The project involves comparisons between di erent sampling methods, di erent estimation methods, numerical results using samples selected from a synthetic nite population, and a nal written report.
(a) Choose two sampling methods from (1) SRSWOR; (2) SRSWR; (3) Systematic sampling; (4) One-stage cluster sampling; and (5) Strati ed sampling (with possibly one of the methods (1)-(4) used within each stratum). Provide detailed descriptions and discussions on the two sampling procedures, including strengths and limitations, and comparisons between the two methods. (Note: you need to look at the structure of the population before choosing your sampling methods) (b) Provide detailed estimation theory on the population mean Y , including point estimator, theoretical variance formula and variance estimator under the choosing designs, and do this for two cases (i) without any auxiliary information; (ii) with known auxiliary population mean X, using either a ratio estimator or a regression estimator. (c) Take two probability samples from the synthetic population described below (available from the UWACE course webpage) using the two methods you chose in (a), with the same overall sample size n. The sampling fraction f = n=N should be no more than 10%. Compute point estimates and associated standard errors following discussions in (b). Repeat this process at least three times, and summarize your results using a table.
(d) The true population mean Y is known under the current arti cial situation. Using the known Y , compute the average bias of the estimates you reported in (c) for a given sampling/estimation scenario, averaging over repeated samples; make commends anddiscussions based on your results.
2. The Synthetic Finite PopulationThe population consists of 250 couples who both have an income last year and this year. The population data le has six columns: 1. Individual ID number; 2. Couple ID number;
3. Gender (Male: 0; Female: 1); 4. xi: reported annual income (in thousand dollars) of last year on tax-returns; 5. yi: this year's actual annual income; 6. Strata indicator variables, created based on values of x (low income: 1; medium income: 2; high income: 3). The truepopulation means are X = 47:1 and Y = 51:1.
3. Final Project Report
(a) A word-processed written report; single-sided with a cover page; minimum 6 pages and maximum 12 pages (excluding the cover page).
(b) The written report should contain (1) a formal title such as A Comparative Study Between SRSWOR and Strati ed SRSWOR; (2) a one-paragraph summary on the main content of the report and major ndings or conclusions (i.e. Abstract); and (3) sections such as Introduction; The Sampling Methods; Estimation Theory; The Finite Population; Numerical Results; and Conclusions. Tables and graphs are welcome.
(c) Repeated simulation through R programming is viewed as a bonus.