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Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2004 Thomson/South-Western Slide 1 Chapter 7 Sampling and Sampling Distributions Simple Random Sampling Point Estimation Introduction to Sampling Distributions Sampling Distribution of x Sampling Distribution of p Sampling Methods © 2004 Thomson/South-Western Slide 2 Statistical Inference The purpose of statistical inference is to obtain information about a population from information contained in a sample. A population is the set of all the elements of interest. A sample is a subset of the population. © 2004 Thomson/South-Western Slide 3 Statistical Inference The sample results provide only estimates of the values of the population characteristics. With proper sampling methods, the sample results can provide “good” estimates of the population characteristics. A parameter is a numerical characteristic of a population. © 2004 Thomson/South-Western Slide 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 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. © 2004 Thomson/South-Western Slide 5 Simple Random Sampling: Finite Population Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often. In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. © 2004 Thomson/South-Western Slide 6 Simple Random Sampling: Infinite Population 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. A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. • Each element selected comes from the same population. • Each element is selected independently. © 2004 Thomson/South-Western Slide 7 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. © 2004 Thomson/South-Western Slide 8 Point Estimation In point estimation we use the data from the sample to compute a value of a sample statistic that serves as an estimate of a population parameter. We refer to mean . x as the point estimator of the population s is the point estimator of the population standard deviation . p is the point estimator of the population proportion p. © 2004 Thomson/South-Western Slide 9 Sampling Error When the expected value of a point estimator is equal to the population parameter, the point estimator is said to be unbiased. The absolute value of the difference between an unbiased point estimate and the corresponding population parameter is called the sampling error. Sampling error is the result of using a subset of the population (the sample), and not the entire population. Statistical methods can be used to make probability statements about the size of the sampling error. © 2004 Thomson/South-Western Slide 10 Sampling Error The sampling errors are: |x | for sample mean |s | for sample standard deviation | p p| for sample proportion © 2004 Thomson/South-Western Slide 11 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. © 2004 Thomson/South-Western Slide 12 Example: St. Andrew’s The director of admissions would like to know the following information: • the average SAT score for the 900 applicants, and • the proportion of applicants that want to live on campus. © 2004 Thomson/South-Western Slide 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 © 2004 Thomson/South-Western Slide 14 Conducting a Census 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. We will assume for the moment that conducting a census is practical in this example. © 2004 Thomson/South-Western Slide 15 Conducting a Census Population Mean SAT Score xi 990 900 Population Standard Deviation for SAT Score 2 ( x ) i 80 900 Population Proportion Wanting On-Campus Housing 648 p .72 900 © 2004 Thomson/South-Western Slide 16 Simple Random Sampling Now suppose that the necessary information on the current year’s applicants was not yet entered in the college’s database. Furthermore, the Director of Admissions must obtain estimates of the population parameters of interest for a meeting taking place in a few hours. She decides a sample of 30 applicants will be used. The applicants were numbered, from 1 to 900, as their applications arrived. © 2004 Thomson/South-Western Slide 17 Simple Random Sampling: Using a Random Number Table Taking a Sample of 30 Applicants • Since the finite population has 900 elements, we will need 3-digit random numbers to randomly select applicants numbered from 1 to 900. • We will use the last three digits of the 5-digit random numbers in the third column of the textbook’s random number table, and continue into the fourth column as needed. © 2004 Thomson/South-Western Slide 18 Simple Random Sampling: Using a Random Number Table Taking a Sample of 30 Applicants • • • 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 has already been used. We will continue to draw random numbers until we have selected 30 applicants for our sample. (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.) © 2004 Thomson/South-Western Slide 19 Simple Random Sampling: Using a Random Number Table Use of Random Numbers for Sampling 3-Digit Applicant Random Number Included in Sample 744 No. 744 436 No. 436 865 No. 865 790 No. 790 835 No. 835 902 Number exceeds 900 190 No. 190 836 No. 836 . . . and so on © 2004 Thomson/South-Western Slide 20 Simple Random Sampling: Using a Random Number Table Sample Data Random No. Number 1 744 2 436 3 865 4 790 5 835 . . . . 