1. 1 1 Slide © 2001 South-Western/Thomson Learning  Anderson  Sweeney  Williams Anderson  Sweeney  Williams  Slides Prepared by JOHN LOUCKS  CONTEMPORARYBUSINESSSTATISTICS WITH MICROSOFT  EXCEL CONTEMPORARYBUSINESSSTATISTICS

2. 2 2 Slide Chapter 7 Sampling and Sampling Distributions n Simple Random Sampling n Point Estimation n Introduction to Sampling Distributions n Sampling Distribution of n Other Sampling Methods

3. 3 3 Slide 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 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 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 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 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 Example: St. Edward’s St. Edward’s University receives 1,500 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 is an in-state resident. The director of admissions would like to know, at least roughly, the following information: the average SAT score for the applicants, andthe average SAT score for the applicants, and the proportion of applicants that are in-state residents.the proportion of applicants that are in-state residents. We will now look at two alternatives for obtaining the desired information.

9. 9 9 Slide n Alternative #1: Take a Census of 1,500 Applicants SAT ScoresSAT Scores Population MeanPopulation Mean Population Standard DeviationPopulation Standard Deviation In-State ApplicantsIn-State Applicants Population ProportionPopulation Proportion Example: St. Edward’s

10. 10 Slide n Alternative #2: Take a Sample of 50 Applicants Excel can be used to select a simple random sample without replacement.Excel can be used to select a simple random sample without replacement. The process is based on random numbers generated by Excel’s RAND function.The process is based on random numbers generated by Excel’s RAND function. RAND function generates numbers in the interval from 0 to 1.RAND function generates numbers in the interval from 0 to 1. Any number in the interval is equally likely.Any number in the interval is equally likely. The numbers are actually values of a uniformly distributed random variable.The numbers are actually values of a uniformly distributed random variable. Example: St. Edward’s

11. 11 Slide Example: St. Edward’s n Using Excel to Select a Simple Random Sample 1500 random numbers are generated, one for each applicant in the population.1500 random numbers are generated, one for each applicant in the population. Then we choose the 50 applicants corresponding to the 50 smallest random numbers as our sample.Then we choose the 50 applicants corresponding to the 50 smallest random numbers as our sample. Each of the 1500 applicants have the same probability of being included.Each of the 1500 applicants have the same probability of being included.

12. 12 Slide Using Excel to Select a Simple Random Sample n Formula Worksheet Note: Rows 10-1501 are not shown.

13. 13 Slide Using Excel to Select a Simple Random Sample n Value Worksheet Note: Rows 10-1501 are not shown.

14. 14 Slide Using Excel to Select a Simple Random Sample n Put Random Numbers in Ascending Order Step 1 Select cells A2:A1501Step 1 Select cells A2:A1501 Step 2 Select the Data pull-down menuStep 2 Select the Data pull-down menu Step 3 Choose the Sort optionStep 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

15. 15 Slide Using Excel to Select a Simple Random Sample n Value Worksheet (Sorted) Note: Rows 10-1501 are not shown.

16. 16 Slide 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. Edward’s

17. 17 Slide Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean. n Expected Value of E ( x ) =  E ( x ) = where:  = the population mean  = the population mean

18. 18 Slide 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

19. 19 Slide 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

20. 20 Slide n Sampling Distribution of for the SAT Scores Example: St. Edward’s

21. 21 Slide n Sampling Distribution of for the SAT Scores What is the probability that a simple random sample of 50 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. Edward’s

22. 22 Slide n Sampling Distribution of for the SAT Scores Using the standard normal probability table with z = 10/11.3 =.88, we have area = (.3106)(2) =.6212. Sampling distribution of Sampling distribution of 1000980990 Area =.3106 Example: St. Edward’s

23. 23 Slide 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 Sampling Distribution of

24. 24 Slide 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.

25. 25 Slide n Sampling Distribution of for In-State Residents The normal probability distribution is an acceptable approximation since np = 50(.72) = 36 > 5 and n (1 - p ) = 50(.28) = 14 > 5. Example: St. Edward’s

26. 26 Slide n Sampling Distribution of for In-State Residents What is the probability that a simple random sample of 50 applicants will provide an estimate of the population proportion of in-state residents 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. Edward’s

27. 27 Slide n Sampling Distribution of for In-State Residents For z =.05/.0635 =.79, the area = (.2852)(2) =.5704. The probability is.5704 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 =.2852 Example: St. Edward’s

28. 28 Slide Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

29. 29 Slide 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.

30. 30 Slide 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.

31. 31 Slide 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

32. 32 Slide 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.

33. 33 Slide 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

34. 34 Slide 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.

35. 35 Slide 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.

36. 36 Slide 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.

37. 37 Slide End of Chapter 7