Choosing the correct Distribution for Acceptance Sampling

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#11
You should also note there is a difference between the concept of the AQL, which some tables use as an input, and the statement "I want to be XX% confident that no more than Y% of the items are defective" which is the input to the spreadsheets I supplied.

Since this is a student project - what courses have you had on acceptance sampling, OC curves, etc?
 
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L

lp026713389

#12
I understand what you mean by conservative, since the sample size would be larger, that would actually work in my favor since I'll be inspecting a larger number of units. Understood. The decision will be more reliable. However, any indication on how I should justify using the binomial even though the N (population size) being at least 10 times greater than n (sample size) condition was not satisfied? Just so I can see how I can include that justification in my report.

Unfortunately, the only course I've taken which has discussed acceptance sampling is Quality Control & Improvement, which only has 1 chapter that discusses single, double, and rectification plans. The rest of my knowledge is all from the research I've done on the topic.

I do understand what you mean about the confidence level. What I meant by AQL is that I want to be XX=95% confident that no more than Y=1% of the items are defective.

Lastly, is there any reliable literature (worth referencing) out there that recommends using an AQL of 1% or another certain value for that matter if there is no contractual agreement between me (the consumer) and my supplier?
 
L

lp026713389

#14
Steve,

Excuse my overly analytical nature haha, I have found the reference but it seems that I have somewhat misinterpreted what it implies:

"Type-A and Type-B OC Curves. The OC curves that were constructed in the previous examples [irrelevant here] are called type-B OC curves. In the construction of the OC curve it was assumed that the samples came from a large lot or that we were sampling from a stream of
lots selected at random from a process. In this situation, the binomial distribution is the exact probability distribution for calculating the probability of lot acceptance. Such an OC curve is referred to as a type-B OC curve.

The type-A OC curve is used to calculate probabilities of acceptance for an isolated lot of finite size. Suppose that the lot size is N, the sample size is n, and the acceptance number is c. The exact sampling distribution of the number of defective items in the sample is the hypergeometric distribution.

Figure 15.6 shows the type-A OC curve for a single-sampling plan with n = 50, c = 1, where the lot size is N = 500. The probabilities of acceptance defining the OC curve were calculated using the hypergeometric distribution. Also shown on this graph is the type-A OC curve for N = 2000, n = 50, and c = 1. Note that the two OC curves are very similar.

Generally, as the size of the lot increases, the lot size has a decreasing impact on the OC curve. In fact, if the lot size is at least 10 times the sample size (n/N ≤ 0.10), the type-A and type-B OC curves are virtually indistinguishable."

Douglas C. Montgomery, Introduction to Statistical Quality Control, 6th Edition.


Now, based on rereading this in this context, I would assume the following (please confirm):

1.) Sample size is irrelevant to lot size. Based on AQL, RQL, alpha, and beta, I can use the binomial/Larson nomogram (for single plans) or ANSI Z1.4 tables to get n & c.

2.) After I get n & c, I compare n to N. If the condition is satisfied, I calculate the probabilities of acceptance using the binomial formula. If not, then I calculate the probabilities of acceptance using the hypergeometric formula (which I have for acceptance probabilities), so no issue here if this is the case (for now at least!).

Issue here: If I use the nomogram for my case, I get n=120, c=3. This does not satisfy the binomial approximation condition, and therefore when plotting the OC curve I would use the hypergeometric formula to calculate probabilities. However, if I use ANSI Z1.4 tables, I might (probably will) get a different n and c values than from the nomogram. Now, what if the n value from the ANSI tables satisfies the binomial condition? (For example, say I get an n=80 from the tables). This would satisfy the condition if my lot size is 821 units, and I would therefore be able to use the binomial to plot the OC curve. Won't this make a difference in my OC curve? Which would be safer to go with (more accurate)?

3.) If the sample size obtained is larger than the lot size, do 100% inspection.

Do I make more sense now? I hope this is correct, because if it is, then I'm sorted for single plans.

However, I am also required to develop a double sampling plan to compare. It would seem to me that for double sampling plans, the nomograph cannot be used. However, from some reading it seems that the (beta/alpha) tables are what is used in this case to get n & c. I have a few questions here.

1- Are there any other standards/systems one can use to develop a double sampling plan other than the beta/alpha tables?

2- Since there will be 2 samples sizes in this case (n1 & n2), what would be the criteria to determine whether the OC curve should be based on the binomial or hypergeometric?

Please keep the discussion going!
 
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Steve Prevette

Deming Disciple
Staff member
Super Moderator
#15
You may find double-sampling plans in MIL-STD-105 / ANSI Z1.4.

An important thing to realize in these tables is that their significance levels are really VERY LOW for a single sample. They rely on repeated sampling of lots as they come in to become effective at detecting defects.

Is your instructor requiring a determination of the use of hypergeometric vs binomial?
 
