1. Sampling and Sampling Distributions Simple Random Sampling Point Estimation Sampling Distribution

2. 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.

3. 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

4. Simple Random Sampling: Finite Population Finite populations are often defined by lists such as: –Organization membership roster –Credit card account numbers –Inventory product numbers

5. A simple random sample of size n 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 Good Sample is a Random Sample

6. 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.

7. 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. Each element is selected independently.

8. Simple Random Sampling: Infinite Population In the case of infinite populations, it is impossible to obtain a list of all elements in the population. The random number selection procedure cannot be used for infinite populations.

9. 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.

10. 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.

11. Sampling Error n The sampling errors are: for sample proportion for sample standard deviation for sample mean

12. Example: St. Andrew’s 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.

13. Example: St. Andrew’s We will now look at three alternatives for obtaining the desired information. Conducting a census of the entire 900 applicants Selecting a sample of 30 applicants, using a random number table Selecting a sample of 30 applicants, using Excel

14. Conducting a Census n If the relevant information for the entire 900 applicants was 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.

15. Population Mean SAT Score Population Standard Deviation for SAT Score Population Proportion Wanting On-Campus Housing Conducting a Census

16. 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 information on Now suppose that the necessary information on the current year’s applicants was not yet entered the current year’s applicants was not yet entered in the college’s database. in the college’s database.

17. 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. (See Table 7.1, p. 275) (See Table 7.1, p. 275) Since the finite population has 900 elements, we Since 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.

18. 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.

19. Use of Random Numbers for Sampling 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 See 3 rd column of Table 7.1, p. 275

20. 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.....

21. Taking a Sample of 30 Applicants Using Excel to Select a Simple Random Sample Each of the 900 applicants has the same probability Each of the 900 applicants has the same probability of being included. of being included. 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. 900 random numbers are generated, one for each 900 random numbers are generated, one for each applicant in the population. applicant in the population. Excel provides a function for generating random Excel provides a function for generating random numbers in its worksheet. numbers in its worksheet.

22. Using Excel to Select a Simple Random Sample Formula Worksheet Note: Rows 10-901 are not shown.

23. Using Excel to Select a Simple Random Sample n Value Worksheet Note: Rows 10-901 are not shown.

24. n Put Random Numbers in Ascending Order Using Excel to Select a Simple Random Sample Step 4 When the Sort dialog box appears: Choose Random Numbers in the Choose Random Numbers in the Sort by text box Sort by text box Choose Ascending Choose Ascending Click OK Click OK Step 3 Choose the Sort option Step 2 Select the Data menu Step 1 Select cells A2:A901

25. Using Excel to Select a Simple Random Sample Value Worksheet (Sorted) Note: Rows 10-901 are not shown.

26. Point Estimates as Point Estimator of  s as Point Estimator of  as Point Estimator of p Note: Different random numbers would have identified a different sample which would have resulted in different point estimates.

27. 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

28. 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

29. where:  = the population mean  = the population mean Sampling Distribution of The sampling distribution of is the probability distribution of all possible values of the sample mean

30. 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

31. Sampling Distribution of 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. 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.

32. Central Limit Theorem In selecting simple random samples of size n, the sampling distribution of the mean can be approximated by a normal probability distribution as the sample size becomes large.

33. Sampling Distribution of for SAT Scores SamplingDistributionof

34. 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

35. 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

36. Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution

37. Sampling Distribution of for SAT Scores 990SamplingDistributionof1000 Area =.7517

38. 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 = 1 -. 7517 = 1 - P ( z <.68) Sampling Distribution of for SAT Scores

39. 980990 Area =.2483 SamplingDistributionof

40. 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

41. 1000980990 Sampling Distribution of for SAT Scores Area =.5034 SamplingDistributionof