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Chapter 7 Systematic Sampling. Selection of every kth case from a list of possible subjects. Systematic Sampling - 2. Definition :
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Chapter 7Systematic Sampling • Selection of every kth case from a list of possible subjects.
Systematic Sampling - 2 Definition: A sample obtained by randomly selecting 1 element from among the first k elements in the frame and every kth element thereafter is called a 1-in-k systematic sample with a random start. (Assumes population is randomly ordered). Does each element in the frame have an equal chance to be selected? Yes If so, what is this equal chance? 1/k Is this a simple random sample? NO!!
Systematic Sampling - 3 1 26 51 76 2 27 52 77 3 28 53 78 429 54 79 5 30 55 80 6 31 56 81 7 32 57 82 8 33 58 83 934 59 84 10 35 60 85 11 36 61 86 12 37 62 87 13 38 63 88 1439 64 89 15 40 65 90 16 41 66 91 17 42 67 92 18 43 68 93 1944 69 94 20 45 70 95 21 46 71 96 22 47 72 97 23 48 73 98 244974 99 25 50 75 100 N = 100 Want n = 20 k = N/n = 5 Select a random number between 1 and 5: For example, choose 4 Start with #4 and select every 5th item
Systematic Sampling - 4 There are actually only 5 distinct systematic random samples which are: 1. {1,6,11,…,91,96} 2. {2,7,12,…,92,97} 3. {3,8,13,…,93,98} 4. {4,9,14,…,94,99} 5. {5,10,15,…,95,100} We are simply choosing 1 of these 5 groups at random 1 26 51 76 2 27 52 77 3 28 53 78 429 54 79 5 30 55 80 6 31 56 81 7 32 57 82 8 33 58 83 934 59 84 10 35 60 85 11 36 61 86 12 37 62 87 13 38 63 88 1439 64 89 15 40 65 90 16 41 66 91 17 42 67 92 18 43 68 93 1944 69 94 20 45 70 95 21 46 71 96 22 47 72 97 23 48 73 98 244974 99 25 50 75 100 N = 100 Want n = 20 k = N/n = 5 Select a random number between 1 and 5: Choose 4 Start with #4 and select every 5th item
Systematic Sampling - 5 • Advantages • Easier to perform in the field, especially if a good frame is not available • Frequently provides more information per unit cost than simple random sampling, in the sense of smaller variances. Example. A systematic sample was drawn from a batch of produced computer chips. The first 400 chips are fine but, due to a fault in the machine later in the production process, the last 300 chips are defective. Systematic sampling will select uniformly over the non-defective and defective items and would give a very accurate estimate of the fraction of defective items.
Systematic Sampling - 6 Value of k? • In general, for a systematic random sample of n elements from a population (or frame) of size N, choose k ≤ N/n. Example: From a population of 90,000 students we desire a sample of 12,000 students. Since 90,000/12,000 = 7.5, we can select a 1-in-7 systematic sample.
Systematic Sampling - 7 Value of k when N unknown? • Must guess the value of k to achieve a sample size n. • If k is too large, in some cases can go back and select another 1-in-k sample until the sample size n is attained.
Systematic Sampling - 8Estimation How many people at this rally?
Systematic Sampling – 11 1. If ρ is close to 0, and N is fairly large, systematic sampling is roughly equivalent to simple random sampling. 2. If ρ is close to 1, then the elements within the sample are quite similar wrt the characteristic being measured, and systematic sampling will yield a higher variance of the sample mean than will simple random sampling. 3. If the elements in the systematic sample tend to be very different, then ρ is negative and systematic sampling may be more precise than simple random sampling.
Systematic Sampling – 12Summary: comparison of systematic and simple random sampling 1. Random order (If ρ is close to 0) Systematic and simple random sampling are approximately equal in precision. 2. Cyclic pattern in the y’s Systematic random sampling is worse than simple random sampling. 3. Increasing or Decreasing order in the y’s Systematic random sampling is better than simple random sampling.
Systematic Sampling – 13 1. Random order: if ρ is close to 0, and N is fairly large, systematic sampling is roughly equivalent to simple random sampling.
Systematic Sampling – 14 2. Cyclic pattern in the y’sSystematic random sampling is worse than simple random sampling.
Systematic Sampling – 15 3. Increasing or Decreasing order in the y’s Systematic random sampling is better than simple random sampling.
Behavior of Consider a systematic sample of n elements as a single cluster sample selected from k possible cluster samples in the population. Recall … 1 26 51 76 2 27 52 77 3 28 53 78 429 54 79 5 30 55 80 6 31 56 81 7 32 57 82 8 33 58 83 934 59 84 10 35 60 85 11 36 61 86 12 37 62 87 13 38 63 88 1439 64 89 15 40 65 90 16 41 66 91 17 42 67 92 18 43 68 93 1944 69 94 20 45 70 95 21 46 71 96 22 47 72 97 23 48 73 98 244974 99 25 50 75 100 N = 100 Want n = 20 k = N/n = 5 There are actually only 5 distinct systematic random samples which are: 1. {1,6,11,…,91,96} 2. {2,7,12,…,92,97} 3. {3,8,13,…,93,98} 4. {4,9,14,…,94,99} 5. {5,10,15,…,95,100} We are simply choosing 1 of these 5 groups at random Select a random number between 1 and 5
Behavior of Systematic sampling involves randomly selecting one of the k clusters (rows). So In general, think of the population as being arranged in a rectangular array. Here N=nk
Behavior of measures the correlation among elements in the same cluster (the intracluster correlation) and lies between [1/(n-1)] and 1.
Behavior of With the population of measurements as shown in the table, we can make analysis-of-variance-type (ANOVA) calculations to determine
Behavior of Total sum of squares Sum of squares between clusters Sum of squares within clusters
Behavior of Total sum of squares
Behavior of Total sum of squares Mean square between clusters Mean square within clusters
Behavior of When N = nk is large: If MSB small compared to MST, then <0 and from simple random sampling A small MSB means the cluster means are nearly equal.
Behavior of Example When N = nk is large: ANOVA
