A researcher wishes to estimate, with $95 \%$ confidence, the population proportion of adults who support labeling legislation for genetically modified organisms (GMOs). Her estimate must be accurate within $3 \%$ 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 $78 \%$ of the respondents said they support labeling legislation for GMOs.
(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 ：The researcher wants to estimate the population proportion with a 95% confidence level. This means that the z-score corresponding to this confidence level is 1.96. The margin of error is 3% or 0.03. Since no preliminary estimate is available, we use the most conservative estimate for the population proportion, which is 0.5. This gives the maximum possible sample size.

Step 2 ：We can use the formula for sample size in estimating a population proportion: \(n = \frac{{Z^2 * p * (1-p)}}{{E^2}}\) where Z is the z-score, p is the estimated proportion of the population, and E is the margin of error.

Step 3 ：Substituting the given values into the formula, we get \(n = \frac{{(1.96)^2 * 0.5 * (1-0.5)}}{{(0.03)^2}}\)

Step 4 ：Solving the equation gives us \(n = 1068\)

Step 5 ：So, the minimum sample size needed, assuming that no prior information is available, is 1068.

Step 6 ：Final Answer: The minimum sample size needed is \(\boxed{1068}\)