[De4rest]

Hi, I don't really understand about optimum stratification. What assumptions do you usually make when the sample size of the population is optimally divided into L=4 strata??

[Zazonker]

Well, this isn't my strongest suit, but since no one else has answered, I give you some thoughts.
Since you are asking about optimum allocation in stratified random sampling, I'm assuming that you know something about random sampling and something about stratified random sampling.
So, in a given situation, assume that you made the determination that stratified random sampling would give you a better answer than random sampling -- almost always true. So you've considered using proportional sampling and feel that you'd prefer an approach that would give you a better answer. Remember, that in any kind of sampling, you are trying to estimate the answer that you would get if you did 100% sampling (like asked everyone in the country who they were going to vote for). But you are using sampling techniques to try to get the best answer that you can as cheaply as you can. After all, the biggest reason NOT to do 100% sampling is that it will cost you a fortune in time and resources for every answer.
So stratified sampling divides the population into groups (how you determine the groups is a big factor in your results. A discussion for another time, but suffice it to say that just because you have Blacks and Whites (people, that is) in your population, doesn't necessarily mean that those are the proper strata in any given case.
So, proportional random sampling says I want to preserve the ratio of the strata in the population in taking my sample.
Optimum says, maybe there is a better way. "Better" being defined as coming closer to the answer the population as a whole would give you at a low cost -i not the "best" answer, because that would be achieved by 100% sampling. Not the cheapest answer - because there are a lot of cheap useless answers.
So, for optimum, you take more sample in strata that have greater variability - more samples says better chance of being closer to the "right" answer. And, you take more samples in strata that are cheap to sample. Obviously, these may run counter to each other, so the Optimum equations are aimed at getting a happy blending of the two.

[De4rest]

Thanks :)