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Confidence Interval and Sample Size

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    Hi.I have conducted an experiment of running 20 units (5 groups) through a process and the result was that all of the units were found good. Assuming that the total population is 5000 units.At what confidence level can i say that the products produced by my process will be good.And also if i want a Confidence Level of 95%, how do I calculate my sample size?ThanksJayang


    Hi Jayang,
    With no defectives found in a sample of 20 units, the following formula can be used to compute an upper 95% confidence bound for the population percentage defective:
    p’ Upper (1 – alpha)% Confidence Bound = 1 – alpha ^ (1/n)
    where p’ is the population percentage defective, alpha is the significance level (hence, 1 – alpha is the confidence level), and n is the sample size. 
    In your case, the desired alpha is .05 and n is 20.  Thus, an upper 95% confidence bound for p’ is 13.9%.
    p’ Upper 95% Confidence Bound = 1 – (.05) ^ (1/20) = 1 – .861 = .139 = 13.9%
    Based on zero defectives found in a sample of 20, you can be 95% confident that there are no more than 695 (5000 x .139) defective units in a group of 5000 units.
    At 50% confidence, the upper bound for p’ is .034 = 3.4%.
    p’ Upper 50% Confidence Bound = 1 – (.50) ^ (1/20) = 1 – .966 = .034 = 3.4%
    This means that you have 50% confidence there are no more than 170 defective units in a group of 5000 (5000 x .034).
    At 1% confidence, the upper bound for p’ is .034 = 3.4%.
    p’ Upper 1% Confidence Bound = 1 – (.99) ^ (1/20) = 1 – .9995 = .0005 = 0.05%
    This result means you have 1% confidence that there are no more than 2.5 defective units in a group of 5000 (5000 x .0005).
    I think you can see from the above calculations that there is practically 0% confidence (based on a sample of only 20) that there are zero defectives in your group of 5000.
    Instead of 20, how large a sample is needed?  To have 95% confidence there is no more than 1 defective in a group of 5000, you would need a 95% upper confidence bound of .0002 (1/5000).  This confidence bound would require a sample size of almost 15,000 units!
    n = log(alpha) / log (1 – Desired Upper Confidence Bound)
    n = log(.05) / log(1 – .0002) = 14,977
    Therefore, to answer your question, even if you checked all 5000 units, you would still not have your desired 95% confidence level.
    I hope this helps.


    Thanks Ross,The input was really useful.I created a small spreadsheet to do the calculation (attached below). Confidence Interval And Sample Size Download [Microsoft Excel, 19 KB]

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    Dr. Steve W.
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    to talk to a PhD level statistician about it. I can probably give an answer but I am worried about the fact that you mentioned 5  groups. There might be issue with independence assumption needed for making  statistical inference.


    Hello everyone,Due to an overwhelming interest by the community, iSixSigma has contacted the author directly to ask that the calculation spreadsheet be posted to this thread. The author is out of the office through the 24th. When Mr. Patel returns, it is hopeful that he will allow us to attach his spreadsheet to his posting, with special thanks.You may want to bookmark this thread and visit during the week of February 24th to see if Mr. Patel has provided his spreadsheet.Update: Jayang Patel was kind enough to forward his confidence interface Excel spreadsheet. See the previous post by Jayang for the download.Sincerely,iSixSigma Community

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