Answer:
Explanation:
(a) When no preliminary estimate is available, we use the worst-case scenario, which is 0.5 for the population proportion. We can use the formula:
n = (z^2 * p * (1-p)) / E^2
where:
z is the z-score corresponding to the desired confidence level. For a 95% confidence level, z = 1.96.
p is the population proportion.
E is the maximum error of estimate, which is 0.01 in this case.
Plugging in the values, we get:
n = (1.96^2 * 0.5 * (1-0.5)) / 0.01^2 = 9604
So the minimum sample size needed is 9604.
(b) When a prior study is available, we can use the sample proportion from the prior study as the preliminary estimate. In this case, the sample proportion is 0.22. We can use the formula:
n = (z^2 * p * (1-p)) / E^2
where:
z is the z-score corresponding to the desired confidence level. For a 95% confidence level, z = 1.96.
p is the sample proportion from the prior study.
E is the maximum error of estimate, which is 0.01 in this case.
Plugging in the values, we get:
n = (1.96^2 * 0.22 * (1-0.22)) / 0.01^2 = 723
So the minimum sample size needed, using the prior study, is 723.
(c) The minimum sample size needed in part (b) is much smaller than that in part (a). This is because the prior study provides some information about the population proportion, which reduces the uncertainty and the required sample size. Therefore, using a preliminary estimate can save time and resources in data collection.