Sample size- true or false?

Question

when chosing the sample size for estimating a population proportion p to within E units with confidence (1-alpha)100%, if you take p=0.5 as the approximation to p, you will always obtain a sample size that is at least as large as required.

True or False

Answer 1

true for sure, coz you cant mov ethe alpha-1

Answer 2

On the whole, true.

Answer 3

True.

Answer 4

true.

If we don't know anything about phat and are still asked to find the sample size we let phat = 0.5.  This maximizes the value of the error term and if n is sufficient for phat = 0.5, the n will be sufficient for all other values of phat.

this is easy to prove with a couple calculus steps.

find the the first and second derivate with respect to phat of

z * sqrt(phat * (1- phat) / n)

you'll find that the function is concave down and has a max when phat = 0.5