As in Example 9.1, let h (p) = p (1 − p).
a. Find the remainder term in the Taylor expansion, ʃxa (x − t) h” (t)dt, and use it to find an exact expression for h(Ṕ).
b. Is the remainder term likely to be smaller than the other terms? Explain.
c. Find an exact expression for V [h (Ṕ)] for a simple random sample with replacement. How does it compare with the approximation in Example 9.1? Use moments of the Binomial distribution to find E (Ṕ 4).