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? $n=\square$ (Round up to the nearest whole number as needed.)

Step 1 ：We are given that the confidence level is 95%, which corresponds to a Z-score of 1.96. The margin of error is 1%, or 0.01. Since no preliminary estimate is available, we use the maximum possible value for the product p * (1-p), which occurs when p = 0.5.

Step 2 ：Substitute these values into the formula for the minimum sample size: \(n = \frac{Z^2 * p * (1-p)}{E^2}\).

Step 3 ：Calculate the minimum sample size: \(n = \frac{(1.96)^2 * 0.5 * (1-0.5)}{(0.01)^2}\).

Step 4 ：Perform the calculation to find the minimum sample size: \(n = 9604\).

Step 5 ：Since the sample size must be a whole number, and we always round up to ensure that our sample is large enough, the minimum sample size needed is \(\boxed{9604}\).