30 498 Applicant Conrad Harris Enrique Romero Fabian Avante Lucila Cruz Chan Chiang . . SAT Score 1025 950 1090 1120 930 . . Live OnCampus Yes Yes No Yes No . . Emily Morse 1010 No © 2004 Thomson/South-Western Slide 21 Using Excel to Select a Simple Random Sample Taking a Sample of 30 Applicants • Excel provides a function for generating random numbers in its worksheet. • 900 random numbers are generated, one for each applicant in the population. • Then we choose the 30 applicants corresponding to the 30 smallest random numbers as our sample. • Each of the 900 applicants has the same probability of being included. © 2004 Thomson/South-Western Slide 22 Using Excel to Select a Simple Random Sample Formula Worksheet 1 2 3 4 5 6 7 8 9 A B C Applicant Number 1 2 3 4 5 6 7 8 SAT Score 1008 1025 952 1090 1127 1015 965 1161 On-Campus Housing Yes No Yes Yes Yes No Yes No D Random Number =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() =RAND() Note: Rows 10-901 are not shown. © 2004 Thomson/South-Western Slide 23 Using Excel to Select a Simple Random Sample Value Worksheet A 1 2 3 4 5 6 7 8 9 B C Applicant SAT On-Campus Number Score Housing 1 1008 Yes 2 1025 No 3 952 Yes 4 1090 Yes 5 1127 Yes 6 1015 No 7 965 Yes 8 1161 No Note: Rows 10-901 are not shown. © 2004 Thomson/South-Western D Random Number 0.84891 0.30181 0.35709 0.99433 0.33072 0.54548 0.46758 0.33167 Slide 24 Using Excel to Select a Simple Random Sample Put Random Numbers in Ascending Order Step 1 Select cells A2:A901 Step 2 Select the Data menu Step 3 Choose the Sort option Step 4 When the Sort dialog box appears: Choose Random Numbers in the Sort by text box Choose Ascending Click OK © 2004 Thomson/South-Western Slide 25 Using Excel to Select a Simple Random Sample Value Worksheet (Sorted) 1 2 3 4 5 6 7 8 9 A B C D Applicant Number 12 773 408 58 116 185 510 394 SAT Score 1107 1043 991 1008 1127 982 1163 1008 On-Campus Housing No Yes Yes No Yes Yes Yes No Random Number 0.00027 0.00192 0.00303 0.00481 0.00538 0.00583 0.00649 0.00667 Note: Rows 10-901 are not shown. © 2004 Thomson/South-Western Slide 26 Point Estimates x as Point Estimator of x x i 30 29, 910 997 30 s as Point Estimator of s (x i x )2 29 163, 996 75.2 29 p as Point Estimator of p p 20 30 .68 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. © 2004 Thomson/South-Western Slide 27 Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value = Population mean 990 x = Sample mean 997 = Population std. 80 s = Sample std. deviation for SAT score 75.2 p = Population proportion wanting campus housing .72 p = Sample pro- .68 SAT score deviation for SAT score © 2004 Thomson/South-Western Point Estimator Point Estimate SAT score portion wanting campus housing Slide 28 Sampling Distribution of x Process of Statistical Inference Population with mean =? The value of x is used to make inferences about the value of . © 2004 Thomson/South-Western A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean x . Slide 29 Sampling Distribution of x The sampling distribution of x is the probability distribution of all possible values of the sample mean x. Expected Value of x E( x ) = where: = the population mean © 2004 Thomson/South-Western Slide 30 Sampling Distribution of x Standard Deviation of x Finite Population N n x ( ) n N 1 Infinite Population x n • A finite population is treated as being infinite if n/N < .05. • ( N n) / ( N 1) is the finite correction factor. • x is referred to as the standard error of the mean. © 2004 Thomson/South-Western Slide 31 Sampling Distribution of x If we use a large (n > 30) simple random sample, the central limit theorem enables us to conclude that the sampling distribution of x can be approximated by a normal probability distribution. When the simple random sample is small (n < 30), the sampling distribution of x can be considered normal only if we assume the population has a normal probability distribution. © 2004 Thomson/South-Western Slide 32 Sampling Distribution of x for SAT Scores Sampling Distribution of x E( x ) 990 © 2004 Thomson/South-Western x 80 14.6 n 30 x Slide 33 Sampling Distribution of x for SAT Scores 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 x will be