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Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western/Thomson Learning 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 n = 100 Properties of Point Estimators Other Sampling Methods n = 30 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. The sample results provide only estimates of the values of the population characteristics. A parameter is a numerical characteristic of a population. With proper sampling methods, the sample results will provide “good” estimates of the population characteristics. Slide 3 Simple Random Sampling Finite Population • A simple random sample from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. • 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. Slide 4 Simple Random Sampling Infinite Population • 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. • The population is usually considered infinite if it involves an ongoing process that makes listing or counting every element impossible. • The random number selection procedure cannot be used for infinite populations. Slide 5 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 x as the point estimator of the population mean . s is the point estimator of the population standard deviation . p is the point estimator of the population proportion p. Slide 6 Sampling Error The absolute 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 to develop estimates. The sampling errors are: | x | for sample mean | s - | for sample standard deviation | p p | for sample proportion Slide 7 Example: St. Andrew’s St. Andrew’s University receives 900 applications annually from prospective students. The application forms contain a variety of information including the individual’s scholastic aptitude test (SAT) score and whether or not the individual desires on-campus housing. Slide 8 Example: St. Andrew’s The director of admissions would like to know the following information: • the average SAT score for the applicants, and • the proportion of applicants that want to live on campus. 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 computer-generated random numbers Slide 9 Example: St. Andrew’s Taking a Census of the 900 Applicants • SAT Scores • Population Mean x i 900 990 • Population Standard Deviation (x ) 2 i 900 80 • Applicants Wanting On-Campus Housing • Population Proportion 648 p .72 900 Slide 10 Example: St. Andrew’s Take a Sample of 30Applicants Using a Random Number Table 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 a random number table. 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. Slide 11 Example: St. Andrew’s 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. 190 436 Number already used etc. etc. Slide 12 Example: St. Andrew’s Sample Data Random No. Number Campus 1 744 2 436 Yes 3 865 4 790 5 835 . . 30 685 Applicant SAT Score Connie Reyman William Fox 1025 Fabian Avante Eric Paxton Winona Wheeler . Kevin Cossack 1090 1120 1015 . 965 OnYes 950 No Yes No . No Slide 13 Example: St. Andrew’s Take a Sample of 30 Applicants Using ComputerGenerated Random Numbers • 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 have the same probability of being included. Slide 14 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. Slide 15 Using Excel to Select a Simple Random Sample Value Worksheet 1 2 3 4 5 6 7 8 9 A B C D 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 Random Number 0.41327 0.79514 0.66237 0.00234 0.71205 0.18037 0.71607 0.90512 Note: Rows 10-901 are not shown. Slide 16 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. Slide 17 Example: St. Andrew’s Point Estimates • x as Point Estimator of x x 30 i 29,910 997 30 • s as Point Estimator of (x x ) s 2 i 29 • p as Point Estimator of p 163,996 75.2 29 p 20 30 .68 Note: Different random numbers would have identified a different sample which would have resulted in different point estimates. Slide 18 Sampling Distribution of x Process of Statistical Inference Population with mean =? The value of x is used to make inferences about the value of . A simple random sample of n elements is selected from the population. The sample data provide a value for the sample mean x. Slide 19 Sampling Distribution of x The sampling distribution of is the probability x distribution of all possible values of the sample mean x. Expected Value ofx E( x ) = where: = the population mean Slide 20 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. Slide 21 Sampling Distribution ofx 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. Slide 22 Example: St. Andrew’s Sampling Distribution ofx for the SAT Scores 80 x 14.6 n 30 E ( x ) 990 x Slide 23 Example: St. Andrew’s Sampling Distribution of x for the 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 plus or minus 10 of the actual population mean ? In other words, what is the probability that x will be between 980 and 1000? Slide 24 Example: St. Andrew’s Sampling Distribution of x for the SAT Scores Sampling distribution of x Area = .2518 Area = .2518 x 980 990 1000 Using the standard normal probability table with z = 10/14.6= .68, we have area = (.2518)(2) = .5036 Slide 25 Sampling Distribution of p The sampling distribution pof is the probability distribution of all possible values of the sample p proportion p Expected Value of E ( p) p where: p = the population proportion Slide 26 Sampling Distribution ofp 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. Slide 27 Example: St. Andrew’s Sampling Distribution of p for In-State Residents .72(1 .72) p .082 30 E( p ) .72 The normal probability distribution is an acceptable approximation since np = 30(.72) = 21.6 > 5 and n(1 - p) = 30(.28) = 8.4 > 5. Slide 28 Example: St. Andrew’s Sampling Distribution of p for In-State Residents What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring oncampus 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? Slide 29 Example: St. Andrew’s Sampling Distribution of p for In-State Residents Sampling distribution of p Area = .2291 Area = .2291 0.67 0.72 0.77 p For z = .05/.082 = .61, the area = (.2291)(2) = .4582. The probability is .4582 that the sample proportion will be within +/-.05 of the actual population proportion. Slide 30 Properties of Point Estimators Before using a sample statistic as a point estimator, statisticians check to see whether the sample statistic has the following properties associated with good point estimators. • Unbiasedness • Efficiency • Consistency Slide 31 Properties of Point Estimators Unbiasedness If the expected value of the sample statistic is equal to the population parameter being estimated, the sample statistic is said to be an unbiased estimator of the population parameter. Slide 32 Properties of Point Estimators Efficiency Given the choice of two unbiased estimators of the same population parameter, we would prefer to use the point estimator with the smaller standard deviation, since it tends to provide estimates closer to the population parameter. The point estimator with the smaller standard deviation is said to have greater relative efficiency than the other. Slide 33 Properties of Point Estimators Consistency A point estimator is consistent if the values of the point estimator tend to become closer to the population parameter as the sample size becomes larger. Slide 34 Other Sampling Methods Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling Slide 35 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. homogeneous group). A simple random sample is taken from each stratum. Formulas are available for combining the stratum sample results into one population parameter estimate. Slide 36 Stratified Random Sampling 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, etc. Slide 37 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. … continued Slide 38 Cluster Sampling 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. Example: A primary application is area sampling, where clusters are city blocks or other well-defined areas. Slide 39 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. This method has the properties of a simple random sample, especially if the list of the population elements is a random ordering. … continued Slide 40 Systematic Sampling 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. Slide 41 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. Advantage: Sample selection and data collection are relatively easy. Disadvantage: It is impossible to determine how representative of the population the sample is. Example: A professor conducting research might use student volunteers to constitute a sample. Slide 42 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. 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. Example: A reporter might sample three or four senators, judging them as reflecting the general opinion of the senate. Slide 43 End of Chapter 7 Slide 44