1. Acceptance Sampling Webinar 201011291 Knowing What to Do Knowing How to Do It Getting Better Every Day

2. Acceptance Sampling Webinar 201011292 Acceptance Sampling I

3. Acceptance Sampling Webinar 201011293 What you will learn  The purpose of Sampling  How to draw a statistically valid Sample  How to Develop a Sampling Plan  How to construct an O-C curve for your sampling plan  How to use (and understand) ANSI/ASQ Z1.4  How to use ANSI/ASQ Z1.9  Assessing Inspection Economics

4. Acceptance Sampling Webinar 201011294 What is Sampling Sampling 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 sample Sampling is an Audit practice

5. Acceptance Sampling Webinar 201011295 Why Sample?  Economy  Less inspection labor  Less time  Less handling damage  Provides check on process control  Fewer errors ???  i.e. inspection accuracy

6. Acceptance Sampling Webinar 201011296 What does Sampling not do?  Does not provide detailed information of lot quality  Does not provide judgment of fitness for use (of rejected items)  Does not guarantee elimination of defectives – any AQL permits defectives

7. Acceptance Sampling Webinar 201011297 Sampling Caveats  Size of sample is more important than percentage of lot  Only random samples are statistically valid  Access to samples does not guarantee randomness  Acceptance sampling can place focus on wrong place  Supplier should provide evidence of quality  Focus should be on process control  Misuse of sampling plans can be costly and misleading.  No such thing as a single representative sample

8. Acceptance Sampling Webinar 201011298 Representative Sample? There is no such thing as a single representative sample Why?  Draw repeated samples of 5 from a normally distributed population.  Record the X-bar (mean) and s (std.dev) for each sample  What is the result?

9. Acceptance Sampling Webinar 201011299 Distribution of Means The Distribution of Means obeys normal distribution – regardless of distribution of parent population.

10. Acceptance Sampling Webinar 2010112910 Standard Error of the Mean Central Limit Theorem The 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

11. Acceptance Sampling Webinar 2010112911 The Random Sample At 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, across  Discard numbers which cannot be applied to the sample

12. Acceptance Sampling Webinar 2010112912 Random Number Table Source: Statistical Quality Control by Grant & Leavenworth

13. Acceptance Sampling Webinar 2010112913 Stratified Sampling  Random 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.

14. Acceptance Sampling Webinar 2010112914 Selecting the Sample  Wrong way to select sample  Judgement: often leads to Bias  Convenience  Right ways to select sample  Randomly  Systematically: e.g. every nth unit; risk of bias occurs when selection routine matches a process pattern

15. Acceptance Sampling Webinar 2010112915 The O-C Curve Operating Characteristic Curve PaPa Percent Defective Ideal O-C Curve

16. Acceptance Sampling Webinar 2010112916 The Typical O-C Curve

17. Acceptance Sampling Webinar 2010112917 Sampling Terms  AQL – 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 “A c ” or “c”  AOQ – Average Outgoing Quality: The expected quality of outgoing product, after sampling, for a given value of percent defective in the incoming product. AOQ = p * P a

18. Acceptance Sampling Webinar 2010112918 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 “R e ”  Sample Size – number of items in sample – usually designated by “n”  Lot Size – number of items in the lot (population) – usually designated by “N”

19. Acceptance Sampling Webinar 2010112919 Sampling Risks  Producers Risk – α: calling the population bad when it is good; also called Type I error  Consumers Risk – β: calling the population good when it is bad; also called Type II error

20. Acceptance Sampling Webinar 2010112920 Sampling Risks (cont)

21. Acceptance Sampling Webinar 2010112921 Acceptance Sampling II

22. Acceptance Sampling Webinar 2010112922 Constructing the O-C curve We will do the following O-C curves  Use Hyper-geometric and Poisson for each of the following N=60, n=6, A c = 2 N=200, n=20, A c = 2 N=1000, n=100, A c = 2 N=1000, n=6, A c = 2 Let’s do k ( A c, c - # of successes ) = 0 first

23. Acceptance Sampling Webinar 2010112923 Hyper-geometric The number of distinct combination of “n” items taken “r” at a time is

24. Acceptance Sampling Webinar 2010112924 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.com Note: 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. = ( D C k Nq C n-k ) / N C n

