A statistician wants to estimate the mean height h (in meters) of a population, based on n independent samples X1, . . . , Xn, chosen uniformly from the entire population. He uses the sample mean Mn = (X1 +···+Xn)/n as the estimate of h, and a rough guess of 1.0 meters for the standard deviation of the samples Xi.
(a) How large should n be so that the standard deviation of Mn is at most 1 centimeter? (b) How large should n be so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters from h, with probability at least 0.99?