L

lp026713389

#16
You may find double-sampling plans in MIL-STD-105 / ANSI Z1.4.

An important thing to realize in these tables is that their significance levels are really VERY LOW for a single sample. They rely on repeated sampling of lots as they come in to become effective at detecting defects.

Is your instructor requiring a determination of the use of hypergeometric vs binomial?
Okay, so you recommend I stay away from the tables and use the nomogram instead for a single sampling plan, and use the ANSI Z1.4 or beta/alpha tables to develop a double sampling plan? That is, if I want to develop both, a single sampling plan and double sampling plan.

What do you mean when you say the significance levels for the tables are very low for a single sampling plans? Didn't quite get that part.

My instructor didn't specify a distribution for me to use, but he wants me to develop a single plan AND a double plan and compare them cost wise, time wise, certainty wise, etc. It's up to me to decide what distribution I'll be using to calculate and draw the OC curve depending on comparing n to N, etc.

The question still remains though, how am I supposed to determine which distribution to use to calculate/draw the OC curve for double sampling plans, since there are two sample sizes not 1?
 

Steve Prevette

Deming Disciple
Staff member
Super Moderator
#17
My instructor didn't specify a distribution for me to use, but he wants me to develop a single plan AND a double plan and compare them cost wise, time wise, certainty wise, etc. It's up to me to decide what distribution I'll be using to calculate and draw the OC curve depending on comparing n to N, etc.

The question still remains though, how am I supposed to determine which distribution to use to calculate/draw the OC curve for double sampling plans, since there are two sample sizes not 1?
If you plug in the sample sizes from what ANSI tells you to sample for a lot in to the binomial (or hypergeometric) you will find they come out at around 60 percent confidence that no more than 10 percent are bad (try it out for yourself!). ANSI and MILSTD are dependent upon over the years you will be sampling from lots of this same material and you will build up confidence levels.

One thing I am keeping in mind in this discussion is you are a student taking a course. I have taught several statistics courses over the years, so I am working towards achieving a balance between doing your homework for you, and pointing you in the right direction so you don't completely stall.

Acheson Duncan's book Quality Control and Industrial Statistics (an oldie but a goodie) has a lot of discussion of MIL STD 105D (D version at that time) and acceptance sampling and how to draw OC curves. Generally, sampling plans stick to the binomial distribution because even if you only have a finite sized lot TODAY, it is assumed there is a larger population "out there" that the manufacturer made and yet to make, and you may end up buying more of the product in the future.

If you can borrow a copy of QC and IC it could be of assistance. Though - I am somewhat concerned that you (or your instructor) have expectations beyond what is the curricula (and text book) for the course.

I do suggest that you do some comparisons of what the sampling tables give you, and what binomial sample size calculation gives you so you can see what is going on. Also, plug it into the hypergeometric and either confirm or deny for yourself that the Binomial is "good enough".
 

David-D

Involved In Discussions
#18
I think almost everyone uses Binomial in practice (I know I do); hypergeometric may be more "correct" but it is rarely worth the extra effort in practice. That being said, you "should" be using the hypergeometric when your sampling affects your population(ex. n/N is pretty big (>.1) or p is small). Take for an extreme example, a sampling plan of 50(3-4) on a population of 200 units with 1% defective (2 units); in reality you can't fail the sampling plan, but if you use the binomial approximation, it would preduct you could. For most realities, it doesn't really matter though.

David
 
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L

lp026713389

#19
Okay gents, thank you for the input. Steve, this is actually part of my co-op graduation project, not just any regular course project.

Let's ignore single sampling for a minute here; I think I got that sorted. I still have a very persisting question on double sampling plans.

What sources are available in order to get the 2 sample sizes along with their respective acceptance numbers for double plans? So far I've seen:

1- Table indexed by the ratio of beta to alpha

2- ANSI Z1.4 tables

Is there any other standard/source? If no, which of these 2 is generally more accurate?


Also, this brings up the following question:

1-) How do I justify/test the binomial condition of (n/N) being >0.1 in this case since I have 2 sample sizes? Do I just added n1 and n2 and then compare that value to N? There is very little literature on this.
 

David-D

Involved In Discussions
#20
Generally, I'm looking to have the OC curve closely match one or more points (AQL, LTPD, and/or 50% point).I generally have an approximate quantity I want to sample and/or a maximum number of defects I'm willing to accept. At that point it be comes a bit of playing arround to get sizes that I like and are well matched. I do the same for double sampling plans. When I personally build double sampling plans I often have the second sample the same size as the first; often this makes the logistics of test/inspection easier as well as limiting/easing the defining of plan options, but it doesn't get you the lowest Average Sample Number (ASN).

You can do some optimization, including using the solver function in excel, but I often find that I'd rather massage the plan to get something I like.

David
 
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