Statistical Thinking Session 2 of 4 10:15 to Noon

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1 Statistical Thinking Session 2 of 4 10:15 to Noon ACCEPTANCE SAMPLING 2014 ConteSolutions

2 Acceptance Sampling Notebook pages 31 to 40 SAMPLING CONCEPTS BY JOHN A. CONTE, P.E.

3 The ASQ CQE BoK IV Product and Process Control C. Acceptance Sampling 1. Sampling Concepts Define, describe, and apply the concepts of producer and consumer risk and related terms, including operating characteristic (OC) curves, acceptable quality limit (AQL), lot tolerance percent defective (LTPD), average outgoing quality (AOQ), average outgoing quality limit (AOQL), etc. (Analyze)

4 The ASQ CQE BoK IV Product and Process Control C. Acceptance Sampling 2. Sampling standards and plans Interpret and apply ANSI/ASQ Z1.4 and Z1.9 standards for attributes and variables sampling. Identify and distinguish between single, double, multiple, sequential, and continuous sampling methods. Identify characteristics of Dodge-Romig sampling tables and when they should be used. (Analyze)

5 Acceptance Sampling Inspection of Lots of either incoming raw materials or outgoing product Decision to accept or reject based on sample All sampling plans have two components The sample size (n) The maximum number of defectives allowed in the sample (c or Ac) for acceptance of the lot

6 Producer s Risk The risk of making an incorrect decision by rejecting a good lot For sampling plans in Z1.4 that risk is set at about 5% for a given AQL AQL is the Acceptable Quality Limit

7 Consumer s Risk The probability of making an incorrect decision by accepting a bad lot For Dodge-Romig sampling plans that risk is set at about 10% for a given LTPD LTPD - Lot Tolerance Percent Defective

8 Acceptance Sampling Risks (page 32)

9 Operating Characteristic Curve First text given to me as a new engineer at Western Electric Company in 1966 Sampling Inspection Tables by Harold Dodge and Harry Romig 115 out of 200 pages devoted to OC curves

11 The Operating Characteristic Curve Allows a sampling plan to be almost completely evaluated at a glace Gives a pictorial view of the probabilities of accepting lots submitted a varying levels of percent nonconforming. The OC Curve illustrate the risks of acceptance sampling

12 The Operating Characteristic Curve Curves are based either the Binomial or Poisson distribution Poisson approximation to the binomial was often used as the binomial computations were quite complex before the age of electronic calculators

13 Probability of Acceptance (P a ) Formula or statistical tables Most Poisson tables are cumulative and provide the probability for c or fewer given np (λ) P a = x = c ( np) x x! x= 0 e np

OC Curve with Poisson Probabilities (λ=0.8, c 3) Percent defective.02.04.06.08.10.12.14 np or λ 40(.02)=.80 40(.04)=1.6 40(.06)=2.4 40(.08)=3.

14 OC Curve with Poisson Probabilities (λ=0.8, c 3) Percent defective np or λ 40(.02)=.80 40(.04)=1.6 40(.06)=2.4 40(.08)=3.2 40(.10)=4.0 40(.12)=4.8 40(.14)=5.6 P(Acceptance) = P(finding 3 or fewer defectives in the sample)

15 OC Curve with Poisson Probabilities (λ=2.4, 3.2, 4.0, 4.8, 5.6, c 3) Percent defective np or λ P(Acceptance)= P(finding 3 or fewer defectives in the sample) (.02)=.80 40(.04)=1.6 40(.06)=2.4 40(.08)=3.2 40(.10)=4.0 40(.12)=4.8 40(.14)=

16 Draw the OC Curve using probabilities 1.0 Operating Characteristic (OC) Curve Sample Size = 40, Acceptance Number = 3 Probability of Acceptance Lot Percent Defective

17 Acceptable Quality Level (AQL) From Z1.4 Acceptance Quality Limit (AQL) 4.2 Definition: The AQL is the quality level that is the worst tolerable process average average whena continuing series of lots is submitted for acceptance sample. Note: The use of the abbreviation AQL to mean Acceptable Quality level is no longer recommended.

18 Acceptable Quality Level From Z1.4 - Although individual lots with quality as bad as the AQL can be accepted with a fairly high probability, the designation of a n AQL does not suggest that this is necessary a desirable quality level. The AQL is a parameter of the sampling scheme The probability of acceptance of a lot with a process average equal to the AQL is normally set at 0.95

19 Minitab Software Create or Compare Sampling Plan Stat Quality Tools Acceptance Sampling by Attributes Default Screen Create based on input of AQL, LTPD, and N Or Compare based on input of n, c, and N Measurement type percent defective Acceptable Quality Level (AQL) 0.05 Rejectable Quality Level (RQL or LTPD) 0.10

20 Minitab Example Compare Input of n, c, and N Probability of Acceptance Operating Characteristic (OC) Curve Lot Percent Defectiv e AOQ (Percent Defective) Average Total Inspection A verage Outgoing Quality (A OQ) Curve Incoming Lot Percent Defectiv e A verage Total Inspection (A TI) Curve Lot Percent Defectiv e Sample Size = 40, Acceptance Number = 3

21 Mintab Example Create Plan Acceptance Sampling by Attributes Measurement type: Go/no go Lot quality in percent defective Lot size: 1000 Use binomial distribution to calculate probability of acceptance Acceptable Quality Level (AQL) 1 Producer's Risk (Alpha) 0.05 Rejectable Quality Level (RQL or LTPD) 5 Consumer's Risk (Beta) 0.1 Generated Plan(s) Sample Size 132 Acceptance Number 3 Accept lot if defective items in 132 sampled <= 3; Otherwise reject. Percent Probability Probability Defective Accepting Rejecting AOQ ATI Average outgoing quality limit (AOQL) = at percent defective. Graphs - Acceptance Sampling by Attributes

22 Minitab Create (AQL=1%, LTPD=5%) Operating Characteristic (OC) Curve A verage Outgoing Quality (A OQ) Curve Probability of Acceptance AOQ (Percent Defective) Average Total Inspection Incoming Lot Percent Defectiv e A verage Total Inspection (A TI) Curve Lot Percent Defectiv e Lot Percent Defectiv e 10.0 Sample Size = 132, Acceptance Number = 3

23 AOQ and AOQL Average Outgoing Quality (AOQ) AOQ dependent on the Incoming Quality Average Outgoing Quality Limit AOQL highest value of AOQ

24 AOQ - Average Outgoing Quality The expected average quality of all products, including all accepted lots, plus all rejected lots that have been sorted 10% and have had all defective units replaced Will always be less than the submitted quality AOQ = P a * p ( 1 n/n ) AOQL is the maximum value of AAOQ

IMPROVE. Chapter 3-2. Acceptance Sampling Institute of Industrial Engineers 3-2-1

IMPROVE. Chapter 3-2. Acceptance Sampling Institute of Industrial Engineers 3-2-1 Chapter 3-2 Acceptance Sampling 2011 Institute of Industrial Engineers 3-2-1 Acceptance Sampling Inspection: Process of comparing actual measurable characteristics with predetermined standard characteristics.