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Thread: confidence population error rate is within a range

  1. #1
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    confidence population error rate is within a range

    Hello. Would really appreciate some help answering the following question and making sure my though process even makes sense.

    Context:
    - I have a population of 4219 items.
    - Some number > 0 of the items are calculated wrong
    - 30 items were randomly selected and it was found 100% of them are correct
    - there is no reason to assume that errors would be clustered

    My questions:
    - What is my current confidence that less than 1% of items contain an error based on my sample of 30 that didn't.
    - What sample size would I need to be very confident (95%) the error rate is less than 1%? 5%?
    - What sample size would be considered to accurately represent the overall population error rate?



    I am really looking to minimize manual checking of the data as much as possible since it takes exceedingly long to do (5 minutes per item or more).

    Appreciate any help.
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  2. #2
    MHF Contributor
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    Re: confidence population error rate is within a range

    $\dfrac{P[\text{observe 0 defects in 30}|p<0.01]}{P[\text{observe 0 defects}]} = $

    $\displaystyle \dfrac{\int_0^{0.01}(1-p)^{30}dp}{\int_0^1(1-p)^{30}dp} =0.2677$

    So given the 30 no defect observations you made the probability that the population has under 1% defects is about 27%,
    and about 63% probable that you'd see 0 defects in 30 with p>0.01

    In other words the sample size is too small to support such a low specified defect rate
    A sample size of say 100 brings the probability of $p<0.01$ up to 63%

    A sample size of 500 brings the probability of $p<0.01$ above 99%

    A sample size of 298 gets you to just above 95%


    define "accurately"
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