One Shot Data

[jmc78]

We have tested 1300 devices and have had one failure. We have an additional 850 units to test, and we would like to know what level of confidence we have that there will be 1 or fewer failures in these next 850 devices.

Using the Binomial distribution equation, I have entered the defect proportion to (1/1300) = .000769 (using the historical data). My number of future samples is 850 and my allowable defects is 1 or fewer. The calculated probability of 1 or fewer failures is 0.86018.

On one of the Binomial calculators I found online, they say that the Confidence Level is 1-P, so in my case it would be 1-0.86018=13.982%. This seems awfully low to me. Is this the correct way to calculate the confidence level or am I doing something wrong here?

[rwisniewski]

Your calculated probability of observing 1 or fewer failures in the sample of 850 as equal to 0.86018 is correct. This calculation is based upon the probability of observing a failure in a single trial as p=1/1300 (taken from the original 1300 item sample).

One way that a confidence interval can be constructed for p is through the Clopper-Pearson (1) method - recommended over the normal distribution approximation in this case due to the observed small number of failures in the original 1300 item sample. This method calculates the upper and lower bounds for p by solving exact two-sided confidence interval equations for Pupper and Plower, respectively. (Although the equations can be tedious, a web search revealed a number of calculators that can accomplish this operation). In this case, the upper and lower bounds of a 90% confidence interval for p are 0.003646 and 0.0000396, respectively.

Substituting the upper and lower bounds for p in the binomial probability distribution equation results in the probability, with 90% confidence, of observing 1 or fewer failures in the 850 item sample as ranging from 0.184 to 0.999, with an expected value of 0.860.

Reference:
(1) C.J. Clopper and E.S. Pearson, "The use of confidence or fiducial limits in the case of the binomial", Biometrika 26:404-413, 1934.

[Highkeas]

You do not say how you tested the 1300 samples or how you will test the next 850 samples.
But this seems like a very inefficient and costly way to test this quantity to find a reliability (unless you testing something like rivet strength).
Have you looked into conducting a statistical sensitivity test (e.g. Neyer)?

Incidentally with 2 failures in 2150 tests yeilds an approx reliability* of .97175 @ 90 confidence. (with no failures R = .9989)

* I use the approx formula given by Ireson & Coombs.