You are the operations manager for an airline and you are considering a higher fare level for passengers in aisle seats. How many randomly selected air passengers must you survey? Assume that you want to be $95 \%$ confident that the sample percentage is within 2.5 percentage points of the true population percentage. Complete parts (a) and (b) below.
a. Assume that nothing is known about the percentage of passengers who prefer aisle seats.
$\mathrm{n}=$
(Round up to the nearest integer.)

Step 1 ：We are given that we want to be 95% confident that the sample percentage is within 2.5 percentage points of the true population percentage. This means that the z-score (Z) is 1.96 and the margin of error (E) is 0.025.

Step 2 ：Since we don't know anything about the percentage of passengers who prefer aisle seats, we'll use 0.5 as a conservative estimate for the estimated proportion of the population (p).

Step 3 ：We can now use the formula for sample size in a proportion, which is \(n = \frac{Z^2 * p * (1-p)}{E^2}\).

Step 4 ：Substituting the given values into the formula, we get \(n = \frac{(1.96)^2 * 0.5 * (1-0.5)}{(0.025)^2}\).

Step 5 ：Calculating the above expression, we get n = 1536.64.

Step 6 ：Since we can't survey a fraction of a person, we round up to the nearest integer, which gives us n = 1537.

Step 7 ：\(\boxed{1537}\) is the number of randomly selected air passengers that must be surveyed.