1. 1 1 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. SLIDES. BY John Loucks St. Edward’s University......................

2. 2 2 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 7 Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to Sampling Distributions Point Estimation Point Estimation Selecting a Sample Selecting a Sample Other Sampling Methods Other Sampling Methods Sampling Distribution of Sampling Distribution of

3. 3 3 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Introduction A population is a collection of all the elements of A population is a collection of all the elements of interest. interest. A population is a collection of all the elements of A population is a collection of all the elements of interest. interest. A sample is a subset of the population. A sample is a subset of the population. An element is the entity on which data are collected. An element is the entity on which data are collected. A frame is a list of the elements that the sample will A frame is a list of the elements that the sample will be selected from. be selected from. A frame is a list of the elements that the sample will A frame is a list of the elements that the sample will be selected from. be selected from. The sampled population is the population from The sampled population is the population from which the sample is drawn. which the sample is drawn. The sampled population is the population from The sampled population is the population from which the sample is drawn. which the sample is drawn.

4. 4 4 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. Introduction The reason is simply that the sample contains only The reason is simply that the sample contains only a portion of the population. a portion of the population. The reason is simply that the sample contains only The reason is simply that the sample contains only a portion of the population. a portion of the population. The reason we select a sample is to collect data to The reason we select a sample is to collect data to answer a research question about a population. answer a research question about a population. The reason we select a sample is to collect data to The reason we select a sample is to collect data to answer a research question about a population. answer a research question about a population.

5. 5 5 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Selecting a Sample n Sampling from a Finite Population n Sampling from an Infinite Population

6. 6 6 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling from a 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 probability each possible sample of size n has the same probability of being selected. of being selected.

7. 7 7 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. Sampling from a Finite Population

8. 8 8 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. St. Andrew’s College received 900 applications for St. Andrew’s College received 900 applications for admission in the upcoming year from prospective students. The applicants were numbered, from 1 to 900, as their applications arrived. The Director of Admissions would like to select a simple random sample of 30 applicants. n Example: St. Andrew’s College Sampling from a Finite Population

9. 9 9 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The random numbers generated by Excel’s The random numbers generated by Excel’s RAND function follow a uniform probability RAND function follow a uniform probability distribution between 0 and 1. distribution between 0 and 1. The random numbers generated by Excel’s The random numbers generated by Excel’s RAND function follow a uniform probability RAND function follow a uniform probability distribution between 0 and 1. distribution between 0 and 1. Step 1: Assign a random number to each of the 900 applicants. applicants. Step 2: Select the 30 applicants corresponding to the 30 smallest random numbers. 30 smallest random numbers. Sampling from a Finite Population n Example: St. Andrew’s College

10. 10 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling from an Infinite Population As a result, we cannot construct a frame for the As a result, we cannot construct a frame for the population. population. Sometimes we want to select a sample, but find it is Sometimes we want to select a sample, but find it is not possible to obtain a list of all elements in the not possible to obtain a list of all elements in the population. population. Hence, we cannot use the random number selection Hence, we cannot use the random number selection procedure. procedure. Most often this situation occurs in infinite population Most often this situation occurs in infinite population cases. cases.

11. 11 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Populations are often generated by an ongoing process where there is no upper limit on the number of units that can be generated. Sampling from an Infinite Population Some examples of on-going processes, with infinite Some examples of on-going processes, with infinite populations, are: populations, are: parts being manufactured on a production line parts being manufactured on a production line transactions occurring at a bank transactions occurring at a bank telephone calls arriving at a technical help desk telephone calls arriving at a technical help desk customers entering a store customers entering a store

12. 12 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling from an Infinite Population n A random sample from an infinite population is a sample selected such that the following conditions sample selected such that the following conditions are satisfied. are satisfied. Each element selected comes from the population Each element selected comes from the population of interest. of interest. In the case of an infinite population, we must select In the case of an infinite population, we must select a random sample in order to make valid statistical a random sample in order to make valid statistical inferences about the population from which the inferences about the population from which the sample is taken. sample is taken. Each element is selected independently. Each element is selected independently.

13. 13 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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. Point estimation is a form of statistical inference. Point estimation is a form of statistical inference.

