Stratified Sampling: Given required bound, calculated $n_h$ is bigger than $N_h$?

I'm dealing with a question that has given me a peculiar result and I would like someone's opinion on how to deal this:

Say you have a population of $$N=550$$ objects: $$N_1=75$$ red and $$N_2=475$$ blue.

Given their $$s_h^2$$ 's and a required bound on a characteristic of interest, I was able to calculate the proper total sample size $$n$$ required, and used the formula to calculate each $$n_h$$.

The problem is that I ended up with a calculated $$n_1>N_1$$ because of the large difference in stratum sample variances. I've still used the calculated $$n$$, set $$n_1=N_1$$ and set $$n_2=n-n_1$$, but I don't think this will guarantee the required bound. What do I do?

• You need to think in terms of variance reduction. One method of dealing with this is to further stratify $N_1$ into group that have homogeneity within the sub-strata and heterogeneity between the sub-strata. Also, you didn't provide the actual formula you used to compute your sample size so it' tough to say what you've done is actually a good approach. – StatsStudent Mar 10 at 14:31