Acceptance Sampling Powerpoint

Knowing What to Do Knowing How to Do It Getting Better Every Day

Acceptance Sampling Webinar 20101129

1

Acceptance Sampling I

Acceptance Sampling Webinar 20101129

2

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 Acceptance Sampling Webinar 20101129

3

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 Acceptance Sampling Webinar 20101129

4

Why Sample?  Economy  Less inspection labor  Less time  Less handling damage

 Provides check on process control  Fewer errors ???  i.e. inspection accuracy

Acceptance Sampling Webinar 20101129

5

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

Acceptance Sampling Webinar 20101129

6

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

Acceptance Sampling Webinar 20101129

7

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? Acceptance Sampling Webinar 20101129

8

Distribution of Means

The Distribution of Means obeys normal distribution – regardless of distribution of parent population. Acceptance Sampling Webinar 20101129

9

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

Acceptance Sampling Webinar 20101129

10

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-thedonkey)  Proceed in a predetermined direction – up, down, across  Discard numbers which cannot be applied to the sample

Acceptance Sampling Webinar 20101129

11

Random Number Table

Acceptance Sampling Webinar 20101129

Source: Statistical Quality Control by Grant & Leavenworth 12

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.

Acceptance Sampling Webinar 20101129

13

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

Acceptance Sampling Webinar 20101129

14

The O-C Curve Operating Characteristic Curve

Ideal O-C Curve Pa

Percent Defective Acceptance Sampling Webinar 20101129

15

The Typical O-C Curve

Acceptance Sampling Webinar 20101129

16

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 “Ac” 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 * Pa Acceptance Sampling Webinar 20101129

17

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 “Re”  Sample Size – number of items in sample – usually designated by “n”  Lot Size – number of items in the lot (population) – usually designated by “N” Acceptance Sampling Webinar 20101129

18

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

Acceptance Sampling Webinar 20101129

19

Sampling Risks (cont)

Acceptance Sampling Webinar 20101129

20

Acceptance Sampling II

Acceptance Sampling Webinar 20101129

21

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, Ac = 2 N=200, n=20, Ac = 2 N=1000, n=100, Ac = 2 N=1000, n=6, Ac = 2

Let’s do k (Ac, c - # of successes) = 0 first

Acceptance Sampling Webinar 20101129

22

Hyper-geometric

The number of distinct combination of “n” items taken “r” at a time is

Acceptance Sampling Webinar 20101129

23

Hyper-geometric (cont) = (DCk

NqCn-k)

/ NCn

Construct the following Table p D=Np P(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. Acceptance Sampling Webinar 20101129

24

Hypergeometric Calculator N= n= p 0% 1% 2% 3% 4% 5% 6% 7%

100 10 D=Np K 0 1 2 3 4 5 6 7

D=Defects in Pop.

Nq=N-Np 100 99 98 97 96 95 94 93

P(k=0) 0 1 0.9 0.809091 0.726531 0.651631 0.583752 0.522305 0.46674

P(k=1) 1 0.1 0.181818 0.247681 0.2996 0.339391 0.368686 0.38895

P(k=2) 2

0.009091 0.025046 0.045961 0.070219 0.096458 0.123549

P(k ≤ 2) 1 1 1 0.999258 0.997192 0.993362 0.987449 0.97924

total successes in Popl.

Acceptance Sampling Webinar 20101129

25

Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10

Acceptance Sampling Webinar 20101129

26

Hypergeometric Calculator Example: p=0.02, k=0, N=100, n=10

Acceptance Sampling Webinar 20101129

27

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

Acceptance Sampling Webinar 20101129

28

Acceptance Sampling Webinar 20101129

29

From QCI-CQE Primer 2005, pVI-9 Acceptance Sampling Webinar 20101129

30

Poisson Construct the following Table, using the Poisson Cumulative Table p np P (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. Acceptance Sampling Webinar 20101129

31

Poisson Calculator Example: p=0.02, n=10, c=0

X=k, the number of successes in the sample, i.e. “c”

Acceptance Sampling Webinar 20101129

32

Poisson Calculator Example: p=0.02, n=10, c=0

Mean = np

Acceptance Sampling Webinar 20101129

33

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)

Acceptance Sampling Webinar 20101129

34

From QCI-CQE Primer 2005, pVI-8 Acceptance Sampling Webinar 20101129

35

From QCI-CQE Primer 2005, pVI-8 Acceptance Sampling Webinar 20101129

36

From QCI-CQE Primer 2005, pVI-9 Acceptance Sampling Webinar 20101129

37

O-C Curve & AOQ Determine the O-C curve.  Prepare the following Table using the Poisson distribution p Pa AOQ = p * Pa 0% 1% 2% 3% etc Graph the results: Pa and AOQ vs p.

Acceptance Sampling Webinar 20101129

38

OC Curve & AOQ (2)

Acceptance Sampling Webinar 20101129

39

OC Curve & AOQ (3)

Acceptance Sampling Webinar 20101129

40

Acceptance Sampling III

Acceptance Sampling Webinar 20101129

41

Questions 1. What if this AOQ is not adequate? 2. What if you would like to add a 2nd sample when

the first sample fails?

