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

2. 2 2 Slide © 2003 South-Western/Thomson Learning™ Chapter 7 Sampling and Sampling Distributions n Simple Random Sampling n Point Estimation n Introduction to Sampling Distributions n Sampling Distribution of n Properties of Point Estimators n Other Sampling Methods n = 100 n = 30

3. 3 3 Slide © 2003 South-Western/Thomson Learning™ Statistical Inference n The purpose of statistical inference is to obtain information about a population from information contained in a sample. n A population is the set of all the elements of interest. n A sample is a subset of the population. n The sample results provide only estimates of the values of the population characteristics. n A parameter is a numerical characteristic of a population. n With proper sampling methods, the sample results will provide “good” estimates of the population characteristics.

4. 4 4 Slide © 2003 South-Western/Thomson Learning™ Simple Random Sampling n Finite Population A simple random sample from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. A simple random sample from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often. Sampling without replacement is the procedure used most often. In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. In large sampling projects, computer-generated random numbers are often used to automate the sample selection process.

5. 5 5 Slide © 2003 South-Western/Thomson Learning™ n Infinite Population A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. Each element selected comes from the same population.Each element selected comes from the same population. Each element is selected independently.Each element is selected independently. The population is usually considered infinite if it involves an ongoing process that makes listing or counting every element impossible. The population is usually considered infinite if it involves an ongoing process that makes listing or counting every element impossible. The random number selection procedure cannot be used for infinite populations. The random number selection procedure cannot be used for infinite populations. Simple Random Sampling

6. 6 6 Slide © 2003 South-Western/Thomson Learning™ Point Estimation n In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to as the point estimator of the population mean . We refer to as the point estimator of the population mean . s is the point estimator of the population standard deviation . s is the point estimator of the population standard deviation . n is the point estimator of the population proportion p.

7. 7 7 Slide © 2003 South-Western/Thomson Learning™ Sampling Error n The absolute difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. n Sampling error is the result of using a subset of the population (the sample), and not the entire population to develop estimates. n The sampling errors are: for sample mean | s -  for sample standard deviation | s -  for sample standard deviation for sample proportion

8. 8 8 Slide © 2003 South-Western/Thomson Learning™ Example: St. Andrew’s St. Andrew’s University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

9. 9 9 Slide © 2003 South-Western/Thomson Learning™ Example: St. Andrew’s The director of admissions would like to know the following information: the average SAT score for the applicants, and the average SAT score for the applicants, and the proportion of applicants that want to live on campus. the proportion of applicants that want to live on campus. We will now look at three alternatives for obtaining the desired information. Conducting a census of the entire 900 applicants Conducting a census of the entire 900 applicants Selecting a sample of 30 applicants, using a random number table Selecting a sample of 30 applicants, using a random number table Selecting a sample of 30 applicants, using computer-generated random numbers Selecting a sample of 30 applicants, using computer-generated random numbers

10. 10 Slide © 2003 South-Western/Thomson Learning™ n Taking a Census of the 900 Applicants SAT Scores SAT Scores Population MeanPopulation Mean Population Standard DeviationPopulation Standard Deviation Applicants Wanting On-Campus Housing Applicants Wanting On-Campus Housing Population ProportionPopulation Proportion Example: St. Andrew’s

11. 11 Slide © 2003 South-Western/Thomson Learning™ Example: St. Andrew’s n Take a Sample of 30 Applicants Using a Random Number Table Since the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. We will use the last three digits of the 5-digit random numbers in the third column of a random number table. The numbers we draw will be the numbers of the applicants we will sample unless the random number is greater than 900 orthe random number is greater than 900 or the random number has already been used.the random number has already been used. We will continue to draw random numbers until we have selected 30 applicants for our sample.

12. 12 Slide © 2003 South-Western/Thomson Learning™ Example: St. Andrew’s n Use of Random Numbers for Sampling 3-DigitApplicant 3-DigitApplicant Random Number Included in Sample Random Number Included in Sample 744 No. 744 744 No. 744 436 No. 436 436 No. 436 865 No. 865 865 No. 865 790 No. 790 790 No. 790 835 No. 835 835 No. 835 902 Number exceeds 900 902 Number exceeds 900 190 No. 190 190 No. 190 436 Number already used 436 Number already used etc. etc. etc. etc.

13. 13 Slide © 2003 South-Western/Thomson Learning™ n Sample Data Random Random No. Number Applicant SAT Score On-Campus 1 744 Connie Reyman 1025 Yes 1 744 Connie Reyman 1025 Yes 2 436 William Fox 950 Yes 2 436 William Fox 950 Yes 3 865 Fabian Avante 1090 No 3 865 Fabian Avante 1090 No 4 790 Eric Paxton 1120 Yes 4 790 Eric Paxton 1120 Yes 5 835 Winona Wheeler 1015 No 5 835 Winona Wheeler 1015 No.......... 30 685 Kevin Cossack 965 No 30 685 Kevin Cossack 965 No Example: St. Andrew’s

14. 14 Slide © 2003 South-Western/Thomson Learning™ Example: St. Andrew’s n Take a Sample of 30 Applicants Using Computer-Generated Random Numbers Excel provides a function for generating random numbers in its worksheet. Excel provides a function for generating random numbers in its worksheet. 900 random numbers are generated, one for each applicant in the population. 900 random numbers are generated, one for each applicant in the population. Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. Each of the 900 applicants have the same probability of being included. Each of the 900 applicants have the same probability of being included.

15. 15 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Select a Simple Random Sample n Formula Worksheet Note: Rows 10-901 are not shown.

16. 16 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Select a Simple Random Sample n Value Worksheet (Unsorted) Note: Rows 10-901 are not shown.

17. 17 Slide © 2003 South-Western/Thomson Learning™ n Put Random Numbers in Ascending Order Step 1 Select cells A2:A901 Step 1 Select cells A2:A901 Step 2 Select the Data pull-down menu Step 2 Select the Data pull-down menu Step 3 Choose the Sort option Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: Step 4 When the Sort dialog box appears: Choose Random Numbers Choose Random Numbers in the Sort by text box Choose Ascending Choose Ascending Click OK Click OK Using Excel to Select a Simple Random Sample

18. 18 Slide © 2003 South-Western/Thomson Learning™ Using Excel to Select a Simple Random Sample n Value Worksheet (Sorted) Note: Rows 10-901 are not shown.

19. 19 Slide © 2003 South-Western/Thomson Learning™ n Point Estimates as Point Estimator of  as Point Estimator of  s as Point Estimator of  s as Point Estimator of  as Point Estimator of p as Point Estimator of p n Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Example: St. Andrew’s

20. 20 Slide © 2003 South-Western/Thomson Learning™ Sampling Distribution of n Making Inferences about a Population Mean Population Population with mean  = ? Population Population with mean  = ? A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean. The sample data provide a value for the sample mean. The value of is used to make inferences about the value of . The value of is used to make inferences about the value of .

21. 21 Slide © 2003 South-Western/Thomson Learning™ n The sampling distribution of is the probability distribution of all possible values of the sample mean. n Expected Value of E ( ) =  E ( ) = where:  = the population mean  = the population mean Sampling Distribution of

22. 22 Slide © 2003 South-Western/Thomson Learning™ Desirable Property of Point Estimators n Unbiasedness 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.

23. 23 Slide © 2003 South-Western/Thomson Learning™ n Standard Deviation of Finite Population Infinite Population Finite Population Infinite Population A finite population is treated as being infinite if n / N <.05. A finite population is treated as being infinite if n / N <.05. is the finite correction factor. is the finite correction factor. is referred to as the standard error of the mean. is referred to as the standard error of the mean. Sampling Distribution of

24. 24 Slide © 2003 South-Western/Thomson Learning™ n If we use a large ( n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of can be approximated by a normal probability distribution. n When the simple random sample is small ( n < 30), the sampling distribution of can be considered normal only if we assume the population has a normal probability distribution. Sampling Distribution of

25. 25 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for the SAT Scores Example: St. Andrew’s

26. 26 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for the SAT Scores What is the probability that a simple random sample of 30 applicants will provide an estimate of the population mean SAT score that is within plus or minus 10 of the actual population mean  ? In other words, what is the probability that will be between 980 and 1000? Example: St. Andrew’s

