Chapter 8

1. Chapter 8 Random Sampling: Planning Ahead for Data Gathering © Andrew F. Siegel, 2003

2. Random Sampling • The basis for statistical inference about a population based on a sample • Example: Build restaurant in neighborhood? • Population: the collection of items you want to understand N items. Example: all people in neighborhood • Sample: a smaller collection of population units n items. Example: 100 neighborhood residents who agree to be interviewed • Which 100 residents? • How to select? © Andrew F. Siegel, 2003

3. Terminology of Sampling • Representative Sample • Same percentages in sample as in population • Example: sample is representative if same percent: • work, are young/old, are single/married, etc … • Biased Sample • Not representative of population in an important way • Example: sample is biased if too many retired people • Frame: Access to population (by number from 1 to N) Example: from phone book: 1. Fred Jones 2. Mary Kipling … … N. Carol Lewis © Andrew F. Siegel, 2003

4. Population Sample Visualizing the Sampling Process • Here is a fairly (but not perfectly) representative sample • Note that neither open triangle was selected © Andrew F. Siegel, 2003

5. Sampling Terminology (continued) • Sampling Without Replacement • No unit may appear more than once in the sample • After a unit is selected, it is removed from the population before another population unit is drawn • Sampling With Replacement • A unit can be represented more than once in the sample • After a unit is selected, it is “replaced” back in the population before another unit is drawn • Census • A sample that consists of the entire population • Often too expensive • Even when affordable, may not be cost-efficient © Andrew F. Siegel, 2003

6. Statistic and Parameter • Sample Statistic • Any number computed from sample data • A random variable. Known • Example: Average weekly food expenditures for 100 sampled residents • Random? Yes! Due to randomness of sample selection • Population Parameter • Any number computed for the entire population • A fixed number. Unknown • Example: mean weekly food expenditures for all 77,386 residents • Do we ever know this? NO! • But we estimate it (with error) © Andrew F. Siegel, 2003

7. Estimator and Estimate • Estimator • A sample statistic used to guess a population parameter • Example: Sample average for 100 selected residents is an estimator of the population mean of all 77,386 residents • Estimate [WRONG! Estimators are usually wrong. Often useful anyway] • The actual number computed from the data • Example: $33.91 is an estimate of neighborhood weekly food expenditures per person • Estimation error • Estimator minus population parameter. Unknown • Example: 33.91 – 35.69 = –1.78 Estimation error (unknown) Mean, all 77,386 residents (unknown) Average, 100 residents © Andrew F. Siegel, 2003

8. Sampling Terminology (continued) • Unbiased Estimator • Correct on average. Neither systematically too high nor too low • Hits the target “on average” although it may miss each time • Pilot Study • Small-scale test before full study is run • Test your questionnaire on a few people, change as needed, and then use it on hundreds more • Sampling Distribution • The probability distribution of a statistic computed using data from a random sample © Andrew F. Siegel, 2003

9. Random Sample • Provides a foundation for statistical inference • A random sample must satisfy: 1. Each population unit must have an equal chance of being selected • This helps assure representation, because all units in the population are equally accessible 2. Units must be selected independently of one another • This guarantees that each item to be selected will bring new, independent information • Properties • Sample is representative of population (on average) • Statistical inference will use randomness of the sample © Andrew F. Siegel, 2003

10. Random Samples of 5 from 36 • On 6 by 6 grid 5 squares shaded in, selected at random, one at a time • Without replacement • Note that adjacent (touching) squares can be selected • To exclude them would make selection non-independent • Perhaps you see systematic patterns • They are not there by design, but by coincidence • But a “checkerboard” pattern would very likely not be random © Andrew F. Siegel, 2003

11. Selecting a Random Sample • Use Table of Random Digits • Establish the frame (population units from 1 to N) • Decide starting place in table of random digits • Read random digits in groups • e.g., if N = 5,281, then use groups of 4 digits (N has 4 digits) • Include number group • if it is from 1 to N, and has not yet been chosen • Shuffle the Population (Spreadsheet) • Arrange the population items in a column from 1 to N • Put random numbers in an adjacent column: =RAND() • Sort population items in order by random numbers © Andrew F. Siegel, 2003

12. 1 2 3 4 5 6 7 8 9 10 1 51449 39284 85527 67168 91284 19954 91166 70918 85957 19492 2 16144 56830 67507 97275 25982 69294 32841 20861 83114 12531 3 48145 48280 99481 13050 81818 25282 66466 24461 97021 21072 4 83780 48351 85422 42978 26088 17869 94245 26622 48318 73850 5 95329 38482 93510 39170 63683 40587 80451 43058 81923 97072 6 11179 69004 34273 36062 26234 58601 47159 82248 95968 99722 7 94631 52413 31524 02316 27611 15888 13525 43809 40014 30667 8 64275 10294 35027 25604 65695 36014 17988 02734 31732 29911 9 72125 19232 10782 30615 42005 90419 32447 53688 36125 28456 10 16463 42028 27927 48403 88963 79615 41218 43290 53618 68082 11 10036 66273 69506 19610 01479 92338 55140 81097 73071 61544 12 85356 51400 88502 98267 73943 25828 38219 13268 09016 77465 13 84076 82087 55053 75370 71030 92275 55497 97123 40919 57479 14 76731 39755 78537 51937 11680 78820 50082 56068 36908 55399 15 19032 73472 79399 05549 14772 32746 38841 45524 13535 03113 16 72791 59040 61529 74437 74482 76619 05232 28616 98690 24011 17 11553 00135 28306 65571 34465 47423 39198 54456 95283 54637 18 71405 70352 46763 64002 62461 41982 15933 46942 36941 93412 19 17594 10116 55483 96219 85493 96955 89180 59690 82170 77643 20 09584 23476 09243 65568 89128 36747 63692 09986 47687 46448 21 81677 62634 52794 01466 85938 14565 79993 44956 82254 65223 22 45849 01177 13773 43523 69825 03222 58458 77463 58521 07273 23 97252 92257 90419 01241 52516 66293 14536 23870 78402 41759 24 26232 77422 76289 57587 42831 87047 20092 92676 12017 43554 25 87799 33602 01931 66913 63008 03745 93939 07178 70003 18158 26 46120 62298 69126 07862 76731 58527 39342 42749 57050 91725 27 53292 55652 11834 47581 25682 64085 26587 92289 41853 38354 28 81606 56009 06021 98392 40450 87721 50917 16978 39472 23505 29 67819 47314 96988 89931 49395 37071 72658 53947 11996 64631 30 50458 20350 87362 83996 86422 58694 71813 97695 28804 58523 31 59772 27000 97805 25042 09916 77569 71347 62667 09330 02152 32 94752 91056 08939 93410 59204 04644 44336 55570 21106 76588 33 01885 82054 45944 55398 55487 56455 56940 68787 36591 29914 34 85190 91941 86714 76593 77199 39724 99548 13827 84961 76740 35 97747 67607 14549 08215 95408 46381 12449 03672 40325 77312 36 43318 84469 26047 86003 34786 38931 34846 28711 42833 93019 37 47874 71365 76603 57440 49514 17335 71969 58055 99136 73589 38 24259 48079 71198 95859 94212 55402 93392 31965 94622 11673 39 31947 64805 34133 03245 24546 48934 41730 47831 26531 02203 40 37911 93224 87153 54541 57529 38299 65659 00202 07054 40168 41 82714 15799 93126 74180 94171 97117 31431 00323 62793 11995 42 82927 37884 74411 45887 36713 52339 68421 35968 67714 05883 43 65934 21782 35804 36676 35404 69987 52268 19894 81977 87764 44 56953 04356 68903 21369 35901 86797 83901 68681 02397 55359 45 16278 17165 67843 49349 90163 97337 35003 34915 91485 33814 46 96339 95028 48468 12279 81039 56531 10759 19579 00015 22829 47 84110 49661 13988 75909 35580 18426 29038 79111 56049 96451 48 49017 60748 03412 09880 94091 90052 43596 21424 16584 67970 49 43560 05552 54344 69418 01327 07771 25364 77373 34841 75927 50 25206 15177 63049 12464 16149 18759 96184 15968 89446 07168 19 17594 10116 55483 96219 85493 96955 20 09584 23476 09243 65568 89128 36747 21 81677 62634 52794 01466 85938 14565 22 45849 01177 13773 43523 69825 03222 23 97252 92257 90419 01241 52516 66293 24 26232 77422 76289 57587 42831 87047 Table of Random Digits Table 8.2.1 • For example • Starting in row 21, column 3 • We find 52794, then 01466 © Andrew F. Siegel, 2003

