1. 2015-7-3 www.uic.edu.hk/~xlpeng 1 STAT 4060 Design and Analysis of Surveys Exam: 60% Mid Test: 20% Mini Project: 10% Continuous assessment: 10%

2. 2015-7-3 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.

3. 2015-7-3 www.uic.edu.hk/~xlpeng 3 Population Parameter

4. 2015-7-3 www.uic.edu.hk/~xlpeng 4 Sample Statistics

5. 2015-7-3 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.

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7. 2015-7-3 www.uic.edu.hk/~xlpeng 7 Proof of (1.9)

8. 2015-7-3 www.uic.edu.hk/~xlpeng 8 Confidence interval for the population mean

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14. 2015-7-3 www.uic.edu.hk/~xlpeng 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.

15. The sample average of ratio unbiased for estimating the population mean Two obvious estimators of R are The ratio of the sample averages is widely used. 2015-7-3 www.uic.edu.hk/~xlpeng 15 but biased for estimating R

16. The bias in estimating R by r The bias in estimating R by r is the expectation of the following difference: (2.3) 2015-7-3 www.uic.edu.hk/~xlpeng 16

17. Discussion about the bias 2015-7-3 www.uic.edu.hk/~xlpeng 17 ≈

18. 2015-7-3 www.uic.edu.hk/~xlpeng 18 (2.5)

19. 2.2 Ratio estimation of a population mean or total 2015-7-3 www.uic.edu.hk/~xlpeng 19

20. Variance of ratio estimator 2015-7-3 www.uic.edu.hk/~xlpeng 20

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22. 2015-7-3 www.uic.edu.hk/~xlpeng 22 The estimate of the ratio R of the present weight to prestudy weight for the herd is: Solution:

23. 2015-7-3 www.uic.edu.hk/~xlpeng 23 This examines when the variance of (2.10) could be less or greater than that of (1.9)

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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. 2015-7-3 www.uic.edu.hk/~xlpeng 25

26. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 26 A practicable simple linear regression model is (2.17). An ideal (perfect) linear relationship is (2.16) (2.18)

27. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 27 Consider the average (mean) of either (2.16) or (2.17), (2.19)

28. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 28 (2.20)

29. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 29 From (2.20), The minimum is obtained with Thus the most efficient regression estimator of is (2.22)

30. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 30 The optimal value of b of (2.22) suggests the obvious estimate: (2.24) (2.25) which enjoys the following asymptotic properties:

31. 2.3 Regression estimation 2015-7-3 www.uic.edu.hk/~xlpeng 31 Asymptotic properties: (2.27) (2.26)

32. 2.4 Comparison of ratio and regression estimators 2015-7-3 www.uic.edu.hk/~xlpeng 32

33. 2015-7-3 www.uic.edu.hk/~xlpeng 33 2.4 Comparison of ratio and regression estimators

34. 2015-7-3 www.uic.edu.hk/~xlpeng 34 Stratified Simple Random Sampling (Chapter 5, Textbook, Barnett, V., 1991) Consider another sampling method:

35. Some Notations 2015-7-3 www.uic.edu.hk/~xlpeng 35 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

36. 2015-7-3 www.uic.edu.hk/~xlpeng 36 Estimation of Population Characteristics in Stratified Populations

37. Estimating 2015-7-3 www.uic.edu.hk/~xlpeng 37 The stratified sample mean is defined as Here we assume the weights W i =N i /N is given (known).

38. The mean and variance of 2015-7-3 www.uic.edu.hk/~xlpeng 38 Note that Since Because it is assumed that “sampling in different strata are independent”, that is

39. 2015-7-3 www.uic.edu.hk/~xlpeng 39 Simple random sampling Stratified sampling with proportional allocation

40. 2015-7-3 www.uic.edu.hk/~xlpeng 40 (a) When stratum size is large enough:

41. 2015-7-3 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!

42. Optimum Choice of Sample Size 2015-7-3 www.uic.edu.hk/~xlpeng 42  To achieve required precision of estimation  Some cost limitation The simplest form assumes that there is some overhead cost, c 0 of administering The survey, and that individual observations from the ith stratum each cost an Amount c i. Thus the total cost is:

43. 2015-7-3 www.uic.edu.hk/~xlpeng 43 I. Minimum variance for fixed cost (Cont.)

44. 2015-7-3 www.uic.edu.hk/~xlpeng 44 I. Minimum variance for fixed cost (Cont.) Then

45. II. Minimum cost for fixed variance 2015-7-3 www.uic.edu.hk/~xlpeng 45 Consider to satisfy for the minimum possible total cost.

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48. Comparison of proportional allocation and optimum allocation 2015-7-3 www.uic.edu.hk/~xlpeng 48 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.

49. 2015-7-3 www.uic.edu.hk/~xlpeng 49 Cluster Sampling (Chapter 6, Textbook, Barnett, V., 1991)

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54. 2015-7-3 www.uic.edu.hk/~xlpeng 54 Comparison of s.r. sampling with cluster sampling

55. Systematic Sampling 2015-7-3 www.uic.edu.hk/~xlpeng 55 Systematic sample can be viewed as a cluster sample of size m=1! Systematic sample mean

56. Systematic Sampling 2015-7-3 www.uic.edu.hk/~xlpeng 56 Comparison of s.r. sampling with systimatic sampling

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58. Two ways of estimating --- 2015-7-3 www.uic.edu.hk/~xlpeng 58

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