleRandomlySystematically: e.g. every nth unit; risk of bias occurs when selection routine matches a process pattern


Acceptance Sampling Webinar20101129*The O-C CurveOperating Characteristic CurvePaPercent DefectiveIdeal O-C Curve


Acceptance Sampling Webinar20101129*The Typical O-C Curve


Acceptance Sampling Webinar20101129*Sampling TermsAQL Acceptable Quality Level: The worst quality level that can be considered acceptable.Acceptance Number: the largest number of defective units permitted in the sample to accept a lot usually designated as Ac or cAOQ Average Outgoing Quality: The expected quality of outgoing product, after sampling, for a given value of percent defective in the incoming product. AOQ = p * Pa


Acceptance Sampling Webinar20101129*Sampling Terms (cont.)AOQL Average Outgoing Quality Level: For a given O-C curve, the maximum value of AOQ.Rejection Number smallest number of defective units in the sample which will cause the lot to be rejected usually designated as ReSample Size number of items in sample usually designated by nLot Size number of items in the lot (population) usually designated by N


Acceptance Sampling Webinar20101129*Sampling RisksProducers Risk : calling the population bad when it is good; also called Type I error

Consumers Risk : calling the population good when it is bad; also called Type II error


Acceptance Sampling Webinar20101129*Sampling Risks (cont)


Acceptance Sampling Webinar20101129*Acceptance Sampling II


Acceptance Sampling Webinar20101129*Constructing the O-C curveWe will do the following O-C curves Use Hyper-geometric and Poisson for each of the followingN=60, n=6, Ac = 2N=200, n=20, Ac = 2N=1000, n=100, Ac = 2N=1000, n=6, Ac = 2

Lets do k (Ac, c - # of successes) = 0 first


Acceptance Sampling Webinar20101129*Hyper-geometricThe number of distinct combination of n items taken r at a time is


Acceptance Sampling Webinar20101129*Hyper-geometric (cont)Construct the following Table p D=NpP(k=0)P(k=1)P(k=2)P(k 2)0 %1%2%3%etc.A Hyper-geometric calculator can be found at www.stattrek.comNote: The Hyper-geometric distribution applies when the population, N, is small compared to the sample size, however, it can always be used. Sampling is done without replacement.

= (DCk NqCn-k) / NCn


Acceptance Sampling Webinar20101129*Hypergeometric Calculator


Sheet1

N =100D=Defects in Pop.

n =10

pD=NpNq=N-NpP(k=0)P(k=1)P(k=2)P(k 2)

K012

0%010011

1%1990.90.11

2%2980.80909090910.18181818180.00909090911

3%3970.72653061220.24768089050.02504638220.999257885

4%4960.65163054910.29960025250.04596140240.997192204

5%5950.58375236690.3393909110.07021880920.9933620871

6%6940.52230474940.36868570540.09645846940.9874489242

7%7930.46674041430.38895034530.12354893320.9792396928

totalsuccessesin Popl.

Acceptance Sampling Webinar20101129*Hypergeometric CalculatorExample: p=0.02, k=0, N=100, n=10


Acceptance Sampling Webinar20101129*Hypergeometric CalculatorExample: p=0.02, k=0, N=100, n=10

P (k=0) = 0.809091P (k=1) = 0.181818P (k=2) = 0.009091-----------------------P(k2) = 1.0


Acceptance Sampling Webinar20101129*


Acceptance Sampling Webinar20101129*From QCI-CQE Primer 2005, pVI-9


Acceptance Sampling Webinar20101129*PoissonConstruct the following Table, using the Poisson Cumulative Table p npP (k 2)0%1%2%3%4%etc.Compare. When is Poisson a good approximationUse the Poisson when n/N0.1 and np 5.


Acceptance Sampling Webinar20101129*Poisson CalculatorExample: p=0.02, n=10, c=0X=k, the number of successes in the sample, i.e. c


Acceptance Sampling Webinar20101129*Poisson CalculatorExample: p=0.02, n=10, c=0Mean = np


Acceptance Sampling Webinar20101129*Poisson CalculatorExample: p=0.02, n=10, c=0TRUE for cumulative, i.e. k; FALSE for probability mass function, i.e.p(x=k)


Acceptance Sampling Webinar20101129*O-C Curve & AOQDetermine the O-C curve. Prepare the following Table using the Poisson distributionp PaAOQ = p * Pa0%1%2%3%etcGraph the results: Pa and AOQ vs p.


Acceptance Sampling Webinar20101129*OC Curve & AOQ (2)


Acceptance Sampling Webinar20101129*OC Curve & AOQ (3)


Acceptance Sampling Webinar20101129*Acceptance Sampling III


Acceptance Sampling Webinar20101129*QuestionsWhat if this AOQ is not adequate?What if you would like to add a 2nd sample when the first sample fails?