between 980 and 1000? © 2004 Thomson/South-Western Slide 34 Sampling Distribution of x for SAT Scores Step 1: Calculate the z-value at the upper endpoint of the interval. z = (1000 - 990)/14.6= .68 Step 2: Find the area under the curve to the left of the upper endpoint. P(z < .68) = .7517 © 2004 Thomson/South-Western Slide 35 Sampling Distribution of x for SAT Scores Cumulative Probabilities for the Standard Normal Distribution 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 . . . . . © 2004 Thomson/South-Western . . . . . . Slide 36 Sampling Distribution of x for SAT Scores Sampling Distribution of x x 14.6 Area = .7517 x 990 1000 © 2004 Thomson/South-Western Slide 37 Sampling Distribution of x for SAT Scores Step 3: Calculate the z-value at the lower endpoint of the interval. z = (980 - 990)/14.6= - .68 Step 4: Find the area under the curve to the left of the lower endpoint. P(z < -.68) = P(z > .68) = 1 - P(z < .68) = 1 - . 7517 = .2483 © 2004 Thomson/South-Western Slide 38 Sampling Distribution of x for SAT Scores Sampling Distribution of x x 14.6 Area = .2483 x 980 990 © 2004 Thomson/South-Western Slide 39 Sampling Distribution of x for SAT Scores Step 5: Calculate the area under the curve between 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 < x < 1000) = .5034 © 2004 Thomson/South-Western Slide 40 Sampling Distribution of x for SAT Scores Sampling Distribution of x x 14.6 Area = .5034 980 990 1000 © 2004 Thomson/South-Western x Slide 41 Relationship Between the Sample Size and the Sampling Distribution of x Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. E(x ) = regardless of the sample size. In our example, E( x) remains at 990. Whenever the sample size is increased, the standard error of the mean x is decreased. With the increase in the sample size to n = 100, the standard error of the mean is decreased to: 80 x 8.0 n © 2004 Thomson/South-Western 100 Slide 42 Relationship Between the Sample Size and the Sampling Distribution of x With n = 100, x 8 With n = 30, x 14.6 E( x ) 990 © 2004 Thomson/South-Western x Slide 43 Relationship Between the Sample Size and the Sampling Distribution of x Recall that when n = 30, P(980 < x < 1000) = .5034. We follow the same steps to solve for P(980 < x < 1000) when n = 100 as we showed earlier when n = 30. Now, with n = 100, P(980 < x < 1000) = .7888. Because the sampling distribution with n = 100 has a smaller standard error, the values of x have less variability and tend to be closer to the population mean than the values of x with n = 30. © 2004 Thomson/South-Western Slide 44 Relationship Between the Sample Size and the Sampling Distribution of x Sampling Distribution of x x 8 Area = .7888 x 980 990 1000 © 2004 Thomson/South-Western Slide 45 Sampling Distribution of p Making Inferences about a Population Proportion Population with proportion p=? The value of p is used to make inferences about the value of p. © 2004 Thomson/South-Western A simple random sample of n elements is selected from the population. The sample data provide a value for the sample proportion p. Slide 46 Sampling Distribution of p The sampling distribution of p is the probability distribution of all possible values of the sample proportion p. Expected Value of p E ( p) p where: p = the population proportion © 2004 Thomson/South-Western Slide 47 Sampling Distribution of p Standard Deviation of p Finite Population p p(1 p) N n n N 1 Infinite Population p p(1 p) n p is referred to as the standard error of the proportion. © 2004 Thomson/South-Western Slide 48 Sampling Distribution of p The sampling distribution of p can be approximated by a normal probability distribution whenever the sample size is large. The sample size is considered large whenever these conditions are satisfied: np > 5 © 2004 Thomson/South-Western and n(1 – p) > 5 Slide 49 Sampling Distribution of p For values of p near .50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed. © 2004 Thomson/South-Western Slide 50 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing For our example, with n = 30 and p = .72, the normal probability distribution is an acceptable approximation because: np = 30(.72) = 21.6 > 5 and n(1 - p) = 30(.28) = 8.4 > 5 © 2004 Thomson/South-Western Slide 51 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Sampling Distribution of p .72(1 .72) p .082 30 E( p ) .72 © 2004 Thomson/South-Western p Slide 52 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus .05 of the actual population proportion? In other words, what is the probability that p will be between .67 and .77? © 2004 Thomson/South-Western Slide 53 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Step 1: Calculate the z-value at the upper endpoint of the interval. z = (.77 - .72)/.082 = .61 Step 2: Find the area under the curve to the left of the upper endpoint. P(z < .61) = .7291 © 2004 Thomson/South-Western Slide 54 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Cumulative Probabilities for the Standard Normal Distribution 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 . . . . . © 2004 Thomson/South-Western . . . . . . Slide 55 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Sampling Distribution of p p .082 Area = .7291 p .72 .77 © 2004 Thomson/South-Western Slide 56 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Step 3: Calculate the z-value at the lower endpoint of the interval. z = (.67 - .72)/.082 = - .61 Step 4: Find the area under the curve to the left of the lower endpoint. P(z < -.61) = P(z > .61) = 1 - P(z < .61) = 1 - . 7291 = .2709 © 2004 Thomson/South-Western Slide 57 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Sampling Distribution of p p .082 Area = .2709 p .67 .72 © 2004 Thomson/South-Western Slide 58 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Step 5: Calculate the area under the curve between 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 : P(.67 < p < .77) = .4582 © 2004 Thomson/South-Western Slide 59 Sampling Distribution of p for the Proportion of Applicants Wanting On-Campus Housing Sampling Distribution of p p .082 Area = .4582 p .67 © 2004 Thomson/South-Western .72 .77 Slide 60 Sampling Methods Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling © 2004 Thomson/South-Western Slide 61 Stratified Random Sampling The population is first divided into groups of elements called strata. Each element in the population belongs to one and only one stratum. Best results are obtained when the elements within each stratum are as much alike as possible (i.e. a homogeneous group). © 2004 Thomson/South-Western Slide 62 Stratified Random Sampling A simple random sample is taken from each stratum. Formulas are available for combining the stratum sample results into one population parameter estimate. Advantage: If strata are homogeneous, this method is as “precise” as simple random sampling but with a smaller total sample size. Example: The basis for forming the strata might be department, location, age, industry type, and so on. © 2004 Thomson/South-Western Slide 63 Cluster Sampling The population is first divided into separate groups of elements called clusters. Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). A simple random sample of the clusters is then taken. All elements within each sampled (chosen) cluster form the sample. © 2004 Thomson/South-Western Slide 64 Cluster Sampling Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas. Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be obtained in a short time). Disadvantage: This method generally requires a larger total sample size than simple or stratified random sampling. © 2004 Thomson/South-Western Slide 65 Systematic Sampling If a sample size of n is desired from a population containing N elements, we might sample one element for every n/N elements in the population. We randomly select one of the first n/N elements from the population list. We then select every n/Nth element that follows in the population list. © 2004 Thomson/South-Western Slide 66 Systematic Sampling This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. Advantage: The sample usually will be easier to identify than it would be if simple random sampling were used. Example: Selecting every 100th listing in a telephone book after the first randomly selected listing © 2004 Thomson/South-Western Slide 67 Convenience Sampling It is a nonprobability sampling technique. Items are included in the sample without known probabilities of being selected. The sample is identified primarily by convenience. Example: A professor conducting research might use student volunteers to constitute a sample. © 2004 Thomson/South-Western Slide 68 Convenience Sampling Advantage: Sample selection and data collection are relatively easy. Disadvantage: It is impossible to determine how representative of the population the sample is. © 2004 Thomson/South-Western Slide 69 Judgment Sampling The person most knowledgeable on the subject of the study selects elements of the population that he or she feels are most representative of the population. It is a nonprobability sampling technique. Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. © 2004 Thomson/South-Western Slide 70 Judgment Sampling Advantage: It is a relatively easy way of selecting a sample. Disadvantage: The quality of the sample results depends on the judgment of the person selecting the sample. © 2004 Thomson/South-Western Slide 71 End of Chapter 7 © 2004 Thomson/South-Western Slide 72