I Combinatorics problem on drawing sample with given mean

[DrDu]

I am faced with a problem in combinatorics while trying to set up a pool. Instead of explaining my real problem, I prefer to give you a simplified example:
Say I am given a population of N persons of varying height $h_i$. The height of each person $i$ in the population is known to me. Now I want to set up a sample of n persons so that the mean value of the height has on average a predefined value which is different from the grand mean of the overall population but otherwise I want the two populations to be as similar as possible. Especially, I don't want to introduce a hard cutoff.
I thought about Kullback Leibler entropy maximization, and this works well if it were possible to sample the same person repeatedly.
But how do you proceed if a person can only be drawn one time?

 

[Stephen Tashi]

so that the mean value of the height has on average a predefined value

Since a mean value is "an average" , are you asking about how to set up a sampling procedure so the sample mean varies and may not always be equal to the predefined value, but the "mean of the sample means" taken over the distribution of samples is equal to the predefined value ?

 

[DrDu]

Yes, exactly. I think this problem becomes muche easier if I allow also the number of persons n to vary. Then I get basically a Fermi-Dirac statistic.
Thank's for your help!