# How to estimate the probability that the mean of an unknown distribution is over a threshold given small sample size

I am trying to quantify my concerns regarding a proposed incinerator in our community. The company is basing its potential to emit dioxins (a class of chlorinated organic compounds with a reference dose of 1.7E-8 grams per 150 lb person per year) on three emission measurements (mg/m^3 exhaust):

0.0002139; 0.0000014; 0.00000186

The company has submitted their potential to emit as the mean of these three values, which is near legal limits. If the plant is built, they may be required to measure dioxins only once on completion, or possibly every year thereafter.

My initial thoughts have been to find the probability that these samples were drawn from a distribution with a mean above a certain threshold (e.g. the legal limit). As sample measurements, I would assume a normal distribution, though I guess with some non-zero skewness. I don't have any prior knowledge about the variance other than the samples.

But, I am now confronted with an infinite number of possible distributions, and am unsure how to proceed. I suspect this is a well studied area of statistics (risk analysis?), and any pointers in common methods would be much appreciated. I am a biochemist, and statistics is not my first language.

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As @Greg Snow points out (+1), there aren't enough data here to do anything definitive. You need a different strategy. There's a huge one out there: focus on the burden of proof. If the incinerator operator has to show that they meet the legal limit, then they will have to collect enough data to demonstrate they are highly likely to continue meeting the limit. This means that an upper bound of their values must lie beneath that limit. If you win this argument, you give the operator a strong incentive to collect lots of data, which will help both sides understand what the risks really are. –  whuber Oct 26 '12 at 19:15
To put it more technically: normally, it is not sufficient to have an average value lie below a regulatory limit. The operator must show that an upper confidence limit of the mean is below that limit. In the US, you should be able to find requirements to use UCLs written into the state and federal guidance documents, the regulations, and in some cases the very laws that govern this situation. (In some documents such limits are vaguely referred to as "UPLs" and even "ULs" ("upper limits").) –  whuber Oct 26 '12 at 19:18
Thanks! What you say makes much sense, but in our state much seems to be left to the DEQ permit writer, and it has been his position that it is not his responsibility to vet the applicant's submitted values but just to require standard testing (though the testing protocol depends on whether criteria threshold are exceeded in the application!). But we are making headway -- permit writer booted for conflict of interest. If application is granted this will be challenged. –  roverred Oct 26 '12 at 21:52