Survey Sampling Theory Proof Question

Proving unbiasedness of estimators for a population total in a stratified sampling

Look at picture & solve both parts of Question 4) with detailed steps for the proof. All steps must be shown with no shortcuts in between.

4. (20 marks) Suppose we have L strata dened in a population, with N1, . . . , N L elements of the pOpulation in each of the L strata, and we take simple random samples of Hg observations in each stratum. (a) (15 marks) Show that L its; = Z Nfgf 3:1 is an unbiased estimator of the population total '1', where is the sample mean in each stratum. (b) (5 marks) Suppose we also measure an auxiliary variable, with values 2733- in each stratum, and estimate the population total using the ratio estimator L a A _ Z ye Trat {ElTab e=1 '5 where 73,13 is the population total for the auxiliary variable in the fth stratum. Is this estimator unbiased? Explain briey, without actually calculating EUR\"). (There were typos in the original exam!)