14. 14 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Recall that St. Andrew’s College received 900 Recall that St. Andrew’s College received 900 applications 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. n Example: St. Andrew’s College Point Estimation At a meeting in a few hours, the Director of At a meeting in a few hours, the Director of Admissions would like to announce the average SAT score and the proportion of applicants that want to live on campus, for the population of 900 applicants.

15. 15 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Point Estimation n Example: St. Andrew’s College However, the necessary data on the applicants have However, the necessary data on the applicants have not yet been entered in the college’s computerized database. So, the Director decides to estimate the values of the population parameters of interest based on sample statistics. The sample of 30 applicants is selected using computer-generated random numbers.

16. 16 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 

17. 17 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing Once all the data for the 900 applicants were entered Once all the data for the 900 applicants were entered in the college’s database, the values of the population parameters of interest were calculated. Point Estimation

18. 18 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score 16971684  = Population std. deviation for deviation for SAT score SAT score 87.4 s = Sample stan- s = Sample stan- dard deviation dard deviation for SAT score for SAT score85.2 p = Population pro- portion wanting portion wanting campus housing campus housing.72.67 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

19. 19 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Practical Advice The target population is the population we want to The target population is the population we want to make inferences about. make inferences about. The target population is the population we want to The target population is the population we want to make inferences about. make inferences about. Whenever a sample is used to make inferences Whenever a sample is used to make inferences about a population, we should make sure that the about a population, we should make sure that the targeted population and the sampled population targeted population and the sampled population are in close agreement. are in close agreement. Whenever a sample is used to make inferences Whenever a sample is used to make inferences about a population, we should make sure that the about a population, we should make sure that the targeted population and the sampled population targeted population and the sampled population are in close agreement. are in close agreement. The sampled population is the population from The sampled population is the population from which the sample is actually taken. which the sample is actually taken. The sampled population is the population from The sampled population is the population from which the sample is actually taken. which the sample is actually taken.

20. 20 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

21. 21 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. The sampling distribution of is the probability The sampling distribution of is the probability distribution of all possible values of the sample mean. Sampling Distribution of where:  = the population mean E ( ) =  Expected Value of Expected Value of When the expected value of the point estimator equals the population parameter, we say the point estimator is unbiased.

22. 22 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of We will use the following notation to define the We will use the following notation to define the standard deviation of the sampling distribution of.  = the standard deviation of  = the standard deviation of the population n = the sample size N = the population size Standard Deviation of Standard Deviation of

23. 23 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of Finite Population Infinite Population is referred to as the standard error of the is referred to as the standard error of the mean. mean. A finite population is treated as being A finite population is treated as being infinite if n / N <.05. infinite if n / N <.05. is the finite population is the finite population correction factor. correction factor. Standard Deviation of Standard Deviation of

24. 24 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. When the population has a normal distribution, the sampling distribution of is normally distributed for any sample size. 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 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. In most applications, the sampling distribution of can be approximated by a normal distribution whenever the sample is size 30 or more. Sampling Distribution of

25. 25 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of The sampling distribution of can be used to provide probability information about how close the sample mean is to the population mean . The sampling distribution of can be used to provide probability information about how close the sample mean is to the population mean .

26. 26 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Central Limit Theorem When the population from which we are selecting When the population from which we are selecting a random sample does not have a normal distribution, the central limit theorem is helpful in identifying the shape of the sampling distribution of. CENTRAL LIMIT THEOREM CENTRAL LIMIT THEOREM In selecting random samples of size n from a In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal mean can be approximated by a normal distribution as the sample size becomes large. distribution as the sample size becomes large. CENTRAL LIMIT THEOREM CENTRAL LIMIT THEOREM In selecting random samples of size n from a In selecting random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal mean can be approximated by a normal distribution as the sample size becomes large. distribution as the sample size becomes large.

27. 27 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.SamplingDistributionof for SAT Scores n Example: St. Andrew’s College Sampling Distribution of

28. 28 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. What is the probability that a simple random 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 +/  10 of the actual population mean  ? n Example: St. Andrew’s College Sampling Distribution of In other words, what is the probability that will In other words, what is the probability that will be between 1687 and 1707?

