A statistician wants to estimate the mean height h(in meters) of a population, based on nindependent samples X1, . . . ,Xn, chosen uniformly from the entire population. He uses thesample mean Mn= (X1+ . . .+ Xn)/nas the estimate of h, and a rough guess of 1.0 meters for the standard deviation of the samples Xi.

(a) How large should nbe so that the standard deviation of Mnis at most 1 centimeter?

(b) How large should nbe so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters from h, with probability at least 0.99?

(c) The statistician realizes that all persons in the population have heights between 1.4 and 2.0 meters, and revises the standard deviation figure that he uses based on the bound of Example 5.3 in the textbook. How should the values of nobtained in parts (a) and (b) be revised?