Let U be a finite population of size N. From U, we select a sample S according to a sampling design with first-order inclusion probabilities _i and second-order inclusion probabilities _ij. To estimate the population mean of a variable yyU=1/NiUyi we propose to use the sample meanyS (a) Show that the bias ofyS is given byBiais(yS)=iU((1/N)i1/N)yi

When is the bias equal to 0? This exercise shows that the sample mean is not unbiased with respect to the population mean, in general, except for certain specific sample designs. (b) Determine the variance ofyS under this design.

Reference: Sampling: Design and Analysis, 3rd Edition, by S.L. Lohr (2022).

Course: MAT3777 University of ottawa