27. 27 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for the SAT Scores Using the standard normal probability table with z = 10/14.6=.68, we have area = (.2518)(2) =.5036 Sampling distribution of Sampling distribution of 1000980990 Area =.2518 Example: St. Andrew’s

28. 28 Slide © 2003 South-Western/Thomson Learning™ n The sampling distribution of is the probability distribution of all possible values of the sample proportion. n Expected Value of where: p = the population proportion n Thus, is an unbiased estimate of the population proportion, p. Sampling Distribution of

29. 29 Slide © 2003 South-Western/Thomson Learning™ Sampling Distribution of n Making Inferences about a Population Proportion Population Population with proportion p = ? Population Population with proportion p = ? A simple random sample of n elements is selected from the population. The sample data provide a value for the sample proportion. The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. The value of is used to make inferences about the value of p.

30. 30 Slide © 2003 South-Western/Thomson Learning™ Sampling Distribution of n Standard Deviation of Finite Population Infinite Population is referred to as the standard error of the proportion. is referred to as the standard error of the proportion.

31. 31 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for In-State Residents The normal probability distribution is an acceptable approximation since np = 30(.72) = 21.6 > 5 and n (1 - p ) = 30(.28) = 8.4 > 5. Example: St. Andrew’s

32. 32 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for In-State Residents What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on- campus housing that is within plus or minus.05 of the actual population proportion? In other words, what is the probability that will be between.67 and.77? Example: St. Andrew’s

33. 33 Slide © 2003 South-Western/Thomson Learning™ n Sampling Distribution of for In-State Residents For z =.05/.082 =.61, the area = (.2291)(2) =.4582. The probability is.4582 that the sample proportion will be within +/-.05 of the actual population proportion. Sampling distribution of Sampling distribution of 0.77 0.670.72 Area =.2291 Example: St. Andrew’s

34. 34 Slide © 2003 South-Western/Thomson Learning™ Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

35. 35 Slide © 2003 South-Western/Thomson Learning™ Stratified Random Sampling n The population is first divided into groups of elements called strata. n Each element in the population belongs to one and only one stratum. n Best results are obtained when the elements within each stratum are as much alike as possible (i.e. homogeneous group). n A simple random sample is taken from each stratum. n Formulas are available for combining the stratum sample results into one population parameter estimate.

36. 36 Slide © 2003 South-Western/Thomson Learning™ Stratified Random Sampling n Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size. n Example: The basis for forming the strata might be department, location, age, industry type, etc.

37. 37 Slide © 2003 South-Western/Thomson Learning™ Cluster Sampling n The population is first divided into separate groups of elements called clusters. n Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). n A simple random sample of the clusters is then taken. n All elements within each sampled (chosen) cluster form the sample. … continued

38. 38 Slide © 2003 South-Western/Thomson Learning™ Cluster Sampling n Advantage: The close proximity of elements can be cost effective (I.e. many sample observations can be obtained in a short time). n Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling. n Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas.

39. 39 Slide © 2003 South-Western/Thomson Learning™ Systematic Sampling n If a sample size of n is desired from a population containing N elements, we might sample one element for every n / N elements in the population. n We randomly select one of the first n / N elements from the population list. n We then select every n / N th element that follows in the population list. n This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. … continued

40. 40 Slide © 2003 South-Western/Thomson Learning™ Systematic Sampling n Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. n Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing.

41. 41 Slide © 2003 South-Western/Thomson Learning™ Convenience Sampling n It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected. n The sample is identified primarily by convenience. n Advantage: Sample selection and data collection are relatively easy. n Disadvantage: It is impossible to determine how representative of the population the sample is. n Example: A professor conducting research might use student volunteers to constitute a sample.

42. 42 Slide © 2003 South-Western/Thomson Learning™ Judgment Sampling n The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population. n It is a nonprobability sampling technique. n Advantage: It is a relatively easy way of selecting a sample. n Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample. n Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate.

43. 43 Slide © 2003 South-Western/Thomson Learning™ End of Chapter 7