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? $n=1068$ (Round up to the nearest whole number as needed.) (b) What is the minimum sample size needed using a prior study that found that $78 \%$ of the respondents support labeling legislation? $n=\sqrt{n}$ (Round up to the nearest whole number as needed.)

Step 1 ：Given that the researcher wishes to estimate the population proportion of adults who support labeling legislation for genetically modified organisms (GMOs) with a 95% confidence level and a margin of error of 3%, we can use the formula for the sample size of a proportion to find the minimum sample size needed.

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 for the desired confidence level, p is the estimated proportion, and E is the desired margin of error.

Step 3 ：For a 95% confidence level, the z-score (Z) is 1.96. The desired margin of error (E) is 3%, or 0.03.

Step 4 ：For part (a), since no preliminary estimate is available, we use p = 0.5. Substituting these values into the formula, we get \(n = \frac{(1.96)^2 * 0.5 * (1-0.5)}{(0.03)^2}\), which simplifies to n = 1068.

Step 5 ：For part (b), we use the estimate from the prior study, p = 0.78. Substituting these values into the formula, we get \(n = \frac{(1.96)^2 * 0.78 * (1-0.78)}{(0.03)^2}\), which simplifies to n = 733.

Step 6 ：Comparing the results from parts (a) and (b), we can see that the minimum sample size needed is larger when no prior information is available.

Step 7 ：Final Answer: The minimum sample size needed assuming that no prior information is available is \(\boxed{1068}\). The minimum sample size needed using a prior study that found that 78% of the respondents support labeling legislation is \(\boxed{733}\).