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SAMPLING METHODS

SAMPLING METHODS. Reasons for Sampling. Samples can be studied more quickly than populations.

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SAMPLING METHODS

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  1. SAMPLING METHODS

  2. Reasons for Sampling • Samples can be studied more quickly than populations. • A study of a sample is less expensive than studying an entire population, because smaller number of items or subjects are examined. This consideration is especially important in the design of large studies that require a length follow-up. • A study of an entire population (census) is impossible in most situations. Sometimes, the process of the study destroys or depletes the item being studied.

  3. Sample results are often more accurate than results based on a population. • If samples are properly selected, probability methods can be used to estimate the error in the resulting statistics. It is this aspect of sampling that permits investigators to make probability statements about observations in a study.

  4. The primary purpose of sampling is to estimate certain population parameters such as means, totals, proportions or ratios. SAMPLING MEHODS Probability Sampling Non-probability Sampling A probabilty sample has the characteristic that every element in the population has a known, nonzero probablity of being included in the sample. A non-probability sample is one, that does not have this feature.

  5. Probability Sampling Methods • Simple Random Sampling • Stratified Random Sampling • Systematic Sampling • Cluster Sampling

  6. Simple Random Sampling A simple random sample is one in which every subject has an equal probability of being selected for the study. The recommended way to select a simple random sample is to use a table of random numbers or a computer-generated list of random numbers.

  7. From a population of size N, in order to select a simple random sample of size n; • List and number each element in the population from 1 to N. • 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ..............................., N-1, N • 2. Determine the required sample size, n. • 3. Select n random numbers by a random process, e.g. Table of random numbers or a sofware, MS Excell • 4. Take subjects from the population corresponding to the selected random numbers. • 5. Estimate the population values (parameters).

  8. From a population of size N=500, select a random sample of size n=10. • Number subjects from 1 to 500. • From a random starting point, 838, move down. Take numbers ≤ 500. • 379, 404, 100, 215, 290, 479, 487, 69, 405, 290th subjects in the population will constitute the sample. • Make observations on selected subjects. • Estimate parameters. 83760 31255 71609 89887 00940 54355 44351 89781 58054 65813 66280 56046 50526 33649 87067 02697 06577 16707 96368 47678 70218 28376 98535 34190 96911 81578 97312 20500 48030 27256 02349 88955 52760 73696 91510 38633 38883 90419 26716 98215 93606 21415 34843 12969 84847 06280 95916 12991 08262 58385 24274 18747 37327 06780 08032 98544 24902 81607 87914 22721 67778 70496 57588 89813 71211 83848 93494 27946 79722 70315 89134 06458 40897 73025 04191 77144 49340 89446 71852 80854 83625 00097 71092 12009 63223 37993 50067 25688 98179 34628 03324 68196 72460 55616 27006 50790 28629 88726 97143 63218 84392 36623 91964 03505 46525 40490 77787 68545 02795 72676 76926 10866 39734 50512 04181 78012 78705 86194 28371 54535 06612 60200 49085 85108 71438 10099 99027 65081 82492 77584 76721 02889 95600 07984 31925 59685 91510 40039 43205 37149 64599 51953 55612 89088 58436 21501 86219 74528 59805 65020 79440 99677 49530 55291 34867 54774 52449 23294 94815 95124 35839 00177 57742 09502 42624 29017 94284 81409 36904 54329 83013 94568 75490 12138 24067 86954 00910 61171 82982 87191 19980 47085 46064 19102 26297 79745 99611 04555 52501 32088 55716 10350 67645 62922 81919 47925 91448 36025 20611 38939 36624 03992 27656 33092 22252 54461 83386 55340 11313 23290 50678 33814 07643 81452 60689 48745 49894 27285 90420 31188 17932 27351 34623 55864 58659 06992 88558 45742 56792 71027 76795 23022 20409 60100 59507 40596 16971 96490 47676 49129 20654 64916 59927 62495 81133 29095 64024 02792 39809 85302 73601 60099 50404 41700 53664 54397 49600 46980 13882 54275 59678 14528 96293 12957 68229 95753 15727 75113 09892 71487 92132 51012 09399 30175 73025 99849 34334 20089 19323 95149 76143 16802 32819 34057 94227 25779 93959 89810 47627 70561 99617 64239 13967 90188 60291 38478 09723 10697 78020 51388 02841 25077 02368 75931 42679 70900 33040 08871 46696 18647 57979 28621 03155 03704 98473 25894 26753 62390 54746 84189 41233 68027 17036 28310 50551 84295 80793 93235 78902 18351 48049 09367 15040 29166 64290 16439 67192 16681 46304 68190 10984 97394 23070 90585 53139 96998 39834 27678 42288 33778 59531 76937 15645 70938 00036 72773 25984 06507 27933 46779 36874 61476 74611 74476 48713 36124 98549 70465 58742 28707 49377 53222 14506 80260 59070 47101 02248 99520 08803 79772 59707 00510 29216 53012 47115 39798 79797 06491 72669 05055 63469 49151 35960 88792 43961 62352 78114 77810 95638 84227

