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

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 7, Part A Sampling and Sampling Distributions Sampling Distribution of Sampling Distribution of Introduction to Sampling Distributions Introduction to Sampling Distributions Point Estimation Point Estimation Simple Random Sampling Simple Random Sampling

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

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Statistical Inference A sample mean provides an estimate of a population A sample mean provides an estimate of a population mean, and a sample proportion provides an estimate mean, and a sample proportion provides an estimate of a population proportion. Finally, a sample variance of a population proportion. Finally, a sample variance provides an estimate of a population variance. A sample mean provides an estimate of a population A sample mean provides an estimate of a population mean, and a sample proportion provides an estimate mean, and a sample proportion provides an estimate of a population proportion. Finally, a sample variance of a population proportion. Finally, a sample variance provides an estimate of a population variance. As with all estimates, we can expect some estimation As with all estimates, we can expect some estimation error. In this chapter, we will provide the basis for error. In this chapter, we will provide the basis for determining how large that error might be by using determining how large that error might be by using confidence intervals and our knowledge of the confidence intervals and our knowledge of the sampling distribution for sample means, proportions, sampling distribution for sample means, proportions, and variances. and variances. As with all estimates, we can expect some estimation As with all estimates, we can expect some estimation error. In this chapter, we will provide the basis for error. In this chapter, we will provide the basis for determining how large that error might be by using determining how large that error might be by using confidence intervals and our knowledge of the confidence intervals and our knowledge of the sampling distribution for sample means, proportions, sampling distribution for sample means, proportions, and variances. and variances.

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. The sample results provide only estimates of the The sample results provide only estimates of the values of the population characteristics. values of the population characteristics. A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. A parameter is a numerical characteristic of a A parameter is a numerical characteristic of a population. population. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. With proper sampling methods, the sample results With proper sampling methods, the sample results can provide “good” estimates of the population can provide “good” estimates of the population characteristics. characteristics. Statistical Inference

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Finite Population n Finite populations are often defined by lists such as: Organization membership roster Organization membership roster Credit card account numbers Credit card account numbers Inventory product numbers Inventory product numbers n A simple random sample of size n from a finite population of size N is a sample selected such that population of size N is a sample selected such that each possible sample of size n has the same each possible sample of size n has the same probability of being selected. probability of being selected.

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Finite Population In large sampling projects, computer-generated In large sampling projects, computer-generated random numbers are often used to automate the random numbers are often used to automate the sample selection process. sample selection process. Sampling without replacement is the procedure Sampling without replacement is the procedure used most often. used most often. Replacing each sampled element before selecting Replacing each sampled element before selecting subsequent elements is called sampling with subsequent elements is called sampling with replacement. replacement. In practice, a population being studied is usually In practice, a population being studied is usually considered infinite if it involves an ongoing process considered infinite if it involves an ongoing process that makes listing or counting every element in the that makes listing or counting every element in the population impossible. population impossible.

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved n Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely. Simple Random Sampling: Infinite Population n A simple random sample from an infinite population is a sample selected such that the following conditions is a sample selected such that the following conditions are satisfied. are satisfied. Each element selected comes from the same Each element selected comes from the same population. population. Each element is selected independently to avoid Each element is selected independently to avoid selection bias. selection bias. Each of the elements remaining in the population has Each of the elements remaining in the population has the same probability of being selected. the same probability of being selected.

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling: Infinite Population The random number selection procedure cannot be The random number selection procedure cannot be used for infinite populations. used for infinite populations. In the case of infinite populations, it is impossible to In the case of infinite populations, it is impossible to obtain a list of all elements in the population. obtain a list of all elements in the population. The number of different simple random samples of The number of different simple random samples of size n that you can select from a finite population of size n that you can select from a finite population of size N is: size N is:N!n!(N-n)!