ExampleOC curve after 1st Sample: p=0.02, n=30, N=500, c (Ac)=0, Re=2OC curve after 2nd Sample (of 30 more): p=0.02, n=60, N=500, c (Ac)= 1, Re=2


Acceptance Sampling Webinar20101129*Hypergeometric Multiple Sampling

N =500500500500n =30606060p D=NpNq=N-NpP(k=0)P(k=0)P(k=1)P(k 1) K0010.0005001110.0154950.730.530.360.890.02104900.540.280.380.660.03154850.390.140.300.440.04204800.280.070.210.280.05254750.200.040.140.170.06304700.150.020.080.100.07354650.110.010.050.06


Acceptance Sampling Webinar20101129*Hypergeometric Multiple Sampling


Chart1

11

0.73296188790.8884082863

0.5354890030.6579682061

0.38992187730.4442980811

0.28296432530.2831423204

0.20463549050.1731988657

0.14746621220.1026671059

0.10588450430.0593209164

N=500, n=30, c=0

N=500, n=60, c=1

Lot defective

Prob of Acceptance

Hypergeometric Multiple Sample

HypoGeomCalc

N =1000D=Defects

n =100


K012

0%0100011

1%109900.34692771480.38936892790.7362966427

2%209800.11900191850.27015191490.28807015420.6772239876

3%309700.04034950240.13897647220.22878523610.4081112108

4%409600.01352014740.06281137010.14066977960.2170012972

5%509500.0044757910.02629724430.07486380990.1056368452

6%609400.00146347590.01044096930.03621478720.0481192323

7%709300.00047250560.00398019190.01633935740.020792055

Total "successes" in Population

HyperGeom_OC_Curves

N-->601202401000

n-->61224100

c-->0000

p

0%1111

1%10.90.8096234310.3469277148

2%0.90.80924369750.65426116420.1190019185

3%0.90.72694772820.47357964810.0403495024

4%0.80847457630.65238898690.38085419680.0135201474

5%0.72483927530.52386737840.27363703390.004475791

6%0.72483927530.46872344380.21894769390.0014634759

7%0.64854040420.41894750290.17482230620.0004725056

HyperGeom_OC_Curves

N=60, n=6, c=0

N=120, n=12, c=0

N=240, n=24, c=0

N=1000, n=100, c=0

Percent Devective

Prob. of Acceptance

Hypergeometric OC Curves for 10% Sampling Plan

Sheet1

N =500D=Defects

n =60


K012

0%050011

1%54950.52628534350.36212294280.8884082863

2%104900.27505751390.38291069220.23533052960.8932987357

3%154850.14273829750.30155978350.29167257750.7359706586

4%204800.07353665450.20960566590.27839804680.5615403671

5%254750.03760476420.13559410150.23021732350.4034161892

6%304700.01908465880.08358244710.17355529980.2762224056

7%354650.00961065130.04971026520.12250465840.1818255748

N =500500500500

n =30606060


K001

0.000500111

0.0154950.730.530.360.89

0.02104900.540.280.380.66

0.03154850.390.140.300.44

0.04204800.280.070.210.28

0.05254750.200.040.140.17

0.06304700.150.020.080.10

0.07354650.110.010.050.06

Total "successes" in Population

Sheet1

N=500, n=30, c=0

N=500, n=60, c=1

Lot defective

Prob of Acceptance

Hypergeometric Multiple Sample

PoissonCalc

0.04

100

xexp(-np)(np)^xx!Prob of exactly x defectives in sample P(r=x)"c": Prob of less than or equal to x defectives in sample P{r

Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.4-1993Mil-Std 105Sampling for Attributes; 95 page DocumentPas from 83% to 99%Information necessary: N, AQL, Inspection Level How to UseCode LettersSingle, Double, Multiple PlansSwitching RulesObtain: n, Ac, Re, O-C Curves


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.4-1993ExercisesN=475, AQL = 0.1%, Single Plan, NormalWhat is Code LetterWhat is Sample Size, What is Ac, ReRepeat for Tightened InspectionRepeat for Reduced Inspection

Note: 0.1% is 1000 ppm


Acceptance Sampling Webinar20101129*Z1.4 Code LettersI-Reduced, II-Normal, III-tightened |||| For N=475, Normal, code letter is H


Acceptance Sampling Webinar20101129*Z1.4 Single Plan Normal Insp.Table II-An=125, New code Letter K


