Suppose an analyst studies three villages having population N1 = 50000, N2 =30000 and N3 = 40000, respectively. A stratified random sample is to be taken with a total sample size of n = 500. Determine the sample size to be selected from each village individually using the method of (i) proportional, and (ii) optimal allocation. From the previous survey, it is known that the standard deviations are S1=30, S2 =15& S3=20.

The method of stratified random sampling is a technique used in statistics to ensure that the sample collected is representative of the entire population, especially when there are significant differences among different subgroups or strata. In this case, the analyst is studying three villages with populations \(N_1 = 50000\), \(N_2 = 30000\), and \(N_3 = 40000\) and wants to take a stratified random sample with a total size \(n = 500\).

### Proportional Allocation:

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