So I have a very large set of data for adding one chemical to another (say adding X to Y) (about a years worth of data so not the entire population, about 87,000 data points). Currently we have an upper and lower limit on the concentration of chemical X in chemical Y. We would like to make our target concentration as close to the lower limit as possible without causing quality issues. Sometimes we mix batches where there would only be about 3 to 20 sample points, so I would like to prove with a certain level of confidence that if I took a random sample (about size 3 to 20) from within that larger sample, the mean of the concentration of chemical X would be above the lower limit. What kind of test should I perform?
Note that the injection of chemical x into chemical y is automated and has some variability, which is why we are doing statistics on it.