Sample size question.

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    It’s been quite awhile since I had to calculate this and all I can find looking is sample size calculations when I know the mean and all the normal jazz.  The cost to 100% sort is huge and we have a 100% machine sorter at our customer, but we need to do something here to get an idea of the problem. 
    My question is if I have a known exposure of scrap to be x% at my customer.  How much of my product should I sort in order to verify we have it contained.  For example…
    I have ~2% scrap at my customer.  I have 30,000 pieces of it in my plant and those pieces are in tubs of 500.  We are going with a sampling plan to pull 50 pieces per tub and if one part is bad then sort the tub.  What I am wondering is how, statistically, I should have came up with the 50 pieces??  



    Is there any reason to believe one tub is different from the next? If
    not, sample inspection is folly.



    Actually there is.  The tubs come from different machines, but get mixed at the customer.  So I know what my percent overall fallout is, but not which machine comes from.  So I went this route trying to track down the machine problem.  Since then we have sampled parts off of each machine to track it down, but I wanted to “clean up my mind” to see how to go about this in the future for sampling purposes.



    Wish I could just tell you 693 samples, but there are a lot of factors.
    The general answer to your question is:

    If items either pass or fail (attribute – two levels of output)
    If there is only one subgroup (you aren’t separating the tubs when you sample)
    If you gather a random sample
    If you “historical” proportion rejected is 2%
    Then, these assumptions will lead you to using a “1 Proportion” calculation for sample size. If you are using Minitab it will be under Stat>Power and Sample Size>1 Proportion.
    Tip: Since your data is proportion and not variable you will need to collect a significant number of samples to show a delta. However, if you can some how quantify the defect using a continuous measurement (length, time, etc.) instead of just “pass/fail” you can dramatically reduce the number of samples needed. (A note for our readers: percent of pass/fail is not continuous data – its attribute data pretending it’s continuous). Of course this might negate your baseline, but you could create a new baseline by sampling existing products with your continuous measurement system. It would be some leg work, but will reduce the number of samples needed by several factors and give you more information for your analysis.
    Assuming, you are keeping with your proportion data, you will need to now decide several things:

    Sample Size – can leave blank since you want to know how many samples, but you can fill in later for any “what if’s” you want to run. Simply put the number of samples here and leave either “alternative values of p” blank or “power value.”
    Alternative values of p – how much less than 2% do you want to be able to detect. If you say “how many samples needed to show that the new level is 1.5% (0.015)?” then you will need a larger sample size than if you want to show the value has been reduced to 1%.
    Power value – What power value should you use? Most Six Sigma courses will teach that 0.8 is an acceptable value. Can be lower if you want to increase the chance of a Type II error (say there is no difference when there actually is; letting a guilty man go free). 

    What alpha value to use. Most courses teach 0.05, however you can go higher. It will reduce the samples needed, but increase the Type 1 error (the chance that you will say there is a difference in populations when none really exists; “sending the innocent to jail”). If you do go above 0.05, you will need to be able to defend it with your champion and peers.   
    If you are only interested in showing a reduction, you can do a one-sided test. Set the alternate hypothesis to “less than.” It reduces the number of samples needed, but makes you blind if the proportion is worse than before.
    So, as you see, simple question, complex answer. But, it is all about being able to confidently draw a practical conclusion that you can stand behind.

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