25. Acceptance Sampling Webinar 2010112925 Hypergeometric Calculator

26. Acceptance Sampling Webinar 2010112926 Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10

27. Acceptance Sampling Webinar 2010112927 Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10

28. Acceptance Sampling Webinar 2010112928 Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10 P (k=0) = 0.809091 P (k=1) = 0.181818 P (k=2) = 0.009091 ----------------------- P(k≤2) = 1.0

29. Acceptance Sampling Webinar 2010112929

30. Acceptance Sampling Webinar 2010112930 From QCI-CQE Primer 2005, pVI-9

31. Acceptance Sampling Webinar 2010112931 Poisson Construct the following Table, using the Poisson Cumulative Table p npP (k ≤ 2) 0% 1% 2% 3% 4% etc. Compare. When is Poisson a good approximation Use the Poisson when n/N ˂ 0.1 and np ˂ 5.

32. Acceptance Sampling Webinar 2010112932 Poisson Calculator Example: p=0.02, n=10, c=0 X=k, the number of successes in the sample, i.e. “c”

33. Acceptance Sampling Webinar 2010112933 Poisson Calculator Example: p=0.02, n=10, c=0 Mean = np

34. Acceptance Sampling Webinar 2010112934 Poisson Calculator Example: p=0.02, n=10, c=0 TRUE for cumulative, i.e. Σk; FALSE for probability mass function, i.e.p(x=k)

35. Acceptance Sampling Webinar 2010112935 From QCI-CQE Primer 2005, pVI-8

36. Acceptance Sampling Webinar 2010112936 From QCI-CQE Primer 2005, pVI-8

37. Acceptance Sampling Webinar 2010112937 From QCI-CQE Primer 2005, pVI-9

38. Acceptance Sampling Webinar 2010112938 O-C Curve & AOQ Determine the O-C curve.  Prepare the following Table using the Poisson distribution p P a AOQ = p * P a 0% 1% 2% 3% etc Graph the results: P a and AOQ vs p.

39. Acceptance Sampling Webinar 2010112939 OC Curve & AOQ (2)

40. Acceptance Sampling Webinar 2010112940 OC Curve & AOQ (3)

41. Acceptance Sampling Webinar 2010112941 Acceptance Sampling III

42. Acceptance Sampling Webinar 2010112942 Questions 1. What if this AOQ is not adequate? 2. What if you would like to add a 2 nd sample when the first sample fails? Example  OC curve after 1 st Sample: p=0.02, n=30, N=500, c (A c )=0, R e =2  OC curve after 2 nd Sample (of 30 more): p=0.02, n=60, N=500, c (A c )= 1, R e =2

43. Acceptance Sampling Webinar 2010112943 Hypergeometric Multiple Sampling N =500 n =3060 pD=NpNq=N-NpP(k=0) P(k=1)P(k ≤ 1) K 001 0.00050011 1 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

44. Acceptance Sampling Webinar 2010112944 Hypergeometric Multiple Sampling

45. Acceptance Sampling Webinar 2010112945 ANSI/ASQC Z1.4-1993 Mil-Std 105  Sampling for Attributes; 95 page Document  P a ’s from 83% to 99%  Information necessary: N, AQL, Inspection Level  How to Use  Code Letters  Single, Double, Multiple Plans  Switching Rules  Obtain: n, A c, R e,  O-C Curves

46. Acceptance Sampling Webinar 2010112946 ANSI/ASQC Z1.4-1993 Exercises  N=475, AQL = 0.1%, Single Plan, Normal  What is Code Letter  What is Sample Size,  What is A c, R e  Repeat for Tightened Inspection  Repeat for Reduced Inspection Note: 0.1% is 1000 ppm

47. Acceptance Sampling Webinar 2010112947 Z1.4 Code Letters I-Reduced, II-Normal, III-tightened |||| For N=475, Normal, code letter is “H”

48. Acceptance Sampling Webinar 2010112948 Z1.4 Single Plan – Normal Insp. Table II-A n=125, New code Letter “K”

49. Acceptance Sampling Webinar 2010112949 Z1.4 O-C Curve for Code Letter “K” Table X-K

50. Acceptance Sampling Webinar 2010112950 Z1.4 Switching Rules

51. Acceptance Sampling Webinar 2010112951 ANSI/ASQC Z1.4-1993 What happens when AQL =. 1% isn’t good enough AQL = 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)

52. Acceptance Sampling Webinar 2010112952 ANSI/ASQC Z1.9-1993 Mil-Std 414 Sampling for Variables; 110 page Document Four Sections in the document  Section A: General description of Plans  Section 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.

