John Loucks St . Edward’s University

1. SLIDES.BY . . . . . . . . . . . John Loucks St. Edward’s University

2. Chapter 22, Part B Sample Survey • Stratified Simple Random Sampling • Cluster Sampling • Systematic Sampling

3. Stratified Simple Random Sampling • The population is first divided into H groups, called strata. • The basis for forming the various strata depends on the judgment of the designer of the sample. • Then for stratum h, a simple random sample of size nh is selected. • The data from the H simple random samples are combined to develop an estimate of a population parameter. • If the variability within each stratum is smaller than the variability across the strata, a stratified simple random sample can lead to greater precision.

4. Stratified Simple Random Sampling • Example: ChemTech International ChemTech International has used stratified simple random sampling to obtain demographic information and preferences regarding health care coverage for its employees and their families. The population of employees has been divided into 3 strata on the basis of age: under 30, 30-49, and 50 or over. Some of the sample data is shown on the next slide.

5. Stratified Simple Random Sampling • Demographic Data Annual Family Dental Expense Proportion Married Stratum Mean St.Dev. Nh nh Under 30 100 30 $250 $75 .60 30-49 250 45 400 100 .70 50 or Over 125 30 425 130 .68 475 105

6. = sample mean for stratum h Stratified Simple Random Sampling • Population Mean • Point Estimator • where: H= number of strata • Nh = number of elements in the • population in stratum h • N = total number of elements • in the population (all strata)

7. Stratified Simple Random Sampling • Population Mean • Estimate of the Standard Error of the Mean

8. Stratified Simple Random Sampling • Population Mean • Interval Estimate • Approximate 95% Confidence Interval Estimate

9. Stratified Simple Random Sampling • Point Estimate of Mean Annual Dental Expense = $375 • Estimate of Standard Error of Mean = $9.27

10. Stratified Simple Random Sampling • Approximate 95% Confidence Interval for Mean Annual Dental Expense

11. Stratified Simple Random Sampling • Population Total • Point Estimator • Estimate of the Standard Error of the Total

12. Stratified Simple Random Sampling • Population Total • Interval Estimate • Approximate 95% Confidence Interval Estimate

13. Stratified Simple Random Sampling • Point Estimate of Total Family Expense For All Employees • Approximate 95% Confidence Interval = $169,318 to $186,932

14. = sample proportion for stratum h Stratified Simple Random Sampling • Population Proportion • Point Estimator • where: H= number of strata • Nh = number of elements in the • population in stratum h • N = total number of elements • in the population (all strata)

15. Stratified Simple Random Sampling • Population Proportion • Estimate of the Standard Error of the Proportion

16. Stratified Simple Random Sampling • Population Proportion • Interval Estimate • Approximate 95% Confidence Interval Estimate

17. Stratified Simple Random Sampling • Point Estimate of Proportion Married • Estimate of Standard Error of Proportion = .0417 • Approximate 95% Confidence Interval for Proportion

18. Stratified Simple Random Sampling • Sample Size When Estimating Population Mean

19. Stratified Simple Random Sampling • Sample Size When Estimating Population Total

20. Stratified Simple Random Sampling • Sample Size When Estimating Population Proportion

21. Stratified Simple Random Sampling • Proportional Allocation of Sample n to the Strata

22. Cluster Sampling • Cluster samplingrequires that the population be divided into N groups of elements called clusters. • We would define the frame as the list of N clusters. • We then select a simple random sample of n clusters. • We would then collect data for all elements in each of the n clusters.

23. Cluster Sampling • A primary application of cluster sampling involves area sampling, where the clusters are counties, city blocks, or other well-defined geographic sections. • Cluster sampling tends to provide better results than stratified sampling when the elements within the clusters are heterogeneous.

24. M = average number of elements in a cluster Cluster Sampling • Notation N= number of clusters in the population n= number of clusters selected in the sample Mi= number of elements in cluster i M = number of elements in the population xi = total of all observations in cluster i ai = number of observations in cluster i with a certain characteristic

25. Cluster Sampling • Population Mean • Point Estimator • Estimate of Standard Error of the Mean

26. Cluster Sampling • Population Mean • Interval Estimate • Approximate 95% Confidence Interval Estimate

27. Cluster Sampling • Population Total • Point Estimator • Estimate of the Standard Error of the Total

28. Cluster Sampling • Population Total • Interval Estimate • Approximate 95% Confidence Interval Estimate

29. Cluster Sampling • Population Proportion • Point Estimator

30. Cluster Sampling • Population Proportion • Estimate of the Standard Error of the Proportion

31. Cluster Sampling • Population Proportion • Interval Estimate • Approximate 95% Confidence Interval Estimate

32. Cluster Sampling • Example: Cooper County Schools There are 40 high schools in Cooper County. School officials are interested in the effect of participation in athletics on academic preparation for college. A cluster sample of 5 schools has been taken and a questionnaire administered to all the seniors on the basketball teams at those schools. There are a total of 1200 high school seniors in the county playing basketball.

33. Cluster Sampling • Data Obtained From the Questionnaire Number of Players Average SAT Score Number Planning to Attend College School 1 45 840 15 2 20 980 16 3 30 905 12 4 38 880 18 5 40 970 23 173 84

34. Cluster Sampling • Point Estimate of Population Mean SAT Score

35. Cluster Sampling • Estimate of Standard Error of the Point Estimator of Population Mean

36. Cluster Sampling • Approximate 95% Confidence Interval Estimate of the Population Mean SAT Score

37. Cluster Sampling • Point Estimator of Population Total SAT Score • Estimate of Standard Error of the Point Estimator of Population Total

38. Cluster Sampling • Approximate 95% Confidence Interval Estimate of the Population Total SAT Score = 1,075,605.28 to 1,099,834.72

39. Cluster Sampling • Point Estimate of Population Proportion Planning to Attend College

40. Cluster Sampling • Estimate of Standard Error of the Point Estimator of the Population Proportion

41. Cluster Sampling • Approximate 95% Confidence Interval Estimate of the Population Proportion Planning College = .4617264 to .5182736

42. Systematic Sampling • Systematic Samplingis often used as an alternative to simple random sampling which can be time-consuming if a large population is involved. • If a sample size of n from a population of size N is desired, we might sample one element for every N/n elements in the population. • We would randomly select one of the first N/n elements and then select every (N/n)th element thereafter. • Since the first element selected is a random choice, a systematic sample is often assumed to have the properties of a simple random sample.

43. End of Chapter 22, Part B