Let Y1,...,Yniidf(y) , denote the population mean E(Yi) by , and consider the sample mean as an estimator of (i.e., let Y=^ ). Now, if Z N(0, 1) and with denoting its CDF, the central limit theorem provides the following approximation:

P(Y>k)=1P(kYk))

= 1P(knZkn)

= 2(kn)

other words, the error of estimation is bounded by k with probability approximately 12(kn)

(a) Verify that P(knZkn)=12(kn)

(b) For the standard normal CDF, (2.575) = 0.005. Find a sample size n, as a function of , such that 0.01 with probability approximately 0.99.

(c) Find the specific values of n that ensure this bound when 2 = 1, 4, 9