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

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.
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.
The basis for forming the various strata depends on the judgment of the designer of the sample.

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 Nh nh
Mean St.Dev.
Under $ $
50 or Over

6. Stratified Simple Random Sampling
Population Mean
Point Estimator
where: H = number of strata
= sample mean for stratum h
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. Stratified Simple Random Sampling
Population Proportion
Point Estimator
where: H = number of strata
= sample proportion for stratum h
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 sampling requires 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
Cluster sampling tends to provide better results than stratified sampling when the elements within the clusters are heterogeneous.
A primary application of cluster sampling involves area sampling, where the clusters are counties, city blocks, or other well-defined geographic sections.

24. 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
M = average number of elements in a cluster
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
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, 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
= to

42. Systematic Sampling
Systematic Sampling is 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