1. 1 1 Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western College Publishing/Thomson Learning

2. 2 2 Slide Chapter 21 Sample Survey n Terminology Used in Sample Surveys n Types of Surveys and Sampling Methods n Survey Errors n Simple Random Sampling n Stratified Simple Random Sampling n Cluster Sampling n Systematic Sampling

3. 3 3 Slide Terminology Used in Sample Surveys n An element is the entity on which data are collected. n A population is the collection of all elements of interest. n A sample is a subset of the population.

4. 4 4 Slide Terminology Used in Sample Surveys n The target population is the population we want to make inferences about. n The sampled population is the population from which the sample is actually selected. n These two populations are not always the same. n If inferences from a sample are to be valid, the sampled population must be representative of the target population.

5. 5 5 Slide Terminology Used in Sample Surveys n The population is divided into sampling units which are groups of elements or the elements themselves. n A list of the sampling units for a particular study is called a frame. n The choice of a particular frame is often determined by the availability and reliability of a list. n The development of a frame can be the most difficult and important steps in conducting a sample survey.

6. 6 6 Slide Types of Surveys n Surveys Involving Questionnaires Three common types are mail surveys, telephone surveys, and personal interview surveys. Three common types are mail surveys, telephone surveys, and personal interview surveys. Survey cost are lower for mail and telephone surveys. Survey cost are lower for mail and telephone surveys. With well-trained interviewers, higher response rates and longer questionnaires are possible with personal interviews. With well-trained interviewers, higher response rates and longer questionnaires are possible with personal interviews. The design of the questionnaire is critical. The design of the questionnaire is critical.

7. 7 7 Slide Types of Surveys n Surveys Not Involving Questionnaires Often, someone simply counts or measures the sampled items and records the results. Often, someone simply counts or measures the sampled items and records the results. An example is sampling a company’s inventory of parts to estimate the total inventory value. An example is sampling a company’s inventory of parts to estimate the total inventory value.

8. 8 8 Slide Sampling Methods n Sample surveys can also be classified in terms of the sampling method used. n The two categories of sampling methods are: Probabilistic sampling Probabilistic sampling Nonprobabilistic sampling Nonprobabilistic sampling

9. 9 9 Slide Nonprobabilistic Sampling Methods n The probability of obtaining each possible sample can be computed. n Statistically valid statements cannot be made about the precision of the estimates. n Sampling cost is lower and implementation is easier. n Methods include convenience and judgment sampling.

10. 10 Slide Nonprobabilistic Sampling Methods n Convenience Sampling The units included in the sample are chosen because of accessibility. The units included in the sample are chosen because of accessibility. In some cases, convenience sampling is the only practical approach. In some cases, convenience sampling is the only practical approach.

11. 11 Slide Nonprobabilistic Sampling Methods n Judgment Sampling A knowledgeable person selects sampling units that he/she feels are most representative of the population. A knowledgeable person selects sampling units that he/she feels are most representative of the population. The quality of the result is dependent on the judgment of the person selecting the sample. The quality of the result is dependent on the judgment of the person selecting the sample. Generally, no statistical statement should be made about the precision of the result. Generally, no statistical statement should be made about the precision of the result.

12. 12 Slide Probabilistic Sampling Methods n The probability of obtaining each possible sample can be computed. n Confidence intervals can be developed which provide bounds on the sampling error. n Methods include simple random, stratified simple random, cluster, and systematic sampling.

13. 13 Slide Survey Errors n Two types of errors can occur in conducting a survey: Sampling error Sampling error Nonsampling error Nonsampling error

14. 14 Slide Survey Errors n Sampling Error It is defined as the magnitude of the difference between the point estimate, developed from the sample, and the population parameter. It is defined as the magnitude of the difference between the point estimate, developed from the sample, and the population parameter. It occurs because not every element in the population is surveyed. It occurs because not every element in the population is surveyed. It cannot occur in a census. It cannot occur in a census. It can not be avoided, but it can be controlled. It can not be avoided, but it can be controlled.

15. 15 Slide Survey Errors n Nonsampling Error It can occur in both a census and a sample survey. It can occur in both a census and a sample survey. Examples include: Examples include: Measurement error Measurement error Errors due to nonresponse Errors due to nonresponse Errors due to lack of respondent knowledge Errors due to lack of respondent knowledge Selection error Selection error Processing error Processing error

16. 16 Slide Survey Errors n Nonsampling Error Measurement Error Measurement Error Measuring instruments are not properly calibrated.Measuring instruments are not properly calibrated. People taking the measurements are not properly trained.People taking the measurements are not properly trained.

17. 17 Slide Survey Errors n Nonsampling Error Errors Due to Nonresponse Errors Due to Nonresponse They occur when no data can be obtained, or only partial data are obtained, for some of the units surveyed.They occur when no data can be obtained, or only partial data are obtained, for some of the units surveyed. The problem is most serious when a bias is created.The problem is most serious when a bias is created.

18. 18 Slide Survey Errors n Nonsampling Error Errors Due to Lack of Respondent Knowledge Errors Due to Lack of Respondent Knowledge These errors on common in technical surveys.These errors on common in technical surveys. Some respondents might be more capable than others of answering technical questions.Some respondents might be more capable than others of answering technical questions.

19. 19 Slide Survey Errors n Nonsampling Error Selection Error Selection Error An inappropriate item is included in the survey.An inappropriate item is included in the survey. For example, in a survey of “small truck owners” some interviewers include SUV owners while other interviewers do not.For example, in a survey of “small truck owners” some interviewers include SUV owners while other interviewers do not.

20. 20 Slide Survey Errors n Nonsampling Error Processing Error Processing Error Data is incorrectly recorded.Data is incorrectly recorded. Data is incorrectly transferred from recording forms to computer files.Data is incorrectly transferred from recording forms to computer files.

21. 21 Slide Simple Random Sampling n A simple random sample of size n from a finite population of size N is a sample selected such that every possible sample of size n has the same probability of being selected. n We begin by developing a frame or list of all elements in the population. n Then a selection procedure, based on the use of random numbers, is used to ensure that each element in the sampled population has the same probability of being selected.

22. 22 Slide Simple Random Sampling We will see in the upcoming slides how to: n Estimate the following population parameters: Population mean Population mean Population total Population total Population proportion Population proportion n Determine the appropriate sample size

23. 23 Slide n In a sample survey it is common practice to provide an approximate 95% confidence interval estimate of the population parameter. n Assuming the sampling distribution of the point estimator can be approximated by a normal probability distribution, we use a value of z = 2 for a 95% confidence interval. n The interval estimate is: Point Estimator +/- 2(Estimate of the Standard Error Point Estimator +/- 2(Estimate of the Standard Error of the Point Estimator) of the Point Estimator) n The bound on the sampling error is: 2(Estimate of the Standard Error of the Point Estimator) 2(Estimate of the Standard Error of the Point Estimator) Simple Random Sampling

24. 24 Slide n Population Mean Point Estimator Point Estimator Estimate of the Standard Error of the Mean Estimate of the Standard Error of the Mean Simple Random Sampling

25. 25 Slide n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Simple Random Sampling

26. 26 Slide n Population Total Point Estimator Point Estimator Estimate of the Standard Error of the Total Estimate of the Standard Error of the Total Simple Random Sampling

27. 27 Slide n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Simple Random Sampling

28. 28 Slide Simple Random Sampling n Population Proportion Point Estimator Point Estimator Estimate of the Standard Error of the Proportion Estimate of the Standard Error of the Proportion

29. 29 Slide Simple Random Sampling n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

30. 30 Slide Determining the Sample Size n An important consideration in sample design is the choice of sample size. n The best choice usually involves a tradeoff between cost and precision (size of the confidence interval). n Larger samples provide greater precision, but are more costly. n A budget might dictate how large the sample can be. n A specified level of precision might dictate how small a sample can be.

31. 31 Slide Determining the Sample Size n Smaller confidence intervals provide more precision. n The size of the approximate confidence interval depends on the bound B on the sampling error. n Choosing a level of precision amounts to choosing a value for B. n Given a desired level of precision, we can solve for the value of n.

32. 32 Slide Simple Random Sampling n Necessary Sample Size for Estimating the Population Mean Hence,

33. 33 Slide Example: Innis Investments n Simple Random Sampling Innis is a financial advisor for 200 clients. A sample of 40 clients has been taken to obtain various demographic data and information about the clients’ investment objectives. Statistics of particular interest are the clients’ age, clients’ total net worth, and the proportion favoring fixed income investments.

34. 34 Slide Example: Innis Investments n Simple Random Sampling For the sample, the mean age was 52 (with a standard deviation of 10), the mean net worth was $480,000 (with a standard deviation of $120,000), and the proportion favoring fixed-income investments was.30.

35. 35 Slide n Estimate of Standard Error of Mean Age n Approximate 95% Confidence Interval for Mean Age Example: Innis Investments

36. 36 Slide n Point Estimate of Total Net Worth (TNW) of Clients n Estimate of Standard Error of TNW = $6,788,400 = $6,788,400 n Approximate 95% Confidence Interval for TNW = $82,423,200 to $109,576,800 = $82,423,200 to $109,576,800 Example: Innis Investments

37. 37 Slide n Point Estimate of Population Proportion Favoring Fixed-Income Investments p =.30 n Estimate of Standard Error of Proportion n Approximate 95% Confidence Interval Example: Innis Investments

38. 38 Slide One year later Innis wants to again survey his clients. He now has 250 clients and wants to set a bound of $30,000 on the error of the estimate of their mean net worth. n Necessary Sample Size He will need a sample size of 51. Example: Innis Investments

39. 39 Slide Stratified Simple Random Sampling n The population is first divided into H groups, called strata. n Then for stratum h, a simple random sample of size n h is selected. n The data from the H simple random samples are combined to develop an estimate of a population parameter. n If the variability within each stratum is smaller than the variability across the strata, a stratified simple random sample can lead to greater precision. n The basis for forming the various strata depends on the judgment of the designer of the sample.

40. 40 Slide Stratified Simple Random Sampling n Population Mean Point Estimator Point Estimator where: H = number of strata = sample mean for stratum h = sample mean for stratum h N h = number of elements in the population in stratum h N = total number of elements N = total number of elements in the population (all strata)

41. 41 Slide Stratified Simple Random Sampling n Population Mean Estimate of the Standard Error of the Mean Estimate of the Standard Error of the Mean

42. 42 Slide n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Stratified Simple Random Sampling

43. 43 Slide Stratified Simple Random Sampling n Population Total Point Estimator Point Estimator Estimate of the Standard Error of the Total Estimate of the Standard Error of the Total

44. 44 Slide Stratified Simple Random Sampling n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

45. 45 Slide n Population Proportion Point Estimator Point Estimator where: H = number of strata h = sample proportion for stratum h h = sample proportion for stratum h N h = number of elements in the population in stratum h N = total number of elements N = total number of elements in the population (all strata) Stratified Simple Random Sampling

46. 46 Slide Stratified Simple Random Sampling n Population Proportion Estimate of Standard Error of the Proportion Estimate of Standard Error of the Proportion

47. 47 Slide n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Stratified Simple Random Sampling

48. 48 Slide Stratified Simple Random Sampling n Total Sample Size When Estimating Population Mean

49. 49 Slide Stratified Simple Random Sampling n Total Sample Size When Estimating Population Total

50. 50 Slide Stratified Simple Random Sampling n Allocating Total Sample Size When Estimating Population Mean or Total

51. 51 Slide Stratified Simple Random Sampling n Total Sample Size When Estimating Population Proportion

52. 52 Slide Stratified Simple Random Sampling n Allocating Total Sample Size When Estimating Population Proportion

53. 53 Slide Example: Mill Creek Co. n Stratified Simple Random Sampling Mill Creek Co. 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.

