A researcher wishes to estimate, with $95 \%$ confidence, the population proportion of adults who think Congress is doing a good or excellent job. Her estimate must be accurate within $1 \%$ of the true proportion.
(a) No preliminary estimate is available. Find the minimum sample size needed.
(b) Find the minimum sample size needed, using a prior study that found that $28 \%$ of the respondents said they think Congress is doing a good or excellent job.
(c) Compare the results from parts (a) and (b).
(a) What is the minimum sample size needed assuming that no prior information is available?
$n=9604$ (Round up to the nearest whole number as needed)
(b) What is the minimum samples size needed using a prior study that found that $28 \%$ of the respondents said they think Congress is doing a good or excellent job?
$n=$
(Round up to the nearest whole number as needed.)
Answer
For part (b), a prior study found that 28% of the respondents said they think Congress is doing a good or excellent job. So, we use p = 0.28. Substituting these values into the formula, we get \(n = \frac{{(1.96)^2 * 0.28 * (1-0.28)}}{{(0.01)^2}} = 7745\). Therefore, the minimum sample size needed using a prior study is \(\boxed{7745}\).
Step 1 ：We are given that the researcher wishes to estimate the population proportion with a 95% confidence level and the estimate must be accurate within 1% of the true proportion. This means that the Z-score is 1.96 (corresponding to a 95% confidence level) and the desired margin of error E is 0.01.
Step 2 ：The formula for the sample size of a proportion is \(n = \frac{{Z^2 * p * (1-p)}}{{E^2}}\), where n is the sample size, Z is the Z-score, p is the estimated proportion of the population, and E is the desired margin of error.
Step 3 ：For part (a), no preliminary estimate is available, so we use p = 0.5 for maximum variability. Substituting these values into the formula, we get \(n = \frac{{(1.96)^2 * 0.5 * (1-0.5)}}{{(0.01)^2}} = 9604\). Therefore, the minimum sample size needed assuming that no prior information is available is \(\boxed{9604}\).
Step 4 ：For part (b), a prior study found that 28% of the respondents said they think Congress is doing a good or excellent job. So, we use p = 0.28. Substituting these values into the formula, we get \(n = \frac{{(1.96)^2 * 0.28 * (1-0.28)}}{{(0.01)^2}} = 7745\). Therefore, the minimum sample size needed using a prior study is \(\boxed{7745}\).
