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

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

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.

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.

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

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)

Stratified Simple Random Sampling Population Mean Estimate of the Standard Error of the Mean

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

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

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

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

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

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

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)

Stratified Simple Random Sampling Population Proportion Estimate of the Standard Error of the Proportion

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

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

Stratified Simple Random Sampling Sample Size When Estimating Population Mean

Stratified Simple Random Sampling Sample Size When Estimating Population Total

Stratified Simple Random Sampling Sample Size When Estimating Population Proportion

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

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.

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.

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

Cluster Sampling Population Mean Point Estimator Estimate of Standard Error of the Mean

Cluster Sampling Population Mean Interval Estimate Approximate 95% Confidence Interval Estimate

Cluster Sampling Population Total Point Estimator Estimate of the Standard Error of the Total

Cluster Sampling Population Total Interval Estimate Approximate 95% Confidence Interval Estimate

Cluster Sampling Population Proportion Point Estimator

Cluster Sampling Population Proportion Estimate of the Standard Error of the Proportion

Cluster Sampling Population Proportion Interval Estimate Approximate 95% Confidence Interval Estimate

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.

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

Cluster Sampling Point Estimate of Population Mean SAT Score

Cluster Sampling Estimate of Standard Error of the Point Estimator of Population Mean

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

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

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

Cluster Sampling Point Estimate of Population Proportion Planning to Attend College

Cluster Sampling Estimate of Standard Error of the Point Estimator of the Population Proportion

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

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.

End of Chapter 22, Part B