1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 7: Sampling and Sampling Distributions Simple Random Sampling Simple Random Sampling Point Estimation Point Estimation Introduction to Sampling Distributions Introduction to Sampling Distributions Other Sampling Methods Other Sampling Methods

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. The purpose of statistical inference is to obtain The purpose of statistical inference is to obtain information about a population from information information about a population from information contained in a sample. contained in a sample. Statistical Inference A population is the set of all the elements of interest. A population is the set of all the elements of interest. A sample is a subset of the population. A sample is a subset of the population.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. Statistical Inference

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Finite Population n Finite populations are often defined by lists such as: Organization membership roster Organization membership roster Credit card account numbers Credit card account numbers Inventory product numbers Inventory product numbers n A simple random sample of size n from a finite population of size N is a sample selected such that population of size N is a sample selected such that each possible sample of size n has the same each possible sample of size n has the same probability of being selected. probability of being selected.

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Finite Population In large sampling projects, computer-generated In large sampling projects, computer-generated random numbers are often used to automate the random numbers are often used to automate the sample selection process. sample selection process. Sampling without replacement is the procedure Sampling without replacement is the procedure used most often. used most often. Replacing each sampled element before selecting Replacing each sampled element before selecting subsequent elements is called sampling with subsequent elements is called sampling with replacement. replacement.

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved n Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely. Simple Random Sampling: Infinite Population n A simple random sample from an infinite population is a sample selected such that the following conditions is a sample selected such that the following conditions are satisfied. are satisfied. Each element selected comes from the same Each element selected comes from the same population. population. Each element is selected independently. Each element is selected independently.

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Infinite Population The random number selection procedure cannot be The random number selection procedure cannot be used for infinite populations. used for infinite populations. In the case of infinite populations, it is impossible to In the case of infinite populations, it is impossible to obtain a list of all elements in the population. obtain a list of all elements in the population.

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. Point Estimation We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . is the point estimator of the population proportion p. is the point estimator of the population proportion p.

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Error Statistical methods can be used to make probability Statistical methods can be used to make probability statements about the size of the sampling error. statements about the size of the sampling error. Sampling error is the result of using a subset of the Sampling error is the result of using a subset of the population (the sample), and not the entire population (the sample), and not the entire population. population. The absolute value of the difference between an The absolute value of the difference between an unbiased point estimate and the corresponding unbiased point estimate and the corresponding population parameter is called the sampling error. population parameter is called the sampling error. When the expected value of a point estimator is equal When the expected value of a point estimator is equal to the population parameter, the point estimator is said to the population parameter, the point estimator is said to be unbiased. to be unbiased.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Error n The sampling errors are: for sample proportion for sample standard deviation for sample mean

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s St. Andrew’s College receives St. Andrew’s College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s The director of admissions The director of admissions would like to know the following information: the average SAT score for the average SAT score for the 900 applicants, and the 900 applicants, and the proportion of the proportion of applicants that want to live on campus.

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s We will now look at two alternatives for obtaining the desired information. n Conducting a census of the entire 900 applicants entire 900 applicants n Selecting a sample of 30 applicants, using Excel applicants, using Excel

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Conducting a Census n If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. n We will assume for the moment that conducting a census is practical in this example.

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling The applicants were numbered, from 1 to 900, as The applicants were numbered, from 1 to 900, as their applications arrived. their applications arrived. She decides a sample of 30 applicants will be used. She decides a sample of 30 applicants will be used. Furthermore, the Director of Admissions must obtain Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for estimates of the population parameters of interest for a meeting taking place in a few hours. a meeting taking place in a few hours. Now suppose that the necessary data on the Now suppose that the necessary data on the current year’s applicants were not yet entered in the current year’s applicants were not yet entered in the college’s database. college’s database.

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved n Sample Data Simple Random Sampling: Using a Random Number Table 1744 Conrad Harris1025 Yes 2436 Enrique Romero 950 Yes 3865 Fabian Avante1090 No 4790 Lucila Cruz1120 Yes 5835 Chan Chiang 930 No..... 30 498 Emily Morse 1010 No No. RandomNumber Applicant SAT Score Score Live On- Campus.....

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved n Taking a Sample of 30 Applicants Then we choose the 30 applicants corresponding Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. to the 30 smallest random numbers as our sample. For example, Excel’s function For example, Excel’s function = RANDBETWEEN(1,900) = RANDBETWEEN(1,900) can be used to generate random numbers between can be used to generate random numbers between 1 and 900. 1 and 900. Computers can be used to generate random Computers can be used to generate random numbers for selecting random samples. numbers for selecting random samples. Simple Random Sampling: Using a Computer

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved as Point Estimator of  as Point Estimator of  n as Point Estimator of p Point Estimation Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. s as Point Estimator of  s as Point Estimator of 

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score 990997  = Population std. deviation for deviation for SAT score SAT score 80 s = Sample std. s = Sample std. deviation for deviation for SAT score SAT score75.2 p = Population pro- portion wanting portion wanting campus housing campus housing.72.68 Summary of Point Estimates Obtained from a Simple Random Sample = Sample mean = Sample mean SAT score SAT score = Sample pro- = Sample pro- portion wanting portion wanting campus housing campus housing

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved Yesterday’s slide: One needs to know that every statistic has a sampling distribution, which shows every possible value the statistic can take on and the corresponding probability of occurrence.

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved n Process of Statistical Inference The value of is used to make inferences about the value of . The sample data provide a value for the sample mean. A simple random sample of n elements is selected from the population. Population with mean  = ? Sampling Distribution of

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved Form of the Sampling Distribution of When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size. In cases where the population is highly skewed or outliers are present, samples of size 50 may be needed. In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size 30 or more.

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved A simple random sample of n elements is selected from the population. Population with proportion p = ? n Making Inferences about a Population Proportion The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. Sampling Distribution of

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever the sample size by a normal distribution whenever the sample size is large. is large. The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal distribution whenever the sample size by a normal distribution whenever the sample size is large. is large. The sample size is considered large whenever these The sample size is considered large whenever these conditions are satisfied: conditions are satisfied: The sample size is considered large whenever these The sample size is considered large whenever these conditions are satisfied: conditions are satisfied: np > 5 n (1 – p ) > 5 and Form of the Sampling Distribution of

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Recall that 72% of the Recall that 72% of the prospective students applying to St. Andrew’s College desire on-campus housing. n Example: St. Andrew’s College Sampling Distribution of What is the probability that What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicant desiring on-campus housing that is within plus or minus.05 of the actual population proportion?

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved P (.67 < <.77) =.4582 Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.61 < z <.61) = P ( z <.61)  P ( z < -.61) =.7291 .2709 =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion : Sampling Distribution of

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Properties of Point Estimators n Before using a sample statistic as a point estimator, statisticians check to see whether the sample statistic has the following properties associated with good point estimators. Consistency Efficiency Unbiased

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved Properties of Point Estimators If the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an unbiased estimator of the population parameter. Unbiased

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved Properties of Point Estimators Given the choice of two unbiased estimators of the same population parameter, we would prefer to use the point estimator with the smaller standard deviation, since it tends to provide estimates closer to the population parameter. The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other. Efficiency

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved Properties of Point Estimators A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger. Consistency

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. Stratified Random Sampling Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group). Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group).

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved Stratified Random Sampling A simple random sample is taken from each stratum. A simple random sample is taken from each stratum. Formulas are available for combining the stratum Formulas are available for combining the stratum sample results into one population parameter sample results into one population parameter estimate. estimate. Formulas are available for combining the stratum Formulas are available for combining the stratum sample results into one population parameter sample results into one population parameter estimate. estimate. Advantage: If strata are homogeneous, this method Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with is as “precise” as simple random sampling but with a smaller total sample size. a smaller total sample size. Advantage: If strata are homogeneous, this method Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with is as “precise” as simple random sampling but with a smaller total sample size. a smaller total sample size. Example: The basis for forming the strata might be Example: The basis for forming the strata might be department, location, age, industry type, and so on. department, location, age, industry type, and so on. Example: The basis for forming the strata might be Example: The basis for forming the strata might be department, location, age, industry type, and so on. department, location, age, industry type, and so on.

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved Cluster Sampling The population is first divided into separate groups The population is first divided into separate groups of elements called clusters. of elements called clusters. The population is first divided into separate groups The population is first divided into separate groups of elements called clusters. of elements called clusters. Ideally, each cluster is a representative small-scale Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). version of the population (i.e. heterogeneous group). Ideally, each cluster is a representative small-scale Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). version of the population (i.e. heterogeneous group). A simple random sample of the clusters is then taken. A simple random sample of the clusters is then taken. All elements within each sampled (chosen) cluster All elements within each sampled (chosen) cluster form the sample. form the sample. All elements within each sampled (chosen) cluster All elements within each sampled (chosen) cluster form the sample. form the sample.

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved Cluster Sampling Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas.

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved Systematic Sampling If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list.

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved Systematic Sampling This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved Convenience Sampling It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. The sample is identified primarily by convenience. The sample is identified primarily by convenience.

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved Advantage: Sample selection and data collection are Advantage: Sample selection and data collection are relatively easy. relatively easy. Advantage: Sample selection and data collection are Advantage: Sample selection and data collection are relatively easy. relatively easy. Disadvantage: It is impossible to determine how Disadvantage: It is impossible to determine how representative of the population the sample is. representative of the population the sample is. Disadvantage: It is impossible to determine how Disadvantage: It is impossible to determine how representative of the population the sample is. representative of the population the sample is. Convenience Sampling

42. 42 Slide © 2008 Thomson South-Western. All Rights Reserved Judgment Sampling The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. It is a nonprobability sampling technique. It is a nonprobability sampling technique. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate.

43. 43 Slide © 2008 Thomson South-Western. All Rights Reserved Judgment Sampling Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample.