Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-distributed about the true value, with a standard deviation of 5mm), while the other is a dimwit (i.e., his measurement is also unbiased, but with a standard deviation of 100mm). What's the optimal way to combine the two measurements to form a final estimate of their true average size? The formal way to ask this question is, "Given a sample from a normally-distributed random variable with known variance, and another sample from a second normally-distributed random variable with known (but potentially different) variance, what is the best guess as to the expected value of the average of the two random variables?" Is the answer "Just average them."? Ideally, I want to prove the answer, whatever it is.
EDIT: I'm not sure if it was clear, but one contractor measures table 1, while the other contractor measures table 2.