53. Acceptance Sampling Webinar 2010112953 ANSI/ASQC Z1.9-1993 Mil-Std 414  Information necessary: N, AQL, Inspection Level  How to Use  Code Letters  Single or Double Limit, Std. Dev or Range Method Plans  Switching Rules  Obtain: Code Letter, n, Accept/Reject criteria, critical statistic (k)  O-C Curves

54. Acceptance Sampling Webinar 2010112954 ANSI/ASQC Z1.9-1993 Exercise (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, k  Should lot be accepted or rejected

55. Acceptance Sampling Webinar 2010112955 Z1.9 Code Letters For N=40, AQL=0.75 |||||| Use AQL=1.0 & Code Letter “D”

56. Acceptance Sampling Webinar 2010112956 Z1.9 – Finding Decision Criteria Std. Dev method – Table B-1  For Code Letter “D”, n=5 & AQL=1, k=1.52

57. Acceptance Sampling Webinar 2010112957 ANSI/ASQC Z1.9-1993 What 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 Q calc > k, there is less of distribution above Q calc than above “k” and lot is accepted. (Compare to “Z” table) Increasing (USL - X-bar) increases P a

58. Acceptance Sampling Webinar 2010112958 ANSI/ASQC Z1.9-1993 Exercise Solution The five reading are 197F, 188F, 184F, 205F, 201F.  X-bar (mean) = 195F  S (Std. Dev) = 8.8F  Q calc = (USL – X-bar)/s = 1.59  Because Q calc = 1.59 is greater than k=1.52, lot is accepted

59. Acceptance Sampling Webinar 2010112959 Z1.9 – OC Curve for “D” Table A-3 (p9)

60. Acceptance Sampling Webinar 2010112960 ANSI/ASQC Z1.9-1993 Another Exercise  Same information as before  AQL = 0.1  Find Code Letter, n, k  Accept or Reject Lot?

61. Acceptance Sampling Webinar 2010112961 Solution – 2 nd Exercise New code letter is “E”, n=7, & k=2.22 The seven reading are 197F, 188F, 184F, 205F, 201F, 193F & 197F.  X-bar (mean) = 195F  S (std. Dev) = 7.3F  Q calc = (USL – X-bar)/s = 1.91  Because Q calc = 1.91 is less than k=2.22, lot is rejected

62. Acceptance Sampling Webinar 2010112962 Inspection Economics  Average Total Inspection : The average number of devices inspected per lot by the defined sampling plan ATI = n P a + N(1- P a ) which assumes each rejected lot is 100% inspected.  Average Fraction Inspected : AFI = ATI/N  Average Outgoing Quality : AOQ = AQL (1 – AFI)

63. Acceptance Sampling Webinar 2010112963 Inspection Economics Exercise (from Grant & Leavenworth, p395)  AQL = 0.5%, N=1000  Which sampling plan would have least ATI.  n = 100, c = 0  n = 170, c = 1  n = 240, c = 2

64. Acceptance Sampling Webinar 2010112964 Inspection Economics Exercise Solution N1000 n100170240 c012 PaPa 0.590.80.92 n P a 59136220.8 N(1- P a )41020080 ATI460336300.8 AFI0.4600.3360.301 AOQ0.00270.00332.00349

65. Acceptance Sampling Webinar 2010112965 Inspection Economics Comparison of Cost Alternatives  No Inspection NpD  100% Inspection NC  Sampling nC + (N-n)pDP a + (N-n)(1-P a )C D = Cost if defective passes; C = Inspection cost/item

66. Acceptance Sampling Webinar 2010112966 Inspection Economics Sample Size Break-Even Point n BE = D/C D = Cost if defective passes; C = Inspection cost/item

67. Acceptance Sampling Webinar 2010112967 Resources  American Society for Quality  Quality Press  www.asq.org www.asq.org  ASQ/NC A&T partnership quality courses  CQIA, CMI, CQT, CQA, CQMgr, CQE, CSSBB  Quality Progress Magazine  And others  Web-Sites  www.stattrek.com – excellent basic stat site www.stattrek.com  http://mathworld.wolfram.com/ - greaqt math and stat site http://mathworld.wolfram.com/