29. 29 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1707  1697)/15.96=.63 P ( z <.63) =.7357 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Sampling Distribution of n Example: St. Andrew’s College

30. 30 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z.00.01.02.03.04.05.06.07.08.09............5.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224.6.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549.7.7580.7611.7642.7673.7704.7734.7764.7794.7823.7852.8.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133.9.8159.8186.8212.8238.8264.8289.8315.8340.8365.8389........... Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution Sampling Distribution of n Example: St. Andrew’s College

31. 31 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 16971707 Area =.7357 Sampling Distribution of n Example: St. Andrew’s College SamplingDistributionof for SAT Scores

32. 32 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = (1687  1697)/15.96= -.63 P ( z < -.63) =.2643 Sampling Distribution of n Example: St. Andrew’s College

33. 33 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of for SAT Scores 16871697 Area =.2643 n Example: St. Andrew’s College SamplingDistributionof for SAT Scores

34. 34 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of for SAT Scores 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 (-.68 < z <.68) = P ( z <.68)  P ( z < -.68) =.7357 .2643 =.4714 The probability that the sample mean SAT score will be between 1687 and 1707 is: P (1687 < < 1707) =.4714 n Example: St. Andrew’s College

35. 35 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 170716871697 Sampling Distribution of for SAT Scores Area =.4714 n Example: St. Andrew’s College SamplingDistributionof for SAT Scores

36. 36 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Suppose we select a simple random sample of 100 Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. applicants instead of the 30 originally considered. E ( ) =  regardless of the sample size. In our E ( ) =  regardless of the sample size. In our example, E ( ) remains at 1697. example, E ( ) remains at 1697. Whenever the sample size is increased, the standard Whenever the sample size is increased, the standard error of the mean is decreased. With the increase error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of in the sample size to n = 100, the standard error of the mean is decreased from 15.96 to: the mean is decreased from 15.96 to: n Example: St. Andrew’s College

37. 37 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100, n Example: St. Andrew’s College

38. 38 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Recall that when n = 30, P (1687 < < 1707) =.4714. Recall that when n = 30, P (1687 < < 1707) =.4714. Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of We follow the same steps to solve for P (1687 < We follow the same steps to solve for P (1687 < < 1707) when n = 100 as we showed earlier when < 1707) when n = 100 as we showed earlier when n = 30. n = 30. Now, with n = 100, P (1687 < < 1707) =.7776. Now, with n = 100, P (1687 < < 1707) =.7776. Because the sampling distribution with n = 100 has a Because the sampling distribution with n = 100 has a smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population mean than the values of with n = 30. mean than the values of with n = 30. n Example: St. Andrew’s College

39. 39 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of170716871697 Area =.7776 n Example: St. Andrew’s College SamplingDistributionof for SAT Scores

40. 40 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

41. 41 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Sampling Distribution of where: p = the population proportion The sampling distribution of is the probability The sampling distribution of is the probability distribution of all possible values of the sample proportion. Expected Value of Expected Value of

42. 42 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. is referred to as the standard error of is referred to as the standard error of the proportion. the proportion. Sampling Distribution of Finite Population Infinite Population Standard Deviation of Standard Deviation of is the finite population is the finite population correction factor. correction factor.

43. 43 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Form of the Sampling Distribution of The sampling distribution of can be approximated by a normal distribution whenever the sample size is large enough to satisfy the two conditions:... because when these conditions are satisfied, the probability distribution of x in the sample proportion, = x / n, can be approximated by normal distribution (and because n is a constant). The sampling distribution of can be approximated by a normal distribution whenever the sample size is large enough to satisfy the two conditions:... because when these conditions are satisfied, the probability distribution of x in the sample proportion, = x / n, can be approximated by normal distribution (and because n is a constant). np > 5 n (1 – p ) > 5 and

44. 44 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Recall that 72% of the prospective students applying 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 a simple random sample 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?

45. 45 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. For our example, with n = 30 and p =.72, the For our example, with n = 30 and p =.72, the normal distribution is an acceptable approximation because: n (1 - p ) = 30(.28) = 8.4 > 5 and np = 30(.72) = 21.6 > 5 Sampling Distribution of n Example: St. Andrew’s College

46. 46 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part.SamplingDistributionof Sampling Distribution of n Example: St. Andrew’s College

47. 47 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 1: Calculate the z -value at the upper endpoint of the interval. of the interval. z = (.77 .72)/.082 =.61 P ( z <.61) =.7291 Step 2: Find the area under the curve to the left of the upper endpoint. the upper endpoint. Sampling Distribution of n Example: St. Andrew’s College

48. 48 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. z.00.01.02.03.04.05.06.07.08.09............5.6915.6950.6985.7019.7054.7088.7123.7157.7190.7224.6.7257.7291.7324.7357.7389.7422.7454.7486.7517.7549.7.7580.7611.7642.7673.7704.7734.7764.7794.7823.7852.8.7881.7910.7939.7967.7995.8023.8051.8078.8106.8133.9.8159.8186.8212.8238.8264.8289.8315.8340.8365.8389........... Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution Sampling Distribution of n Example: St. Andrew’s College

49. 49 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part..77.77.72 Area =.7291 SamplingDistributionof Sampling Distribution of n Example: St. Andrew’s College

50. 50 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = (.67 .72)/.082 = -.61 P ( z < -.61) =.2709 Sampling Distribution of n Example: St. Andrew’s College

51. 51 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part..67.72 Area =.2709 SamplingDistributionof Sampling Distribution of n Example: St. Andrew’s College

52. 52 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 n Example: St. Andrew’s College

53. 53 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part..77.67.72 Area =.4582 SamplingDistributionof Sampling Distribution of n Example: St. Andrew’s College

54. 54 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Other Sampling Methods n Stratified Random Sampling n Cluster Sampling n Systematic Sampling n Convenience Sampling n Judgment Sampling

55. 55 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 (i.e. a each stratum are as much alike as possible (i.e. a homogeneous group). 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 (i.e. a each stratum are as much alike as possible (i.e. a homogeneous group). homogeneous group).

56. 56 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

57. 57 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

58. 58 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

59. 59 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

60. 60 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

61. 61 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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 Example: A professor conducting research might use student volunteers to constitute a sample. use student volunteers to constitute a sample. Example: A professor conducting research might Example: A professor conducting research might use student volunteers to constitute a sample. use student volunteers to constitute a sample. The sample is identified primarily by convenience. The sample is identified primarily by convenience.

62. 62 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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

63. 63 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Judgment Sampling The person most knowledgeable on the subject of The person most knowledgeable on the subject of the study selects elements of the population that he the study selects elements of the population that he or she feels are most representative of the population. or she feels are most representative of the population. The person most knowledgeable on the subject of The person most knowledgeable on the subject of the study selects elements of the population that he the study selects elements of the population that he or she feels are most representative of the population. or 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.

64. 64 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. 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.

65. 65 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. Recommendation It is recommended that probability sampling methods It is recommended that probability sampling methods (simple random, stratified, cluster, or systematic) be (simple random, stratified, cluster, or systematic) be used. used. It is recommended that probability sampling methods It is recommended that probability sampling methods (simple random, stratified, cluster, or systematic) be (simple random, stratified, cluster, or systematic) be used. used. For these methods, formulas are available for For these methods, formulas are available for evaluating the “goodness” of the sample results in evaluating the “goodness” of the sample results in terms of the closeness of the results to the population terms of the closeness of the results to the population parameters being estimated. parameters being estimated. For these methods, formulas are available for For these methods, formulas are available for evaluating the “goodness” of the sample results in evaluating the “goodness” of the sample results in terms of the closeness of the results to the population terms of the closeness of the results to the population parameters being estimated. parameters being estimated. An evaluation of the goodness cannot be made with An evaluation of the goodness cannot be made with non-probability (convenience or judgment) sampling non-probability (convenience or judgment) sampling methods. methods. An evaluation of the goodness cannot be made with An evaluation of the goodness cannot be made with non-probability (convenience or judgment) sampling non-probability (convenience or judgment) sampling methods. methods.

66. 66 Slide © 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 7