Find the sample size for three strata by proportional allocation from the following information.

a)  

     Find the sample size for three strata by proportional allocation
from the following information.

Stratum	`N_h`	`N_h^2`
A	4397	30759.85
B	751	319594.07
C	1009	277243.00

The required sample size is
250.

Solution:

To find the sample size for three strata by proportional allocation, you need to allocate the sample size proportionally based on the size of each stratum. Here's how you can calculate the sample sizes for each stratum:

Calculate the total population size across all strata:

Total population = Population size in Stratum A + Population size in Stratum B + Population size in Stratum C

Calculate the proportion of each stratum:

Proportion of Stratum A = Population size in Stratum A / Total population

Proportion of Stratum B = Population size in Stratum B / Total population

Proportion of Stratum C = Population size in Stratum C / Total population

Calculate the sample size for each stratum:

Sample size for Stratum A = Proportion of Stratum A * Required sample size

Sample size for Stratum B = Proportion of Stratum B * Required sample size

Sample size for Stratum C = Proportion of Stratum C * Required sample size

Using the provided information:

Total population = 4397 + 751 + 1009 = 6157

Proportion of Stratum A = 4397 / 6157 ≈ 0.7134

Proportion of Stratum B = 751 / 6157 ≈ 0.1219

Proportion of Stratum C = 1009 / 6157 ≈ 0.1647

Sample size for Stratum A = 0.7134 * 250 ≈ 178.35

Sample size for Stratum B = 0.1219 * 250 ≈ 30.475

Sample size for Stratum C = 0.1647 * 250 ≈ 41.175

Therefore, the sample sizes for the three strata, rounded to whole numbers, would be approximately:

Stratum A: 178

Stratum B: 30

Stratum C: 41