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.
 A: This is a case where a confidence interval is probably better than the significance test.  You can construct a confidence interval using t procedures if you are willing to assume normality.  With a positive skew (assuming you think that because the values cannot go below 0) you might take the log of the data, calculate the confidence interval using the t procedure, then transform the limits back to the original scale.  The confidence interval gives a plausible region for where the true mean emmissions could be given the data.  If there are values within the interval that are above the legal limit then that would argue at least for more data.
Possibly even better would be a prediction interval which says what interval you expect most of the individual measurements to be in, not just the mean.  I remember seeing a comic that had 2 people standing on the side of a crater that is clearly the site of a recent explosion and one of them is saying something like "this factory is still the safest on average".  Health concerns should look at how high the emmissions could get, not just what their average is.
With only 3 datapoints it would be best to see if there is other data on other similar plants or other information that could be used to justify assumptions of normality or other possible distributions used.  You may also want to consult with a local statistician, the other side certainly will if they don't like your results and "some guy on the internet said ..." will probably not carry much weight.
