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:5F3110 419EIEJEI Final-Assignment-Q. 1 (12 marks) In a list of N individuals, we are interested in a variable y. The individuals are identied by their order on the list, so their order goes from 1 (for the rst) to N (for the last). We use systematic sampling with interval k to select 11 individuals from the list. We assume that: k = g E N. Let S? = 237:1(3425 3302 be the sample variance for the ith cluster, where g,- 2 23.21 yij denote the ith cluster mean. (a) (4 marks) Show that everything happens as if we selected a unique cluster of individuals from a population predivided into clusters. We will specify what the clusters are, what their size is, and how many there are in the population. (b) (4 marks) Let 3],) be the value of y for j'h record counted in cluster i, and M denote the population mean for y. (i) What is the unbiased estimator of a? Show that is effectively unbiased. (ii) What is the expression of the true variance of , as a function of u, the im cluster mean g, and k. (c) (2 marks) Considering the natural splitting of the population into k clusters, show that the general expression of the population variance 0'; can be decomposed by k k 1 711 2 2 2 22 i E S 0y k 7221(3), ILL) + TL X k; i:1 1, (d) (1 mark) Show that if N is large, and if we denote: ssw 1 k " MSW== garg2? k(n 1) k(n 1) gags 5' ) then we have: k( 1) A _ 2 71 Var(,u) y TMSW (e) (1 mark) Show that systematic sampling is more precise than simple random sampling if and only if: a: