please solve the following question in description

Question

A political candidate has asked you to conduct a poll to determine what percentage of people support him. If the candidate only wants a 3% margin of error at a 97.5% confidence level, what size of sample is needed? When finding the z-value, round it to four decimal places.

Answer 1

For any distribution, the formula for n is σ²z_α/2²/ME²,

For a proportion, σ²= p(1-p)

The maximum value that p(1-p) can have occurs at p = 1/2, and p(1-p) = 1/2(1-1/2) = 1/4

Thus, the formula may be rewritten for proportions as z_α/2²/(4ME²)

In this particular problem, ME = 3% = .03

α = 1 - confidence level = 1 - .975 = .025

Then, z_α/2 = norm.s.inv(1 - α/2) = norm.s.inv(1 - .025/2) = norm.s.inv(.9875) = 2.241403

n= 2.241403²/(4 *.03²) = 1395.524

As n must be an integer, we take the ceiling of 1395.524 = 1396.