Applied statistics

Question

A population 1000 units consists of three strata with N1=4000, N2=2500, and N3=3500. Respective standard deviations were S1=15, N2=25, N3=5. How should a sample of size 100 be allocated under proportional allocation?

Answer 1

We are asked for proportional allocation, which means the sample from each stratum is proportional to the size of the stratum. The formula is:

n_h = N_h⁄ N • n

N

Where:

1. n_h​ = sample size for stratum h
2. N_h​ = population of stratum h
3. N = total population
4. n = total sample size

Step 1: Find total population, N:

N=N₁ + N₂ + N₃ = 4000 + 2500 + 3500 = 10000

Step 2: Apply the proportional allocation formula:

n₁ = N₁ • n = 4000⁄10000 • 100

N

n₂ = N₂ • n = 2500⁄10000 • 100

n₃ = N₃ • n = 3500⁄10000 • 100

Now, can you calculate n₁, n₂, and n₃ from these fractions?

Let’s calculate them digit by digit:

n₁ = 4000⁄10000 • 100 = 0.4 • 100 = 40

n₂ = 2500⁄10000 • 100 = 0.25 • 100 = 25

n₃ = 3500⁄10000 • 100 = 0.35 • 100 = 35

So under proportional allocation, the sample sizes would be:

1. n₁ = 40
2. n₂ = 25
3. n₃ = 35