Right way to sample with prior information on another variable 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):

Example 1:
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).
Example 2:
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.
 A: I would suggest Bayesian methods. Specify priors over these values and compute the posterior. This will take into account the amount of data you have, as requested, and it will incorporate a notion of uncertainty into the assignment of people to each of your discrete groups.
A: You can use the information on height in various ways.
In case 1, you should try to sample people with probability approximately proportional to their weight.  You might (in the absence of more information) guess from the definition of Body Mass Index as weight/height${}^2$ that $\mathrm{weight}\propto\mathrm{height^2}$, and sample with probability proportional to $\mathrm{height}^2$.
In case 2 you want stratified sampling. The optimum way to sample is with the number in each group proportional to the number in the population in that group and inversely proportional to the standard deviation of weight in each group (or your best guess at it).  However, the benefit of stratified sampling is smallest in the scenario where the groups have (as far as you know) the same distribution weight, going to zero when they truly have the same distribution of weight.
