Question

(?) We want to understand the statistics of the height h (in meters) of a population, based on n independent samples X1, . . . , Xn chosen uniformly from the entire population. We will use the sample mean Mn = (X1+···+Xn)/n as the estimate of h. (a) According to Markov's inequality, what's an upper bound on the probability that Mn is greater than five times h? (b) Assume we have a rough guess that the standard deviation of each sample Xi is approximately 1 meter. How large should n be so that the standard deviation is at most 1 centimeter? (c) Assuming the same guess that the standard deviation of each sample Xi is approximately 1 meter, how large should n be so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters of h with probability at least 0.99? (d) After collecting our data, we notice that all of the people sampled have heights between 1.4 and 2.0 meters. If we assume that all heights are within the range [a, b], we can show that ?^2 ? (b ? a)^2/4 (see example 5.3 on p. 268 of the book for this argument). Using this fact, revise the estimate of ?^2 and repeat part (c).

          (?) We want to understand the statistics of the height h (in meters) of a population, based on n independent samples X1, . . . , Xn chosen uniformly from the entire population. We will use the sample mean Mn = (X1+···+Xn)/n as the estimate of h.

(a) According to Markov's inequality, what's an upper bound on the probability that Mn is greater than five times h?
(b) Assume we have a rough guess that the standard deviation of each sample Xi is approximately 1 meter. How large should n be so that the standard deviation is at most 1 centimeter?
(c) Assuming the same guess that the standard deviation of each sample Xi is approximately 1 meter, how large should n be so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters of h with probability at least 0.99?
(d) After collecting our data, we notice that all of the people sampled have heights between 1.4 and 2.0 meters. If we assume that all heights are within the range [a, b], we can show that ?^2 ? (b ? a)^2/4 (see example 5.3 on p. 268 of the book for this argument). Using this fact, revise the estimate of ?^2 and repeat part (c).

(?) We want to understand the statistics of the height h (in meters) of a population, based on n independent samples X1, . . . , Xn chosen uniformly from the entire population. We will use the sample mean Mn = (X1+···+Xn)/n as the estimate of h.

(a) According to Markov's inequality, what's an upper bound on the probability that Mn is greater than five times h?
(b) Assume we have a rough guess that the standard deviation of each sample Xi is approximately 1 meter. How large should n be so that the standard deviation is at most 1 centimeter?
(c) Assuming the same guess that the standard deviation of each sample Xi is approximately 1 meter, how large should n be so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters of h with probability at least 0.99?
(d) After collecting our data, we notice that all of the people sampled have heights between 1.4 and 2.0 meters. If we assume that all heights are within the range [a, b], we can show that ?^2 ? (b ? a)^2/4 (see example 5.3 on p. 268 of the book for this argument). Using this fact, revise the estimate of ?^2 and repeat part (c).

Added by Veronica V.

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