54. 54 Slide n Data Annual Family Annual Family Dental Expense Proportion Dental Expense Proportion Stratum N h n h Mean St.Dev. Married Stratum N h n h Mean St.Dev. Married Under 3010030$250 $75.60 30-4925045400 100.70 30-4925045400 100.70 50 or Over12530425 130.68 50 or Over12530425 130.68 475 105 475 105 Example: Mill Creek Co.

55. 55 Slide n Point Estimate of Mean Annual Dental Expense = $375 = $375 n Estimate of Standard Error of Mean = 9.27 Example: Mill Creek Co.

56. 56 Slide n Approximate 95% Confidence Interval for Mean Annual Dental Expense An approximate 95% confidence interval for mean annual family dental expense is $356.46 to $393.54. Example: Mill Creek Co.

57. 57 Slide n Point Estimate of Total Family Expense for All Employees n Approximate 95% Confidence Interval = $169,318 to $186,932 Example: Mill Creek Co.

58. 58 Slide n Point Estimate of Proportion Married n Estimate of Standard Error of Proportion =.0417 =.0417 n Approximate 95% Confidence Interval for Proportion Example: Mill Creek Co.

59. 59 Slide Cluster Sampling n Cluster sampling requires that the population be divided into N groups of elements called clusters. n We would define the frame as the list of N clusters. n We then select a simple random sample of n clusters. n We would then collect data for all elements in each of the n clusters.

60. 60 Slide Cluster Sampling n Cluster sampling tends to provide better results than stratified sampling when the elements within the clusters are heterogeneous. n A primary application of cluster sampling involves area sampling, where the clusters are counties, city blocks, or other well-defined geographic sections.

61. 61 Slide Cluster Sampling n Notation N = number of clusters in the population N = number of clusters in the population n = number of clusters selected in the sample n = number of clusters selected in the sample M i = number of elements in cluster i M = number of elements in the population M = number of elements in the population M = average number of elements in a cluster M = average number of elements in a cluster x i = total of all observations in cluster i x i = total of all observations in cluster i a i = number of observations in cluster i with a i = number of observations in cluster i with a certain characteristic

62. 62 Slide n Population Mean Point Estimator Point Estimator Estimate of Standard Error of the Mean Estimate of Standard Error of the Mean Cluster Sampling

63. 63 Slide n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Cluster Sampling

64. 64 Slide n Population Total Point Estimator Point Estimator Estimate of Standard Error of the Total Estimate of Standard Error of the Total Cluster Sampling

65. 65 Slide n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Cluster Sampling

66. 66 Slide n Population Proportion Point Estimator Point Estimator Cluster Sampling

67. 67 Slide Cluster Sampling n Population Proportion Estimate of Standard Error of the Proportion Estimate of Standard Error of the Proportion

68. 68 Slide Cluster Sampling n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

69. 69 Slide Example: Cooper County Schools n Cluster Sampling 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 football team at those schools. There are a total of 1200 high school seniors in the county playing football. Data obtained from the questionnaire are shown on the next slide.

70. 70 Slide Example: Cooper County Schools n Data Number Average Number Planning School of Players SAT Score to Attend College Number Average Number Planning School of Players SAT Score to Attend College 14584015 22098016 33090512 43888018 54097023 17384

71. 71 Slide n Point Estimate of Mean SAT Score Example: Cooper County Schools

72. 72 Slide Example: Cooper County Schools n Estimate of Standard Error of the Mean

73. 73 Slide n Approximate 95% Confidence Interval n Point Estimate of Proportion Planning to Attend College Example: Cooper County Schools

74. 74 Slide Systematic Sampling n Systematic Sampling is often used as an alternative to simple random sampling which can be time- consuming if a large population is involved. n 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. n We would randomly select one of the first N / n elements and then select every ( N / n )th element thereafter. n Since the first element selected is a random choice, a systematic sample is often assumed to have the properties of a simple random sample.

75. 75 Slide End of Chapter 21