  9. From the sample calculate statistics to estimate paramaters (population values). Point Estimates μ P P

  10. Interval Estimates Confidence interval for the population mean: Where S is the standard deviation and t is the tabulated t value.

  11. a: One Tail: 0.250 0.100 0.050 0.025 0.010 0.005 1 1.000 3.078 6.314 12.706 31.821 63.657 a: Two Tails: 0.500 0.200 0.100 0.050 0.020 0.010 2 0.816 1.886 2.920 4.303 6.965 9.925 3 0.765 1.638 2.353 3.182 4.541 5.841 4 0.741 1.533 2.132 2.776 3.747 4.604 5 0.727 1.476 2.015 2.571 3.365 4.032 6 0.718 1.440 1.943 2.447 3.143 3.707 7 0.711 1.415 1.895 2.365 2.998 3.499 8 0.706 1.397 1.860 2.306 2.896 3.355 9 0.703 1.383 1.833 2.262 2.821 3.250 10 0.700 1.372 1.812 2.228 2.764 3.169 11 0.697 1.363 1.796 2.201 2.718 3.106 12 0.695 1.356 1.782 2.179 2.681 3.055 13 0.694 1.350 1.771 2.160 2.650 3.012 14 0.692 1.345 1.761 2.145 2.624 2.977 15 0.691 1.341 1.753 2.131 2.602 2.947 16 0.690 1.337 1.746 2.120 2.583 2.921 17 0.689 1.333 1.740 2.110 2.567 2.898 18 0.688 1.330 1.734 2.101 2.552 2.878 19 0.688 1.328 1.729 2.093 2.539 2.861 20 0.687 1.325 1.725 2.086 2.528 2.845 25 0.684 1.316 1.708 2.060 2.485 2.787 30 0.683 1.310 1.697 2.042 2.457 2.750 40 0.681 1.303 1.684 2.021 2.423 2.704 50 0.679 1.299 1.676 2.009 2.403 2.678 60 0.679 1.296 1.671 2.000 2.390 2.660 70 0.678 1.294 1.667 1.994 2.381 2.648 80 0.678 1.292 1.664 1.990 2.374 2.639 90 0.677 1.291 1.662 1.987 2.368 2.632 100 0.677 1.290 1.660 1.984 2.364 2.626  0.674 1.282 1.645 1.960 2.326 2.576

  12. Example A researcher wishes to estimate the average age of the mother at first birth. He selects 10 mothers at random, and gathers the following data:

