Answer: Assuming that the desired level of confidence and the population size remain constant, the sample size required for a given margin of error is proportional to the square of the standard normal deviate corresponding to the desired level of confidence.
That is, if n is the current sample size and m is the current margin of error, and we want to divide the margin of error by 3 while keeping the same level of confidence, then we need to find the new sample size, n', that satisfies the equation:
n'/n = (zα/2 / z'α/2)²
where zα/2 is the standard normal deviate corresponding to half the desired level of confidence, and z'α/2 is the standard normal deviate corresponding to half the desired level of confidence and one-third the margin of error.
Assuming a desired level of confidence of 95%, we have:
zα/2 = 1.96
z'α/2 = 1.96 / 3 = 0.6533 (rounded to four decimal places)
Substituting these values and the given values of n and m, we get:
n'/275 = (1.96 / 0.6533)²
Solving for n', we get:
n' = 275 * (1.96 / 0.6533)²
Using a calculator, we get:
n' ≈ 887
Therefore, the EPA would need to poll approximately 887 people to achieve the same level of confidence with a margin of error divided by three.
Explanation: