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STAT 4060 Design and Analysis of Surveys

STAT 4060 Design and Analysis of Surveys. Exam: 6 0% Mid Test: 20% Mini Project: 10 % Continuous assessment : 10 %. What we have learned:. 1. Simple random sampling, confidence interval and choice of sample size.

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STAT 4060 Design and Analysis of Surveys

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  1. STAT 4060 Design and Analysis of Surveys • Exam: 60% • Mid Test: 20% • Mini Project: 10% • Continuous assessment:10% www.uic.edu.hk/~xlpeng

  2. What we have learned: 1. Simple random sampling, confidence interval and choice of sample size. 2. Ratio and regression estimators, systematic sampling. 3. Stratified random sampling, allocation of stratum weights. 4. Cluster sampling. www.uic.edu.hk/~xlpeng

  3. Population Parameter www.uic.edu.hk/~xlpeng

  4. Sample Statistics www.uic.edu.hk/~xlpeng

  5. Simple random sampling • We shall consider the use of simple random samples for estimating the three population characteristics: the population mean the population total and the proportion P. • We shall discuss how any estimators behave in terms of their sampling distributions. The variance is often a crucial measure. www.uic.edu.hk/~xlpeng

  6. www.uic.edu.hk/~xlpeng

  7. Proof of (1.9) www.uic.edu.hk/~xlpeng

  8. Confidence interval for the population mean www.uic.edu.hk/~xlpeng

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  14. Ratio Estimation and Regression Estimation(Chapter 4, Textbook, Barnett, V., 1991) 2.1 Estimation of a population ratio: The ratio estimator • In some situations it is useful to estimate a (positive) ratio of two population characteristics: the totals, or means, of two (positive) variables X and Y. www.uic.edu.hk/~xlpeng

  15. Two obvious estimators of R are The sample average of ratio unbiased for estimating the population mean The ratio of the sample averages is widely used. but biased for estimating R www.uic.edu.hk/~xlpeng

  16. The bias in estimating R by r The bias in estimating R by r is the expectation of the following difference: (2.3) www.uic.edu.hk/~xlpeng

  17. Discussion about the bias ≈ www.uic.edu.hk/~xlpeng

  18. (2.5) www.uic.edu.hk/~xlpeng

  19. 2.2 Ratio estimation of a population mean or total www.uic.edu.hk/~xlpeng

  20. Variance of ratio estimator www.uic.edu.hk/~xlpeng

  21. www.uic.edu.hk/~xlpeng

  22. Solution: The estimate of the ratio R of the present weight to prestudy weight for the herd is: www.uic.edu.hk/~xlpeng

  23. This examines when the variance of (2.10) could be less or greater than that of (1.9) www.uic.edu.hk/~xlpeng

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  25. 2.3 Regression estimation Condition (2.15.1) demands that X and Y be linearly related, but, if the linear relationship does not pass through the origin, then, it suggests considering an alternative estimator known as regression estimator. www.uic.edu.hk/~xlpeng

  26. 2.3 Regression estimation . An ideal (perfect) linear relationship is (2.16) A practicable simple linear regression model is (2.17) (2.18) www.uic.edu.hk/~xlpeng

  27. 2.3 Regression estimation Consider the average (mean) of either (2.16) or (2.17), (2.19) www.uic.edu.hk/~xlpeng

  28. 2.3 Regression estimation (2.20) www.uic.edu.hk/~xlpeng

  29. 2.3 Regression estimation From (2.20), (2.22) The minimum is obtained with Thus the most efficient regression estimator of is www.uic.edu.hk/~xlpeng

  30. 2.3 Regression estimation The optimal value of b of (2.22) suggests the obvious estimate: (2.24) (2.25) which enjoys the following asymptotic properties: www.uic.edu.hk/~xlpeng

  31. 2.3 Regression estimation Asymptotic properties: (2.26) (2.27) www.uic.edu.hk/~xlpeng

  32. 2.4 Comparison of ratio and regression estimators www.uic.edu.hk/~xlpeng

  33. 2.4 Comparison of ratio and regression estimators www.uic.edu.hk/~xlpeng

  34. Stratified Simple Random Sampling(Chapter 5, Textbook, Barnett, V., 1991) Consider another sampling method: www.uic.edu.hk/~xlpeng

  35. Some Notations To estimate the population mean of a finite population, we assume that the population is stratified, that is to say it has been divided into k non-overlapping groups, or strata, of sizes: The stratum means and variances are denoted by and www.uic.edu.hk/~xlpeng

  36. Estimation of Population Characteristicsin Stratified Populations www.uic.edu.hk/~xlpeng

  37. Estimating The stratified sample mean is defined as Here we assume the weights Wi=Ni /N is given (known). www.uic.edu.hk/~xlpeng

  38. The mean and variance of Note that Since Because it is assumed that “sampling in different strata are independent”, that is www.uic.edu.hk/~xlpeng

  39. Simple random sampling Stratified sampling with proportional allocation www.uic.edu.hk/~xlpeng

  40. (a) When stratum size is large enough: www.uic.edu.hk/~xlpeng

  41. (b) When stratum size is not large enough: The stratified sample mean will be more efficient than the s.r. sample mean If and only if variation between the stratum means is sufficiently large compared with within-strata variation! www.uic.edu.hk/~xlpeng

  42. Optimum Choice of Sample Size • To achieve required precision of estimation • Some cost limitation The simplest form assumes that there is some overhead cost, c0 of administering The survey, and that individual observations from the ith stratum each cost an Amount ci. Thus the total cost is: www.uic.edu.hk/~xlpeng

  43. I. Minimum variance for fixed cost (Cont.) www.uic.edu.hk/~xlpeng

  44. I. Minimum variance for fixed cost (Cont.) Then www.uic.edu.hk/~xlpeng

  45. II. Minimum cost for fixed variance Consider to satisfy for the minimum possible total cost. www.uic.edu.hk/~xlpeng

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  48. Comparison of proportional allocation and optimum allocation Thus the extent of the potential gain from optimum (Neyman) allocation Compared with proportional allocation depends on the variability of the stratum variances: the larger this is, the greater the relative advantage Of optimum allocation. www.uic.edu.hk/~xlpeng

  49. Cluster Sampling(Chapter 6, Textbook, Barnett, V., 1991) www.uic.edu.hk/~xlpeng

  50. www.uic.edu.hk/~xlpeng

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