1. Sampling Design and Analysis MTH 494 Ossam Chohan Assistant Professor CIIT Abbottabad

2. LECTURE-7 2

3. Simple Random Sampling (SRS) Simplest sampling design Def-1: If a sample of size n is drawn from a population of size N in such a way that every possible sample of size n has the same chance (probability) of being selected the sampling procedure is called Simple Random Sampling. The sample thus obtained is called a simple random sample. We will use simple random sampling to obtain estimators for population means, totals, and proportions. 3

4. Problem A federal auditor is to examine the accounts for a city hospital. The hospital records obtained from a computer shows a particular accounts receivable total, and the auditor must verify this total. Issue????? Suppose that all N= 28,000 patients records are recorded on computer cards and sample of size n=100 is to be drawn. The sample is called a simple random sample if every possible sample of size =100 records has the same chance of being selected. 4

5. How to draw a SR Sample This is not as difficult as it looks But selection is important because it leads to – Investigator bias – Poor estimation The procedure for selecting a Simple Random Sample is as follows: – List all the units in the population (construct a sampling frame if one does not exist already), say from 1,..,N 5

6. 6 How to select the sample In other words, give each element a unique Identification (ID) starting from 1 to the number of elements in the population N N is the total number of units in the population Using random numbers or any other random mechanism (eg Lottery or goldfish bowl), select the sample of n units from the list of N units one at a time without replacement

7. 7 How to select the sample There many different random number tables. One example is given on the next slide.

8. 8 Random Numbers 8442 5653 8775 1891 7666 6483 9711 6941 8092 3875 4200 6543 9063 1003 8754 2564 8890 4195 8888 6490 3476

9. 9 How to use a random number table? Decide on the minimum number of digits Start anywhere in the table and going in any direction choose a number/(s) The sequence of reading the numbers should be maintained until the desired sample size is attained If a particular number is not included in your range of population values, choose another number Keep selecting the numbers till you have the required number of elements in your sample

10. 10 How to use a random number table? This process of selecting a large sample using random number tables is tedious Usually we use computer generated random numbers

11. 11 Example Suppose we want to select 5 elements from a population of 8000. We number the population from 1 to 8000 We use the random number table given on couple of slides back Suppose for convenience we start at the top left hand corner and read across

12. 12 Example We need to use four digits at a time as there is a minimum of four digits in our sample ID’s. Let us start. The first set of 4 digits is 8442 8442 not in our population-ignore 5653-use for sample -1 8775-ignore 1891-use for sample -2

13. 13 Example 7666- use for sample -3 6483- use for sample -4 9711-ignore 6941-use for sample- 5

14. 14 The sample So our sample consists of the units numbered 5653,1891,6483,7666, and 6941. Here we have used sampling without replacement. Sampling with replacement (WR) - units can be selected more than once. Sampling without replacement (WOR) - units cannot be selected more than once

15. 15 Need to explain the lottery method of sampling This involves writing the unique numbers on identical items which are then put in an urn. Then one item is drawn at a time and the unit to which the drawn number corresponds is included in the sample. This can also be done with or without replacement

16. 16 Advantages/Disadvantages of Simple Random Sampling Advantages: – Sample is easy to select in cases where the population is small Disadvantages: – Costs of enumeration may be high because by the luck of the draw, the sampled units may be widely spread across the population – By bad luck, the sample may not be representative because it may not be evenly spread across all sections of the population

17. 17 What to do next with samples Having selected the sample, we now need to produce estimates from the sample to make certain statements about the population Usually we want to provide estimates of certain parameters in the population eg mean, medians, totals or proportions

18. Example Assume there are N=1000 patient records from which a simple random sample of n=20 is to be drawn. We know that a simple random sample will be obtained if every possible example of n=20 records has the same chance of being selected. Using random number table (you can found online easily), the possible sample could be: 18

19. Example-Cont… 104779289510 223995635023 241963094010 421895103521 375854071070 19

20. Estimation of a Population Mean and Total We stated previously that objective of Survey sampling is to draw inferences about a population on the basis of sample evidence. There are two approaches to draw inferences: – Estimation – Hypothesis Testing 20

21. Suppose that a simple random sample of n accounts is drawn and we are to estimate the mean value per account for the total population of hospital records. Intuitively, we would employ the sample average To estimate µ (Parameter) 21

22. Single value of is not sufficient to estimate parameter. Goodness of estimator must be evaluated. How? is an unbiased estimator. has a variance that decreases as the sample size n increases. 22

23. Suppose we have a population of n=4 measurements given by {1,2,3,4}. If a simple observation y is selected at random from this population, then y can take on any of the four possible values, each with probability ¼. Thus 23

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26. Some other Estimators Estimator of the population mean µ Estimated variance of 26

27. Bound of the Error of Estimation 27

28. Example A simple random sample of n=9 hospital records is drawn to estimate the average amount of money due on N=484 open accounts. The sample values for these nine records are listed in the following table. Estimate µ, the average amount outstanding, and place a bound on your error of estimation. 28

29. Example-Data Sample NumberValue y133.50 y232.00 y352.00 y443.00 y540.00 y641.00 y745.00 y842.50 y939.00 29

30. Example Solution 30

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32. Conclusion To summarize, the estimate of the mean amount of money owed per account, µ, is $40.89. although we cannot be certain how close is to µ, we are reasonably confident that the error of estimation is less than $3.94 32

33. Estimating Population Totals Many sample surveys are designed to obtain population total. Recall that the mean for a population of size N is sum of all observations in the population divided by N. the population total---that is, the sum of all observations in the population----is denoted by the symbol τ (sound like taw). Hence Nµ = τ Intuitively, we expect the estimator of τ to be N times the estimator of µ, which it is. 33

34. Estimators for totals Estimator of the population total τ Estimated Variance of τ 34

35. Bound on the error of estimation 35

36. Example An industrial firm is concerned about the time per week spent by scientists on certain trivial tasks. The time log sheets of a simple random sample of size n=50 employees show the average amount of time spent on these tasks is 10.31 hours with a sample variance s 2 = 2.25. The company employs N=750 scientists. Estimate the total number of man-hours lost per week on trivial tasks and place a bound on the error of estimation 36

37. Example Solution 37

38. Conclusion Thus the estimate of total time lost is τ = 7732.5 hours. We are reasonably confident that the error of estimation is less than 307.4 hours. 38