Question

Which of the following SRS designs will give the most precision (smallest standard error) for estimating a population mean? Assume that each population has the same value of the population variance S^2?
1) An SRS of size 400 from a population of size 4000
2) An SRS of size 30 from a population of size 300
3) An SRS of size 3000 from a population of size 300,000,000

Answer 1

Answer:

Design 3: An SRS of size 3000 from a population of size 300,000,000

Step-by-step explanation:

To check the SRS designs will give the most precision (smallest standard error) for estimating a population mean, we'll make use of the following formula:

V(y) = S²/n( 1 - n/N)

Where S² is a constant for the three SRS designs

Check the first design

n = 400

N = 4000

So, V(y) = S²/400 (1 - 400/4000)

V(y) = S²/400(1 - 0.1)

V(y) = 0.0025S²(0.9)

V(y) = 0.00225S²

V(y) = 2.25S²E-3

The second design

n = 30

N = 300

So, V(y) = S²/30 (1 - 30/300)

V(y) = S²/30(1 - 0.1)

V(y) = S²/30(0.9)

V(y) = 0.03S²

V(y) = 3S²E-2

The third design

n = 3,000

N = 300,000,000

So, V(y) = S²/3,000 (1 - 3,000/300,000,000)

V(y) = S²/3,000(1 - 0.00001)

V(y) = S²/3,000(0.99999)

V(y) = 0.00033333

V(y) = 3.33S²E-4