Cochran's sample size formula p variable meaning and intuition

Question

I want to calculate a sample size for a large population of about 50 million. I came across Cochran's formula and the finite population correction. In short, Cochran's formula is the following:

$$ n_\infty = \frac{z^2 p(1-p)}{e^2} $$ I have found multiple resources that describe p as a sample proportion or as estimated proportion of an attribute that is present in the population. Most of the time, p is considered by default 0.5.

1) What is p and how can I understand it intuitively with an example?

2) Is p found exclusively via experiments? If not, what is its formula?

3) Most people use p = 0.5 because it gives the maximum sample size. Is this true?

Answer 1

1) What is p and how can I understand it intuitively with an example?
You answered your own question here. "p as a sample proportion or as estimated proportion of an attribute that is present in the population" If example if you want to estimate the percentage of women attending the University, that would be equal to p and 1-p is the percentage of men attending.

2) Is p found exclusively via experiments? If not, what is its formula?
Yes, p in context is a measured variable from the population.

3) Most people use p = 0.5 because it gives the maximum sample size. Is this true?
Yes is case assuming p=0.5 will provide the worst case assessment and thus require the largest sample size. Perform the calculation your self by comparing: 0.5*(1-0.5) versus 0.1*(1-0.1)