Acceptance Sampling Webinar20101129*Z1.4 O-C Curve for Code Letter KTable X-K


Acceptance Sampling Webinar20101129*Z1.4 Switching Rules


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.4-1993

What happens when AQL = . 1% isnt good enoughAQL = 0.1% => 1000 ppm

Is Z1.4 Adequate?How would you decide?If not, what would you do?Construct O-C curve for n=1000, c=0 (Poisson). Use 100ppm < p < 5000 ppm (see slides 38 & 39)


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993Mil-Std 414Sampling for Variables; 110 page Document

Four Sections in the documentSection A: General description of PlansSection B: Plans used when variability is unknown (Std. deviation method is used)Section C: Plans used when variability is unknown (range method is used)Section D: Plans used when the variability is known.


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993Mil-Std 414Information necessary: N, AQL, Inspection LevelHow to UseCode LettersSingle or Double Limit, Std. Dev or Range Method PlansSwitching RulesObtain: Code Letter, n, Accept/Reject criteria, critical statistic (k)O-C Curves


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993Exercise (From QCI, CQE Primer, pVI-37)The specified max. temp for operation of a device is 209F. A lot of 40 is submitted for inspection. Use Normal (Level II) with AQL = 0.75%. The Std. Dev. is unknown. Use Std. Dev. Method, variation unknown Find Code Letter, Sample Size, kShould lot be accepted or rejected


Acceptance Sampling Webinar20101129*Z1.9 Code LettersFor N=40, AQL=0.75 |||||| Use AQL=1.0 & Code Letter D


Acceptance Sampling Webinar20101129*Z1.9 Finding Decision CriteriaStd. Dev method Table B-1For Code Letter D, n=5 & AQL=1, k=1.52


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993What is k

k is a critical statistic (term used in hypothesis testing). It defines the maximum area of the distribution which can be above the USL.When Qcalc > k, there is less of distribution above Qcalc than above k and lot is accepted. (Compare to Z table)Increasing (USL - X-bar) increases Pa


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993Exercise SolutionThe five reading are 197F, 188F, 184F, 205F, 201F.X-bar (mean) = 195FS (Std. Dev) = 8.8FQcalc = (USL X-bar)/s = 1.59Because Qcalc = 1.59 is greater than k=1.52, lot is accepted


Acceptance Sampling Webinar20101129*Z1.9 OC Curve for DTable A-3 (p9)


Acceptance Sampling Webinar20101129*ANSI/ASQC Z1.9-1993

Another Exercise

Same information as beforeAQL = 0.1Find Code Letter, n, kAccept or Reject Lot?


Acceptance Sampling Webinar20101129*Solution 2nd ExerciseNew code letter is E, n=7, & k=2.22

The seven reading are 197F, 188F, 184F, 205F, 201F, 193F & 197F.X-bar (mean) = 195FS (std. Dev) = 7.3FQcalc = (USL X-bar)/s = 1.91Because Qcalc = 1.91 is less than k=2.22, lot is rejected


Acceptance Sampling Webinar20101129*Inspection EconomicsAverage Total Inspection: The average number of devices inspected per lot by the defined sampling planATI = n Pa + N(1- Pa)which assumes each rejected lot is 100% inspected.Average Fraction Inspected: AFI = ATI/NAverage Outgoing Quality: AOQ = AQL (1 AFI)


Acceptance Sampling Webinar20101129*Inspection EconomicsExercise (from Grant & Leavenworth, p395)AQL = 0.5%, N=1000Which sampling plan would have least ATI. n = 100, c = 0n = 170, c = 1n = 240, c = 2


Acceptance Sampling Webinar20101129*Inspection EconomicsExercise Solution

N100010001000n100170240c012Pa0.590.80.92n Pa59136220.8N(1- Pa)41020080ATI460336300.8AFI0.4600.3360.301AOQ0.00270.00332.00349


Acceptance Sampling Webinar20101129*Inspection EconomicsComparison of Cost AlternativesNo InspectionNpD100% InspectionNCSamplingnC + (N-n)pDPa + (N-n)(1-Pa)C

D = Cost if defective passes; C = Inspection cost/item


Acceptance Sampling Webinar20101129*Inspection EconomicsSample Size Break-Even Point

nBE = D/C

D = Cost if defective passes; C = Inspection cost/item


Acceptance Sampling Webinar20101129*ResourcesAmerican Society for QualityQuality Presswww.asq.orgASQ/NC A&T partnership quality coursesCQIA, CMI, CQT, CQA, CQMgr, CQE, CSSBBQuality Progress MagazineAnd others Web-Siteswww.stattrek.com excellent basic stat sitehttp://mathworld.wolfram.com/ - greaqt math and stat site


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