  13. Point estimate of the population mean: Sample standard deviation: Estimated standard eror of the mean:

  14. If the researcher wishes to be 95% confident in his estimate:

  15. a: One Tail: 0.250 0.100 0.050 0.025 0.010 0.005 1 1.000 3.078 6.314 12.706 31.821 63.657 a: Two Tails: 0.500 0.200 0.100 0.050 0.020 0.010 2 0.816 1.886 2.920 4.303 6.965 9.925 3 0.765 1.638 2.353 3.182 4.541 5.841 4 0.741 1.533 2.132 2.776 3.747 4.604 5 0.727 1.476 2.015 2.571 3.365 4.032 6 0.718 1.440 1.943 2.447 3.143 3.707 7 0.711 1.415 1.895 2.365 2.998 3.499 8 0.706 1.397 1.860 2.306 2.896 3.355 9 0.703 1.383 1.833 2.262 2.821 3.250 10 0.700 1.372 1.812 2.228 2.764 3.169 11 0.697 1.363 1.796 2.201 2.718 3.106 12 0.695 1.356 1.782 2.179 2.681 3.055 13 0.694 1.350 1.771 2.160 2.650 3.012 14 0.692 1.345 1.761 2.145 2.624 2.977 15 0.691 1.341 1.753 2.131 2.602 2.947 16 0.690 1.337 1.746 2.120 2.583 2.921 17 0.689 1.333 1.740 2.110 2.567 2.898 18 0.688 1.330 1.734 2.101 2.552 2.878 19 0.688 1.328 1.729 2.093 2.539 2.861 20 0.687 1.325 1.725 2.086 2.528 2.845 25 0.684 1.316 1.708 2.060 2.485 2.787 30 0.683 1.310 1.697 2.042 2.457 2.750 40 0.681 1.303 1.684 2.021 2.423 2.704 50 0.679 1.299 1.676 2.009 2.403 2.678 60 0.679 1.296 1.671 2.000 2.390 2.660 70 0.678 1.294 1.667 1.994 2.381 2.648 80 0.678 1.292 1.664 1.990 2.374 2.639 90 0.677 1.291 1.662 1.987 2.368 2.632 100 0.677 1.290 1.660 1.984 2.364 2.626  0.674 1.282 1.645 1.960 2.326 2.576

  16. CONFIDENCE INTERVAL FOR A POPULATION PROPORTION When P, population proportion is unknown, its estimate, the sample proportion, p can be used.

  17. Example A researcher wishes to estimate, with 95% confidence, the proportion of woman who are at or below 20 years of age at first birth.

  18. Point estimate of the population proportion: p=a/n=4/10=0.4 Estimated standard error of the mean:

  19. In the above example if the sample size were 100 instead of 10, then the 95% confidence interval would be:

  20. Among 250 students of Hacettepe University interwieved 185 responded that they reqularly read a daily newspaper. With 95% confidence, find an interval within which the proportion of students who regularly read a newspaper in Hacettepe University lie. Point estimate of the proportion of students who read a newspaper. The standard error of the estimate is 0.028.

  21. In oder words, the standard deviation of the proportions that can be computed from all possible samples of size 250 is 0.028. The 95% Confidence Interval is:

  22. 1 2 3 4 … i … k … i+k … i+2k … i+3k … N N/n Systematic Sampling A systematic random sample is one in which every kth item is selected; k is determined by dividing the number of items in the population by the desired sample size.

  23. Stratified Sampling A stratified random sample is one in which the population is first divided into relevant strata (subgroups), which are internally homogenous with respect to the variable of interest and a random sample is then selected from each stratum. Characteristics used to stratify should be related to the measurement of interest, in which case stratified random sampling is the most efficient, meaning that it requires the smallest sample size.

  24. Strata Strata size Sample size 1 N1 n1 2 N2 n2 k Nk nk TOTAL N n From each starta, select random samples independently, whose sizes are proportional to the size of that strata.

  25. Estimation of the parameters

  26. Cluster Sampling A cluster random sample results from a two-stage process in which the population is divided into clusters and a subset of the clusters is randomly selected. Clusters are commonly based on geographic areas or districts, so this approach is used more often in epidemiologic research than clinical studies.

  27. Non-probability Sampling The sampling methods just discussed are all based on probability, but nonprobability sampling methods also exist, such as convenience samples or quota samples. Nonprobability samples are those in which the probability that a subject is selected is unknown. Nonprobability samples often reflect selection biases of the person doing the study and do not fulfill the requirements of randomness needed to estimate sampling error. When we use the term “sample” in the context of observational studies, we will assume that the sample has been randomly selected in an appropriate way.

  28. DETERMINATION OF THE SAMPLE SIZE How large a sample is needed for estimating a) Population mean, : i) When population size, N, is unknown ii) When population size, N, is known

  29. Example If we wish, with 95% confidence, to estimate the average birth weight of infants, within 250 gr around the unknown population mean, how largea sample should we select? (Assume =700 gr) When N=60 When d=400 gr, required sample size, n is 9.97~10.

  30. b) Population proportion, P: i) When population size, N, is unknown ii) When population size, N, is known

  31. Example If we wish, with 95% confidence, the proportion of infants with low birth weight within 10% around the unknown population proportion, how many infants should be selected?

  32. Example If we know that the population size from which we will sample is 100, how many infants should be selected?

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