In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. Find the sample size needed to estimate that percentage. Use a 0.05 margin of error and use a confidence level of $95 \%$. Complete parts (a) through (c) below. a. Assume that nothing is known about the percentage to be estimated. \[ \mathrm{n}= \] (Round up to the nearest integer.)

Step 1 ：We are given a problem where we need to estimate the sample size needed to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. We are given a margin of error of 0.05 and a confidence level of 95%.

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

Step 3 ：Since nothing is known about the percentage to be estimated, we'll use 0.5 as a conservative estimate for p. The z-score for a 95% confidence level is approximately 1.96, and the margin of error E is given as 0.05.

Step 4 ：Substituting these values into the formula, we get: \(n = \frac{(1.96)^2 * 0.5 * (1-0.5)}{(0.05)^2}\)

Step 5 ：Solving this equation, we find that the sample size needed to estimate the percentage with a 0.05 margin of error and a confidence level of 95% is \(\boxed{385}\).