A reasearcher wishes to be 95% confident that her estimate of the proportion of two individuals who travel over sea is within 4% of the true proportion. Find the sample necessary if in a prior study a sample of 200 people showed that 40 traveled overseas last year. if no estimate of the sample proportion is available how large should the sample be?
To determine the sample size required, we can use the formula for sample size estimation for proportions. The formula is:
n = (Z^2 * p * q) / E^2
where:
n = required sample size
Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to Z = 1.96 since the population size is not provided)
p = estimated proportion from the prior study
q = 1 - p
E = maximum allowed error or margin of error
First, let's calculate the estimated proportion from the prior study. Out of 200 people, 40 traveled overseas. So the estimated proportion is:
p = 40 / 200 = 0.2
Next, substitute the values into the formula:
n = (1.96^2 * 0.2 * (1 - 0.2)) / (0.04^2)
Simplifying the equation, we get:
n = (3.8416 * 0.2 * 0.8) / 0.0016
n = 0.61472 / 0.0016
n ≈ 384.2
Round up to the nearest whole number since we can't have a fraction of a person:
n ≈ 385
Therefore, if we have the estimate from the prior study, the required sample size is approximately 385.
However, if no estimate of the sample proportion is available, we can assume a conservative estimate of p = 0.5 (maximum variability). In this case, we only need to know the desired confidence level and margin of error.
Substituting the values into the formula:
n = (1.96^2 * 0.5 * (1 - 0.5)) / (0.04^2)
Simplifying gives:
n = (3.8416 * 0.5 * 0.5) / 0.0016
n = 0.4802 / 0.0016
n ≈ 300.125
Again, round up to the nearest whole number:
n ≈ 301
Therefore, if no estimate of the sample proportion is available, the required sample size is approximately 301.