When f is acontinuous pdf with median M, forsimplerandomsamplingthesamplemedian M has approximatelya N(M, 1~4[f(M)]2n) sampling distribution. (a)

When f is acontinuous pdf with median M, forsimplerandomsamplingthesamplemedian

ÃM has approximatelya N(M, 1~4[f(M)]2n) sampling distribution.

(a) Considersamplingfromanormalpopulation,forwhich μ = M. Usingformula(2.8),show that ˆM has asymptoticstandarderror

»

π~2(σ~

º

n) (for π = 3.14...).

(b) Showthattheratio of variancesof Y and ÃM is 2~π, so Y is amuch more efficientestimator, since Y achievesthesame standard erroras ÃM with only 100(2~π) = 63.7% as muchdata.

(c) Findtheratioofvariancesof Y and ÃM for simplerandomsamplesfrom a uniform distribution (2.4)over[0, θ], andinterpret.