I can't seen to answer this question! Any help is much appreciated

Final-AssignmentQ.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 [1 individuals from the list. We assume that: k- = g E N. Let S? n_1 Z? _1(y.;J- 3132 be the sample variance for the 2% cluster, Where g. : %Z?=1 yij denote the it'll cluster mean. (a) (4 marks) Show that everything happens as if we selected a unique cluster of individuals from a population pre-divided into clusters. \\Ve will specify what the clusters are, what their size is, and how many there are in the population. (b) (4 marks) Let ygj be the value of y for jth record counted in cluster 1, and n denote the population mean for y. (i) \"l hat is the unbiased estimator H of a? Show that E is effectively unbiased. (ii) What is the expression of the true variance of )3, as a function of a, the ith cluster mean g, and k. (C) (2 marks} Considering the natural splitting of the population into k clusters, show that the 2 general expression of the population variance 0 can be decomposed by y k =%Z('y_ Fogl 1:1 ((1) (1 mark) Show that if N is large, and if we denote: SSW MSW = ' 2 )izlj: then we have: varm) = a; wMSW (e) (1 mark) Show that systematic sampling is more precise than simple random sampling if and only if: a;