# Unequal sample size for 2 proportion calculation group

I have a question. It’s about the uneven sample size between 2 groups. Let say I want to compare & analysis these 2 groups when 1 of them have small number of sample.

To give you some light, here are the example of the specific objectives of the research: 1) To look for the current proportion of the said interest in Group A. Group A has a finite population of 83. 2) To look for the current proportion of the said interest in Group B. Group B is a general population and is not finite. 3) To see the comparison in the said interest between Group A and Group B.

Proportion of individuals in the sample with the characteristics of interest in previous study was 94% and 85.9% respectively for Group A and Group B. I have calculated each objective sample size and came out as below:

Obj 1) Using the single proportion calculator with finite population,(P1: 94%, Precision: 5%, Significance: 5%, Finite population: 83, Drop out: 20%) = 54 samples.

Obj 2) Using the single proportion calculator only,(P0: 85.9%, Precision: 5%, Significance: 5%, Drop out: 20%) = 234 samples.

Obj 3) Using the two proportion calculator, ,(P1: 94%, P0: 85.9%, Power: 80%, Significance: 5%, Drop out: 20%) = 270 samples per group. However, from the above calculation the biggest sample size which is 270 each group cannot be done due to the limitation of the finite sample of Group A population.

1) So my question is how to calculate the estimation sample size for this research that covers all objective overall which are with high precision and powered? 2) Is there any formula to calculate 2 proportion group sample size with one of the groups or both group has finite population?

It would be grateful if you can share some closure/solution in this situation. Thank you. Regards.