1 1 Slide © 2003 South-Western/Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University

2 2 Slide © 2003 South-Western/Thomson Learning™ Chapter 18 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ Survey Errors n Two types of errors can occur in conducting a survey: Sampling error Sampling error Nonsampling error Nonsampling error

14 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Simple Random Sampling

26 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Simple Random Sampling

28 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ Simple Random Sampling n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

30 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ Simple Random Sampling n Necessary Sample Size for Estimating the Population Mean Hence,

33 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ 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 Slide © 2003 South-Western/Thomson Learning™ n Estimate of Standard Error of Mean Age n Approximate 95% Confidence Interval for Mean Age Example: Innis Investments

36 Slide © 2003 South-Western/Thomson Learning™ n Point Estimate of Total Net Worth (TNW) of Clients n Estimate of Standard Error of TNW = $3,394,113 = $3,394,113 n Approximate 95% Confidence Interval for TNW = $89,211,774 to $102,788,226 = $89,211,774 to $102,788,226 Example: Innis Investments

37 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Simple Random Sampling: Population Total n Formula Worksheet Note: Rows are not shown.

38 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Simple Random Sampling: Population Total n Value Worksheet Note: Rows are not shown.

39 Slide © 2003 South-Western/Thomson Learning™ 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

40 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Simple Random Sampling: Population Proportion n Formula Worksheet Note: Rows are not shown.

41 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Simple Random Sampling: Population Proportion n Value Worksheet Note: Rows are not shown.

42 Slide © 2003 South-Western/Thomson Learning™ 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

43 Slide © 2003 South-Western/Thomson Learning™ 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.

44 Slide © 2003 South-Western/Thomson Learning™ 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.

45 Slide © 2003 South-Western/Thomson Learning™ 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 $250 $ or Over or Over Example: Mill Creek Co.

46 Slide © 2003 South-Western/Thomson Learning™ 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)

47 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Population Mean Estimate of the Standard Error of the Mean Estimate of the Standard Error of the Mean

48 Slide © 2003 South-Western/Thomson Learning™ n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Stratified Simple Random Sampling

49 Slide © 2003 South-Western/Thomson Learning™ n Point Estimate of Mean Annual Dental Expense = $375 = $375 n Estimate of Standard Error of Mean = 9.27 Example: Mill Creek Co.

50 Slide © 2003 South-Western/Thomson Learning™ n Approximate 95% Confidence Interval for Mean Annual Dental Expense An approximate 95% confidence interval for mean annual family dental expense is $ to $ Example: Mill Creek Co.

51 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Stratified Simple Random Sampling: Population Mean n Formula Worksheet =SUM(D2:D4) =SUM(E2:E4)

52 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Stratified Simple Random Sampling: Population Mean n Value Worksheet

53 Slide © 2003 South-Western/Thomson Learning™ 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

54 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

55 Slide © 2003 South-Western/Thomson Learning™ n Point Estimate of Total Family Expense for All Employees n Approximate 95% Confidence Interval = $169,318 to $186,932 Example: Mill Creek Co.

56 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Stratified Simple Random Sampling: Population Total n Formula Worksheet =SUM(D2:D4) =SUM(E2:E4)

57 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Stratified Simple Random Sampling: Population Total n Value Worksheet

58 Slide © 2003 South-Western/Thomson Learning™ 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

59 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Population Proportion Estimate of Standard Error of the Proportion Estimate of Standard Error of the Proportion

60 Slide © 2003 South-Western/Thomson Learning™ n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Stratified Simple Random Sampling

61 Slide © 2003 South-Western/Thomson Learning™ 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.

62 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Total Sample Size When Estimating Population Mean

63 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Total Sample Size When Estimating Population Total

64 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Allocating Total Sample Size When Estimating Population Mean or Total

65 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Total Sample Size When Estimating Population Proportion

66 Slide © 2003 South-Western/Thomson Learning™ Stratified Simple Random Sampling n Allocating Total Sample Size When Estimating Population Proportion

67 Slide © 2003 South-Western/Thomson Learning™ 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.

68 Slide © 2003 South-Western/Thomson Learning™ 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.

69 Slide © 2003 South-Western/Thomson Learning™ 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

70 Slide © 2003 South-Western/Thomson Learning™ n Population Mean Point Estimator Point Estimator Estimate of Standard Error of the Mean Estimate of Standard Error of the Mean Cluster Sampling

71 Slide © 2003 South-Western/Thomson Learning™ n Population Mean Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Cluster Sampling

72 Slide © 2003 South-Western/Thomson Learning™ n Population Total Point Estimator Point Estimator Estimate of Standard Error of the Total Estimate of Standard Error of the Total Cluster Sampling

73 Slide © 2003 South-Western/Thomson Learning™ n Population Total Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate Cluster Sampling

74 Slide © 2003 South-Western/Thomson Learning™ n Population Proportion Point Estimator Point Estimator Cluster Sampling

75 Slide © 2003 South-Western/Thomson Learning™ Cluster Sampling n Population Proportion Estimate of Standard Error of the Proportion Estimate of Standard Error of the Proportion

76 Slide © 2003 South-Western/Thomson Learning™ Cluster Sampling n Population Proportion Interval Estimate Interval Estimate Approximate 95% Confidence Interval Estimate Approximate 95% Confidence Interval Estimate

77 Slide © 2003 South-Western/Thomson Learning™ 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.

78 Slide © 2003 South-Western/Thomson Learning™ 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

79 Slide © 2003 South-Western/Thomson Learning™ n Point Estimator of Population Mean SAT Score Example: Cooper County Schools

80 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Estimate of Standard Error of the Point Estimator of Population Mean

81 Slide © 2003 South-Western/Thomson Learning™ n Approximate 95% Confidence Interval Estimate of the Population Mean SAT Score Example: Cooper County Schools

82 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Point Estimator of Population Total SAT Score n Estimate of Standard Error of the Point Estimator of Population Total

83 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Approximate 95% Confidence Interval Estimate of the Population Total SAT Score of the Population Total SAT Score = 1,075, to 1,099, = 1,075, to 1,099,834.72

84 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Point Estimate of Population Proportion Planning to Attend College

85 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Estimate of Standard Error of the Point Estimator of the Population Proportion

86 Slide © 2003 South-Western/Thomson Learning™ Example: Cooper County Schools n Approximate 95% Confidence Interval Estimate of the Population Proportion Planning College of the Population Proportion Planning College = to = to

87 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Cluster Sampling: Population Mean n Formula Worksheet

88 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Cluster Sampling: Population Mean n Value Worksheet

89 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Cluster Sampling: Population Mean n Formula Worksheet Note: Rows 1-8 are hidden. =SQRT(((F9-B7)/(F9*B7*F11^2))*(E7/(F12-1)))

90 Slide © 2003 South-Western/Thomson Learning™ Using Excel for Cluster Sampling: Population Mean n Value Worksheet Note: Rows 1-8 are hidden.

91 Slide © 2003 South-Western/Thomson Learning™ 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.

92 Slide © 2003 South-Western/Thomson Learning™ End of Chapter 18