Final answer:
The sample size needed to estimate a population proportion with a margin of error can be determined using the formula n = (Z^2 * p * q) / E^2, where Z is the Z-score corresponding to the desired confidence level, p is the estimated proportion, q is 1 - p, and E is the margin of error.
Step-by-step explanation:
The formula to determine the sample size needed to estimate a population proportion with a given margin of error is: n = (Z^2 * p * q) / E^2, where:
n = sample size needed
Z = Z-score (critical value) corresponding to the desired confidence level
p = estimated proportion of individuals who possess the characteristic of interest
q = 1 - p
E = margin of error
In this case, the preliminary estimate of the proportion who smoke is 0.30 and the margin of error is 0.02.
As the confidence level is not provided, we will assume a confidence level of 95% (which corresponds to a Z-score of approximately 1.96).
Substituting the given values into the formula:
n = (1.96^2 * 0.30 * (1 - 0.30)) / 0.02^2 = 752.25
Rounding up to the next higher value, the sample size should be 753 in order to be 90 percent confident that the estimated proportion is within 3 percentage points of the true population proportion.