# Confidence interval on mean of mixture distribution with normal components

I have 50 sacks, each containing material from one of 10 varieties of hemp. Each sack has a different weight. I don't know what variety is in each sack, or how much material I have of each variety. Each variety has a different, unknown, average potency (how much CBD is in it, expressed as a percentage), which I assume to be normally distributed.

I would like to collect samples (by which I mean a small amount of material) from these sacks, submit them to a lab for potency testing, and then compute a confidence interval on the overall potency of all of the material in the sacks. I can collect any number of samples from each sack, but each test costs $50. How do I compute this confidence interval? Would bootstrapping be reasonable given all of the unknowns (not saying I know how to do this)? Please note it has been several years since I've taken a statistics class, so simple explanations are appreciated. I would also appreciate any suggestions on the title and keywords. Thank you! • Since the sample average is asymptotically Normal by the CLT, the confidence interval can be derived from this asymptotic distribution. – Xi'an Nov 3 '19 at 9:38 • Here are some key words for google: "sample size normal distribution power curve". You should estimate (or guess) the parameters of your normal distribution and then calculate the needed sample size. However, be aware that the model is only as good as its assumptions. – Semoi Nov 3 '19 at 11:36 • I'm familiar with some terminology, but I am not a statistician. I don't know what to do with this advice. Can you please express it more simply and explicitly? – user359996 Nov 3 '19 at 23:53 • @Xi'an In this setting the CLT appears to be of little help, because the underlying distribution is a mixture of ten Normals. Unless most of the sacks are sampled, there's a risk of not encountering one or more of the mixture components; and unless most of the sacks are repeatedly sampled, there's a risk of mis-characterizing one or more of the components. These considerations suggest collecting several aliquots from each sack and analyzing a well-mixed composite. Total cost is \$50 and the uncertainty is basically the lab measurement error. Test several composites to obtain a CI. – whuber Nov 4 '19 at 18:04
• @whuber: I agree that the CLT is a crude approach when cost is an issue. I would not call the distribution a mixture though in that it is always possible to pick the sack from where to sample. The quantity of interest is the average of the 50 means. – Xi'an Nov 4 '19 at 19:47