1. Chapter 7 Systematic Sampling n Selection of every kth case from a list of possible subjects.

2. 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? If so, what is this equal chance? Is this a simple random sample? Yes 1/k NO!!

3. Systematic Sampling - 3 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100 N = 100

4. Systematic Sampling - 4 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100 N = 100 Want n = 20

5. Systematic Sampling - 5 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100 N = 100 Want n = 20 k = N/n = 5

6. Systematic Sampling - 6 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100 N = 100 Want n = 20 k = N/n = 5 Select a random number between 1 and 5: Choose 4

7. Systematic Sampling - 7 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 5 th item 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100

8. Systematic Sampling - 8 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 5 th item 1265176 2275277 3285378 4295479 5305580 6315681 7325782 8335883 9345984 10356085 11366186 12376287 13386388 14396489 15406590 16416691 17426792 18436893 19446994 20457095 21467196 22477297 23487398 24497499 255075100 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

9. Systematic Sampling - 9 n 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.

10. Systematic Sampling - 10 n 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. Value of k?

11. Systematic Sampling - 11 n Must guess the value of k to achieve a sample size n. 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. Value of k when N unknown?

12. Systematic Sampling - 12 Estimation How many people at this rally?

13. Systematic Sampling – 13 Estimation of population mean 

14. Systematic Sampling – 14 Estimation of population mean 

15. Systematic Sampling – 15 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. 1. If ρ is close to 0, and N is fairly large, systematic sampling is roughly equivalent to simple random sampling.

16. Systematic Sampling – 16 Summary: comparison of systematic and simple random sampling 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. 1. Random order (If ρ is close to 0) Systematic and simple random sampling are approximately equal in precision.

17. Systematic Sampling – 17 1. Random order: if ρ is close to 0, and N is fairly large, systematic sampling is roughly equivalent to simple random sampling.

18. Systematic Sampling – 18 2. Cyclic pattern in the y’s Systematic random sampling is worse than simple random sampling.

19. Systematic Sampling – 19 3. Increasing or Decreasing order in the y’s Systematic random sampling is better than simple random sampling.

20. Systematic Sampling – 20 Estimation of population total 

21. Systematic Sampling – 21 Estimation of population proportion p

22. Systematic Sampling – 22 Required Sample Size for Bound B