I was confused on the formula for the margin of error. In the derivation, we have a (k/n)(1-k/n) term, which the book then says has a maximum value of 1/4 when k=n/2, and replaces this part of the expression with 1/4.

I am confused on why we do this. If we already know k (number of successes) and n (total samples), why don’t we just use them to arrive at a more accurate margin of error, since we do in fact use them when calculating confidence intervals. I get that the expression is only an estimate for the standard deviation of the binomial distribution, but why did we choose to use the expression when calculating confidence intervals, but not use it for the margin of error?