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . s is the point estimator of the population standard s is the point estimator of the population standard deviation . deviation . In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. In point estimation we use the data from the sample In point estimation we use the data from the sample to compute a value of a sample statistic that serves to compute a value of a sample statistic that serves as an estimate of a population parameter. as an estimate of a population parameter. Point Estimation We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . We refer to as the point estimator of the population We refer to as the point estimator of the population mean . mean . is the point estimator of the population proportion p. is the point estimator of the population proportion p. To estimate the value of a population parameter, we To estimate the value of a population parameter, we compute a corresponding characteristic of the sample, compute a corresponding characteristic of the sample, referred to as a sample statistic. referred to as a sample statistic. To estimate the value of a population parameter, we To estimate the value of a population parameter, we compute a corresponding characteristic of the sample, compute a corresponding characteristic of the sample, referred to as a sample statistic. referred to as a sample statistic.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Error Statistical methods can be used to make probability Statistical methods can be used to make probability statements about the size of the sampling error. statements about the size of the sampling error. Sampling error is the result of using a subset of the Sampling error is the result of using a subset of the population (the sample), and not the entire population (the sample), and not the entire population. population. The absolute value of the difference between an The absolute value of the difference between an unbiased point estimate and the corresponding unbiased point estimate and the corresponding population parameter is called the sampling error. population parameter is called the sampling error. When the expected value of a point estimator is equal When the expected value of a point estimator is equal to the population parameter, the point estimator is said to the population parameter, the point estimator is said to be unbiased. to be unbiased.

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Error n The sampling errors are: for sample proportion for sample standard deviation for sample mean

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s St. Andrew’s College receives St. Andrew’s College receives 900 applications annually from prospective students. The application form contains a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing.

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s The director of admissions The director of admissions would like to know the following information: the average SAT score for the average SAT score for the 900 applicants, and the 900 applicants, and the proportion of the proportion of applicants that want to live on campus.

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Example: St. Andrew’s We will now look at two alternatives for obtaining the desired information. n Conducting a census of the entire 900 applicants entire 900 applicants n Selecting a sample of 30 applicants, using Excel applicants, using Excel

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved Conducting a Census n If the relevant data for the entire 900 applicants were in the college’s database, the population parameters of interest could be calculated using the formulas presented in Chapter 3. n We will assume for the moment that conducting a census is practical in this example.

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Conducting a Census n Population Mean SAT Score n Population Standard Deviation for SAT Score n Population Proportion Wanting On-Campus Housing

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Simple Random Sampling The applicants were numbered, from 1 to 900, as The applicants were numbered, from 1 to 900, as their applications arrived. their applications arrived. She decides a sample of 30 applicants will be used. She decides a sample of 30 applicants will be used. Furthermore, the Director of Admissions must obtain Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for estimates of the population parameters of interest for a meeting taking place in a few hours. a meeting taking place in a few hours. Now suppose that the necessary data on the Now suppose that the necessary data on the current year’s applicants were not yet entered in the current year’s applicants were not yet entered in the college’s database. college’s database.

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved n Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table We will use the last three digits of the 5-digit We will use the last three digits of the 5-digit random numbers in the third column of the random numbers in the third column of the textbook’s random number table, and continue textbook’s random number table, and continue into the fourth column as needed. into the fourth column as needed. Because the finite population has 900 elements, we Because the finite population has 900 elements, we will need 3-digit random numbers to randomly will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. select applicants numbered from 1 to 900.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved n Taking a Sample of 30 Applicants Simple Random Sampling: Using a Random Number Table (We will go through all of column 3 and part of (We will go through all of column 3 and part of column 4 of the random number table, encountering in the process five numbers greater than 900 and one duplicate, 835.) We will continue to draw random numbers until We will continue to draw random numbers until we have selected 30 applicants for our sample. we have selected 30 applicants for our sample. The numbers we draw will be the numbers of the applicants we will sample unless The numbers we draw will be the numbers of the applicants we will sample unless the random number is greater than 900 or the random number is greater than 900 or the random number has already been used. the random number has already been used.

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved n Use of Random Numbers for Sampling Simple Random Sampling: Using a Random Number Table 744 436 865 790 835 902 190 836... and so on 3-Digit 3-Digit Random Number Applicant Included in Sample No. 436 No. 865 No. 790 No. 835 Number exceeds 900 No. 190 No. 836 No. 744

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved n Sample Data Simple Random Sampling: Using a Random Number Table 1744 Conrad Harris1025 Yes 2436 Enrique Romero 950 Yes 3865 Fabian Avante1090 No 4790 Lucila Cruz1120 Yes 5835 Chan Chiang 930 No..... 30 498 Emily Morse 1010 No No. RandomNumber Applicant SAT Score Score Live On- Campus.....

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved n Taking a Sample of 30 Applicants Then we choose the 30 applicants corresponding Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. to the 30 smallest random numbers as our sample. For example, Excel’s function For example, Excel’s function = RANDBETWEEN(1,900) = RANDBETWEEN(1,900) can be used to generate random numbers between can be used to generate random numbers between 1 and 900. 1 and 900. Computers can be used to generate random Computers can be used to generate random numbers for selecting random samples. numbers for selecting random samples. Simple Random Sampling: Using a Computer

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved as Point Estimator of  as Point Estimator of  n as Point Estimator of p Point Estimation Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. s as Point Estimator of  s as Point Estimator of 

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved PopulationParameterPointEstimatorPointEstimateParameterValue  = Population mean SAT score SAT score 990997  = Population std. deviation for deviation for SAT score SAT score 80 s = Sample std. s = Sample std. deviation for deviation for SAT score SAT score75.2 p = Population pro- portion wanting portion wanting campus housing campus housing.72.68 Summary of Point Estimates Obtained from a Simple Random Sample = Sample mean = Sample mean SAT score SAT score = Sample pro- = Sample pro- portion wanting portion wanting campus housing campus housing

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved n Process of Statistical Inference The value of is used to make inferences about the value of . The sample data provide a value for the sample mean. A simple random sample of n elements is selected from the population. Population with mean  = ? Sampling Distribution of

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved The sampling distribution of is the probability distribution of all possible values of the sample mean. Sampling Distribution of where:  = the population mean  = the population mean E ( ) =  Expected Value of

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved 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 correction factor. is the finite correction factor. Standard Deviation of In general, the term standard error refers to In general, the term standard error refers to the standard deviation of a point estimator. the standard deviation of a point estimator.

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved Central Limit Theorem 1. In selecting simple random samples of size n from a population, the sampling distribution of the sample mean can be approximated by a normal distribution as the sample size becomes large. 2. The practical reason we are interested in the sampling distribution of the sample mean is that it can be used to provide probability information about the difference between the sample mean and the population mean. 3. The mean of all possible values of the sample mean is equal to the population mean regardless of the sample size. 4. As the sample size is increased, the standard error of the mean decreases. As a result, the larger sample size provides a higher probability that the sample mean is within a specified distance of the population mean.

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

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Distribution of for SAT Scores SamplingDistributionof

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved 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 mean SAT score that is within +/  10 of the actual population mean  ? In other words, what is the probability that will be In other words, what is the probability that will be between 980 and 1000? Sampling Distribution of for SAT Scores

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = (1000  990)/14.6=.68 P ( z <.68) =.7517 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint. Sampling Distribution of for SAT Scores

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Distribution of for SAT Scores Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =.7517

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved 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 = (980  990)/14.6= -.68 P ( z < -.68) =.2483 Sampling Distribution of for SAT Scores

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved Sampling Distribution of for SAT Scores 980990 Area =.2483 SamplingDistributionof

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved 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) =.7517 .2483 =.5034 The probability that the sample mean SAT score will be between 980 and 1000 is: P (980 < < 1000) =.5034

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved 1000980990 Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved 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 990. example, E ( ) remains at 990. 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 the in the sample size to n = 100, the standard error of the mean is decreased to: mean is decreased to:

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

42. 42 Slide © 2008 Thomson South-Western. All Rights Reserved Recall that when n = 30, P (980 < < 1000) =.5034. Recall that when n = 30, P (980 < < 1000) =.5034. Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of We follow the same steps to solve for P (980 < < 1000) We follow the same steps to solve for P (980 < < 1000) when n = 100 as we showed earlier when n = 30. when n = 100 as we showed earlier when n = 30. Now, with n = 100, P (980 < < 1000) =.7888. Now, with n = 100, P (980 < < 1000) =.7888. 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.

43. 43 Slide © 2008 Thomson South-Western. All Rights Reserved Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of1000980990 Area =.7888 SamplingDistributionof

44. 44 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 7, Part A