Final answer:
To estimate a population proportion with a 95% confidence level and a margin of error of 4%, using a worst-case scenario assumption of p=0.50, the smallest sample size necessary is 601 people.
Step-by-step explanation:
To determine the minimum sample size needed for the study, we can use the formula for the sample size of a proportion in a worst-case scenario, which is when p (the proportion of successes) is 0.50. This formula is particularly useful when no preliminary estimate of the proportion is available. The formula for calculating the sample size n needed to estimate a population proportion with a specific margin of error E at a certain confidence level is:
n = (Z^2 × p × (1 - p)) / E^2
For a 95% confidence level, the Z-value (Z-score) is 1.96. The usual worst-case scenario proportion p to maximize the sample size is 0.50, as it gives the highest variance. An acceptable margin of error E is 0.04, or 4%, as specified in the question.
Substituting these values into the formula:
n = (1.96^2 × 0.50 × (1 - 0.50)) / 0.04^2
n = (3.8416 × 0.25) / 0.0016
n = 0.9604 / 0.0016
n = 600.25
Since we can't survey a fraction of a person, we round up to the next whole number. Therefore, the smallest number of people you should survey to achieve a 95% confidence level with a 4% margin of error is 601 people.