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I have a set of data, which, from an initial test, is expected to be non-normal. My initial test shows a cluster of data gathered around 90, and another cluster gathered around 120. Neither cluster looks particularly normal either (but they both contain a very small amount of values, so it's hard to tell if they follow a precise distribution). From my knowledge of the problem, I'm expecting my real data to follow a similar pattern: a couple of clusters of data centered anywhere between 50 and 150.

I'm trying to design an experiment to prove that the data will always be greater than zero. Intuitively, it should be reasonably easy to prove, given that all my data is generously above the limit. But I can't figure out a statistically sound way of doing this that doesn't require a prohibitive amount of samples.

I've estimated, using Hahn & Meeker's formula as suggested in the NIST handbook in section 7.2.6.4, that in order to guarantee that 99.7% of my data is covered in the interval between the lowest and highest value, with 95% confidence, I'd need 1580 samples, which is not technically feasible for me. But that's not really what I'm trying to prove either.

Is there a way I can take into account the large tolerance margin I have between my lowest sample, and the lower bound I'm trying to prove? Or am I inherently limited by the fact that I'm using to use non-parametric methods?

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You are essentially using two-sided nonparametric tolerance intervals but your problem is inherently one-sided. So you gain something by going to a one-sided tolerance bound. Because you are trying to use a nonparametric bound on the lower tail of a probability distribution there is a need for a large sample size but maybe only about half of what you need for two-sided intervals. If you want to see detailed explanations and tables for the tolerance bounds you may want to also look at Hahn and Meeker's text "Statistical Intervals".

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