Suppose the U.S. president wants an estimate of the proportion of the population who support his current policy toward revisions in the Social Security system. The president wants the estimate to be within 0.028 of the true proportion. Assume a 99 percent level of confidence. The president's political advisers estimated the proportion supporting the current policy to be 0.62
(a) How large of a sample is required?
(b) How large of a sample would be necessary if no estimate were available for the proportion that support current policy?
(a)Formula to find sample size:
n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 99% confidence, p = 0.62, q = 1 - p, ^2 means squared, * means to multiply, and E = 0.028.
Plug values into the formula and calculate n.
(b) Use p = 0.5 and q = 0.5
Recalculate.
I hope this will help get you started.
To answer these questions, we need to use the formula for sample size calculation in order to achieve a desired margin of error at a specific level of confidence. The formula for sample size is:
n = (Z * Z * p * (1-p)) / E^2
Where:
n = required sample size
Z = Z-score for the desired level of confidence
p = estimated proportion from the sample (prior estimate)
E = desired margin of error
(a) How large of a sample is required?
Given:
p = 0.62 (estimated proportion from the president's political advisers)
E = 0.028 (desired margin of error)
Confidence level = 99% (which corresponds to a Z-score of approximately 2.576)
Substituting these values into the formula:
n = (2.576^2 * 0.62 * (1-0.62)) / 0.028^2
n = (6.646976 * 0.62 * 0.38) / 0.000784
n ≈ 9728
Therefore, a sample size of approximately 9728 would be required to estimate the proportion with a 99% confidence level and a margin of error of 0.028 given the estimated proportion of 0.62.
(b) How large of a sample would be necessary if no estimate were available for the proportion that support the current policy?
If no estimate is available, we can assume the most conservative estimate, which is a proportion of 0.5 (since we don't have any prior information to suggest otherwise). So, p = 0.5.
Using the same formula, and substituting the values:
n = (2.576^2 * 0.5 * (1-0.5)) / 0.028^2
n = (6.646976 * 0.5 * 0.5) / 0.000784
n ≈ 1067
Therefore, a sample size of approximately 1067 would be necessary to estimate the proportion with a 99% confidence level and a margin of error of 0.028, assuming no estimate is available for the proportion.