Theformula Y = ~ n for thestandarderrorof Y treats thepopulation size as infinitely large relativetothesamplesize n. Withafinitepopulationsize N, separateobservationsare veryslightlynegativelycorrelated,sovar(i Yi) < ivar(Yi),

Theformula σ

Y

= σ~

º

n for thestandarderrorof Y treats thepopulation size as infinitely large relativetothesamplesize n. Withafinitepopulationsize N, separateobservationsare veryslightlynegativelycorrelated,sovar(Σi Yi) < Σivar(Yi), andactually

The term »
(N − n)~(N − 1) is calledthe finite population correction.

(a) Inpractice, n is usuallysmallrelativeto N, sothecorrectionhaslittleinfluence.Illustrate for sampling n = 300 studentsfromacollegestudentbodyofsize N = 30, 000.

(b) If n = N, showthat σY = 0. Thatis,ifwe sampletheentirepopulation,nosamplingerror occurs,because Y = μ.

(c) For n = 1, explainwhy the samplingdistributionof Y and itsstandarderrorare identical to thepopulationdistributionanditsstandarddeviation.