Example  OC curve after 1st Sample: p=0.02, n=30, N=500, c (Ac)=0, Re=2

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

Acceptance Sampling Webinar 20101129

42

Hypergeometric Multiple Sampling

p

D=Np

N=

500

500

500

500

n=

30

60

60

60

Nq=N-Np

K

P(k=0)

P(k=0)

P(k ≤ 1)

P(k=1)

0

0 1

1

0.00

0

500

1

0.01

5

495

0.73

0.53

0.36

0.89

0.02

10

490

0.54

0.28

0.38

0.66

0.03

15

485

0.39

0.14

0.30

0.44

0.04

20

480

0.28

0.07

0.21

0.28

0.05

25

475

0.20

0.04

0.14

0.17

0.06

30

470

0.15

0.02

0.08

0.10

0.07

35

465

0.11

0.01

0.05

0.06

Acceptance Sampling Webinar 20101129

1

43

Hypergeometric Multiple Sampling Hypergeometric Multiple Sample

Prob of Acceptance

N=500, n=30, c=0

N=500, n=60, c=1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Lot defective

Acceptance Sampling Webinar 20101129

44

ANSI/ASQC Z1.4-1993 Mil-Std 105

 Sampling for Attributes; 95 page Document  Pa’s from 83% to 99%  Information necessary: N, AQL, Inspection Level  How to Use  Code Letters  Single, Double, Multiple Plans  Switching Rules

 Obtain: n, Ac, Re,  O-C Curves Acceptance Sampling Webinar 20101129

45

ANSI/ASQC Z1.4-1993 Exercises

 N=475, AQL = 0.1%, Single Plan, Normal  What is Code Letter  What is Sample Size,  What is Ac, Re

 Repeat for Tightened Inspection  Repeat for Reduced Inspection

Note: 0.1% is 1000 ppm

Acceptance Sampling Webinar 20101129

46

Z1.4 Code Letters

I-Reduced, II-Normal, III-tightened |||| For N=475, Normal, code letter is “H”

Acceptance Sampling Webinar 20101129

47

Z1.4 Single Plan – Normal Insp. Table II-A

n=125,

New code Letter “K”

Acceptance Sampling Webinar 20101129

48

Z1.4 O-C Curve for Code Letter “K” Table X-K

Acceptance Sampling Webinar 20101129

49

Z1.4 Switching Rules

Acceptance Sampling Webinar 20101129

50

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) Acceptance Sampling Webinar 20101129

51

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. Acceptance Sampling Webinar 20101129

52

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

Acceptance Sampling Webinar 20101129

53

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

Acceptance Sampling Webinar 20101129

54

Z1.9 Code Letters

For N=40, AQL=0.75 |||||| Use AQL=1.0 & Code Letter “D”

Acceptance Sampling Webinar 20101129

55

Z1.9 – Finding Decision Criteria Std. Dev method – Table B-1

 For Code Letter “D”, n=5 & AQL=1, k=1.52 Acceptance Sampling Webinar 20101129

56

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 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 Webinar 20101129

57

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  Qcalc = (USL – X-bar)/s = 1.59  Because Qcalc = 1.59 is greater than k=1.52, lot is accepted

Acceptance Sampling Webinar 20101129

58

Z1.9 – OC Curve for “D” Table A-3 (p9)

Acceptance Sampling Webinar 20101129

59

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

Acceptance Sampling Webinar 20101129

60

Solution – 2nd 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  Qcalc = (USL – X-bar)/s = 1.91  Because Qcalc = 1.91 is less than k=2.22, lot is rejected

Acceptance Sampling Webinar 20101129

61

Inspection Economics  Average Total Inspection: The average number of devices inspected per lot by the defined sampling plan ATI = n Pa + N(1- Pa) which assumes each rejected lot is 100% inspected.  Average Fraction Inspected: AFI = ATI/N  Average Outgoing Quality: AOQ = AQL (1 – AFI) Acceptance Sampling Webinar 20101129

62

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

Acceptance Sampling Webinar 20101129

63

Inspection Economics Exercise Solution N

1000

1000

1000

n

100

170

240

c

0

1

2

Pa

0.59

0.8

0.92

n Pa

59

136

220.8

N(1- Pa)

410

200

80

ATI

460

336

300.8

AFI

0.460

0.336

0.301

AOQ

0.0027 0.00332

Acceptance Sampling Webinar 20101129

.00349 64

Inspection Economics Comparison of Cost Alternatives  No Inspection NpD

 100% Inspection NC  Sampling nC + (N-n)pDPa + (N-n)(1-Pa)C D = Cost if defective passes; C = Inspection cost/item Acceptance Sampling Webinar 20101129

65

Inspection Economics

Sample Size Break-Even Point nBE = D/C D = Cost if defective passes; C = Inspection cost/item

Acceptance Sampling Webinar 20101129

66

Resources  American Society for Quality  Quality Press  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  http://mathworld.wolfram.com/ - greaqt math and stat site Acceptance Sampling Webinar 20101129

67