Question

Measurements of a sample of weights were determined as 8.3, 10.6, 9.7, 8.8, 10.2, and 9.4 lb, respectively. Determine unbiased and efficient estimates of (a) the population mean, and (b) the population variance. (c) Compare the sample standard deviation with the estimated population standard deviation.

Answer 1

: Solution :- p Sample Weiguos are 8.3,10.6,9.7,8.4, 10.2,914, db. (as unbiased and efficient estimator of Population mean of cample mean as €CF) = E(x=) - 5 £ ECx;) ECx;) - el. Hence the bicised estimate of Population mean is =9.5 Since this is a regular estimation Case and fince for o= l (0) = - is seen to be an efficient estimator which is 9.56b: (6) In case of Estimating the population Variance unbiasedly we take sample Variance with divison 7 as a oce (n-1 ton xnol

where to & (xp-x) in case of efficient estimator, we will like that unbiased estimator of Population variance for which Craner.Rao lower bound is attained. The col lower bound to the Variance of an unbiased estimator of pr as 26" (using result ) 204 Now, e(t) =p2 and won ) = a (na). var (4") et une Coming Hor ($) 2.6(n-1) (n-1) () vou (196) 2012 But the other not an unbiased estimator of one for 6 6 = 1 E (x,-1)" is unbiased and ethicient as well. But for that it is the population mean and in should be known. (Ov)

. unbiased Estimation - co (207 m2 satna") [545.8-632.93) 2.6% 5 Tuováosed Estimation = 0.93. (C) Comple clandard deviation - 2x0.93 (0.779