1 1 Slide STATISTICS FOR BUSINESS AND ECONOMICS Seventh Edition AndersonSweeneyWilliams Slides Prepared by John Loucks © 1999 ITP/South-Western College Publishing

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 Properties of Point Estimators n Other Sampling Methods

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 in a study. 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 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 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 Slide Point Estimation 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 . is the point estimator of the population proportion p. is the point estimator of the population proportion p.

7 7 Slide Example: St. Edward’s St. Edward’s University receives 7,000 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 is for the applicants, and the average SAT score is 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.

8 8 Slide n Alternative #1: Take a Census of 7,000 Applicants SAT Scores SAT Scores Population Mean Population Mean Population Standard Deviation Population Standard Deviation In-State Applicants In-State Applicants Population Proportion Population Proportion Example: St. Edward’s

9 9 Slide n Alternative #2: Take a Sample of 50 Applicants Since the finite population has 7,000 elements, we will need 4-digit random numbers to randomly select applicants numbered from 1 to 7,000. We will use the last four digits of the 5-digit random numbers in the third column of Table 8 of Appendix B. The numbers we draw will be the numbers of the applicants we will sample unless the random number is greater than 7,000 or the random number has already been used. We will continue to draw random numbers until we have selected 50 applicants for our sample. Example: St. Edward’s

10 Slide n Use of Random Numbers for Sampling 4-DigitApplicant 4-DigitApplicant Random Number Included in Sample Random Number Included in Sample 1744 No No No No No No No No No No Number exceeds 7, Number exceeds 7, Number exceeds 7, Number exceeds 7, No No etc. etc. etc. etc. Example: St. Edward’s

11 Slide n Data Collected from Simple Random Sample No.Applicant SAT Score In-State No.Applicant SAT Score In-State 1Bonnie Reight1025Yes 2Willie Neilson 950Yes 3Fannie Lennox1090 No 4Derek Clapton1120Yes 5Winona Driver1015Yes Kevin Costmore 965 No 50Kevin Costmore 965 No Total 49, Yes Total 49, Yes Example: St. Edward’s

12 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

13 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

14 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

15 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

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

17 Slide n Question 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

18 Slide n Answer Using the standard normal probability table with z = 10/11.3 =.88, we have area = (.3106)(2) = There is a.6212 probability that the sample mean will be within +/-10 of the actual population mean. Sampling distribution of Sampling distribution of Area =.3106 Example: St. Edward’s

19 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

20 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.

21 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

22 Slide n Question 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

23 Slide n Answer For z =.05/.0635 =.79, we have area = (.2825)(2) = The probability is.5704 that the sample proportion will be within +/-.05 of the actual population proportion. Sampling distribution of Sampling distribution of Area =.2825 Example: St. Edward’s

24 Slide Properties 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. n Efficiency The point estimator with the smallest standard deviation is said to have greatest relative efficiency. n Consistency A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size increases.

25 Slide Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

26 Slide Stratified Random Sampling n The population is first divided into groups of elements called strata. 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. n Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size.

27 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. n This method generally requires a larger total sample size than simple or stratified random sampling. n The close proximity of elements can be cost effective.

28 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.

29 Slide Convenience Sampling n The sample is identified primarily by convenience. n It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected. 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.

30 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.

31 Slide The End of Chapter 7