13. Example: Sample Selection • Select random sample using random number table • Of size n = 4 from a population of size N = 861, starting at row 21, column 3 of the Table of Random Digits • Start with random digits 52794 01466 85938 14565 79993 … • Group by 3 (because N = 861 has 3 digits) 527 940 146 685 938 145 657 … • Omit 000, and also 862, 863, …, 999 • Omit duplicates, until n = 4 are obtained in the sample 527 146 685 145 • The random sample includes the following population units (numbered by frame): 527, 146, 685, 145 © Andrew F. Siegel, 2003

14. The Population Visualizing the Sampling Process • The process: Sample n, compute statistic is the same as the process: Choose 1 from sampling distribution … you do … Sample n units Sample n units Sample n units Sample n units you imagine … Statistic (estimator) Statistic (estimator) Statistic (estimator) Statistic (estimator) … A histogram of these imagined values represents the sampling distribution of this statistic © Andrew F. Siegel, 2003

15. Central Limit Theorem • Helps you find probabilities for an average of n independent individuals by giving you • The mean • The standard deviation • The right to use the normal probability tables: • If n is large, then the average is approximately normal, even if individuals are skewed • Works for totals also by giving you • The mean • The standard deviation • The right to use the normal probability tables for total © Andrew F. Siegel, 2003

16. 300 0 0 5 10 200 100 0 0 5 10 100 0 0 5 10 Central Limit Theorem (continued) • Individuals in population • Highly non-normal distribution • Mean , standard deviation  • Averages of n = 3 individuals • Non-normal, but less so • Same mean  • Lower std. deviation: • Averages of n = 10 individuals • Close to normal • Same mean  • Lower std. deviation    © Andrew F. Siegel, 2003

17. Central Limit Theorem (continued) • What good to us is ? • Why bother choosing a larger sample to estimate ? • Even sampling just n = 1 (or a few) we have an unbiased estimator that is, on average, equal to  • But the error can be very large! • When we sample n = 100 or n = 1,000, what we gain for our trouble is the LOWER ERROR , which says that our estimator, , is closer to the unknown population mean  • It tells us how the variability of the sample average is related to the variability of individuals in the population • This will be useful soon in defining the standard error © Andrew F. Siegel, 2003

18. -3 -2 -1 0 1 2 3 –1.37 Example: Average Weight of a Box • The problem: Apples have a mean weight of 7 ounces and a standard deviation of 2 ounces. They are chosen at random (independently) and put in boxes of 30. Find the probability that the average weight of apples in a box is less than 6.5 ounces. • The question: Find Prob( < 6.5) where =7, =2, n=30 • Standardize, using CLT: • Find • Draw picture, use normal table • Answer is 0.085 © Andrew F. Siegel, 2003

19. 0.46 -3 -2 -1 0 1 2 3 Example (continued) • The problem: Find the probability that the total weight of a box exceeds 215 ounces • By central limit theorem, “total weight” is approximately normal with • Standardize, using CLT: • Find • Draw picture, use normal table • Answer is 0.32 © Andrew F. Siegel, 2003

20. Standard Error • Definition: the Estimated Standard Deviation (of the sampling distribution) of a statistic • Standard Error of the Average • Indicates approximately how far the sample average is from the population mean  • Advantage: can be computed using sample data • Standard Deviation of the Average • Also indicates approximately how far the sample average is from the population mean  • Problem: cannot be computed without population parameters © Andrew F. Siegel, 2003

21. Standard Error for Binomial • Standard Error of p (binomial percent) is • Indicates approximately how far the sample estimate p is from its population value  • Advantage: can be computed using sample data • Standard Deviation of p is • Also indicates approximately how far the sample estimate p is from its population value  • Problem: cannot be computed without population parameters © Andrew F. Siegel, 2003

22. Example of Standard Error n = 100 Sample Size: Number of people in sample = $23.91 Sample Average: Typical expenditure for people in sample. An estimate of typical expenditure  for people in the population S = $11.49 Standard Deviation: Individuals in the sample are approximately S = $11.49 away from = $23.91. Individuals in the population are approximately =?away from =? Standard Error: The sample average is approximately from =? Variability of individuals Estimated by S = 11.49 Variability of the sampleaverage © Andrew F. Siegel, 2003

23. Example: Binomial Standard Error X = 83 out of n = 268 interviewed said they would buy the product p = X/n = 83/268 = 0.3097 or 31.0% is the sample percent, an estimate of , the (unknown) population percent How far is the (known) p = 31.0% from the (unknown)  =? Answer: about one standard error The observed sample percentage, 31.0%, is approximately 2.82% (in percentage points) away from the unknown population percentage  © Andrew F. Siegel, 2003