Answer: To determine the appropriate sample size for the US EIA to achieve the desired margins of error at a 95% confidence level, we can use the formula for sample size calculation for estimating a population mean with a specified margin of error:
n = (Z^2 * σ^2) / E^2
where:
n = sample size
Z = Z-score for the desired confidence level (standard value for 95% confidence level is 1.96)
σ = standard deviation of the population
E = desired margin of error
Given:
Standard deviation (σ) = $0.25
Desired margin of error (E) for each case:
Margin of error = $0.05
Margin of error = $0.10
Let's calculate the appropriate sample sizes for each case:
Case 1: Margin of error = $0.05
n = (1.96^2 * 0.25^2) / 0.05^2
n = (3.8416 * 0.0625) / 0.0025
n = 0.2401 / 0.0025
n = 96.04
The appropriate sample size for a margin of error of $0.05 at 95% confidence level is approximately 96.
Case 2: Margin of error = $0.10
n = (1.96^2 * 0.25^2) / 0.10^2
n = (3.8416 * 0.0625) / 0.01
n = 0.2401 / 0.01
n = 24.01
The appropriate sample size for a margin of error of $0.10 at 95% confidence level is approximately 24.
Therefore, the US EIA should use a sample size of 96 if they wish to report a margin of error of $0.05 and a sample size of 24 if they wish to report a margin of error of $0.10 at 95% confidence level.