I have a real-life problem similar to the following two subproblems, which are about maximizing samples' representativeness (and maybe getting smaller variance than with simple random sampling, but due to the small sample size, the stratification gains cannot be used):
Given is a population of 1000 people, their height is known. The sample size shall be
n=5. Their average weight is to be estimated.
What is a reasonable way to draw a sample, taking into account the small sample size and the prior information? Intuitively, better than simple random sampling is drawing people equally distributed from small to large (small, semi-small, medium, semi-large, large).
Given is a population of 1000 people again, the sample size is n=5. Now 45% of the population belong to group 1, 25% to group 2, 15% to group 3, 10% to group 4 and 5 % to group 5. One can assume that members of the same group have similar weight.
What is a good way now to draw a sample of this population? Intuitively about 45% of the sample should be of group 1, 25% of group 2... (But how to calculate mean and variance then?)
The sampling procedures have to include a random component still, and –- in addition to the expectation estimates -- variance estimates are needed for constructing confidence intervals for the unknown average weight.