EXERCISE: SIMPLE RANDOM SAMPLING STATISTICS AND DATA ANALYSIS ABSTRACT. Draw a simple random sample using a random number table and the acceptance-rejection technique. 1. USING A RANDOM NUMBER TABLE In many field and laboratory activities, it is useful to conduct randomizations without the need of computing equipment. Fortunately, this is readily accomplished using a pre-printed random number table.

1 1.1. The Initial Draw. The key to using the table is to remember, the numbers are already random, so don't use the table randomly. Instead, perform only one random act: choose an entry as a starting point. Then proceed systematically: decide which, and how many, digits you will use from each entry choose a primary direction to scan through the table: up, down, left, or right choose a direction to move when you get to the edge: left, right, up, or down using those directions, extract adjacent numbers from the table as needed For example, an agronomist is using the random number table in the Appendix. He closes his eyes a puts his finger on the table at the entry 2016 (16th row, 3 rd column). He decides to use the 3 leading digits, and will move down the column, and shift to the left should he reach the end of the column. He reads off the sequence 201, 592, 905, 635, 704, 495, 496, 097, 823, 267, . . . Notice the "shift left" to the top of the next (left-ward) column after 704. 1.2. Applying the Acceptance-Rejection Technique. Sounds pretty simple, but there's a catch. In most applications, the necessary range of random numbers isn't 0, . . . , 9, or 0, . . . , 99, or 0, . . . , 999, or any other range with such convenient limits. Instead, you may have ranges like 10, . . . , 80 or 1, . . . , 50, as in this exercise. So what to do? The answer is quite simple: discard the numbers that fall outside the desired range. Just throw them out! This has a fancy technical name, it's called the acceptance-rejection technique.

2. DRAWING THE RANDOM SAMPLE Here's the application. During a pandemic emergency, a city is planning for overflow hospital beds. Public health officials have identifed N = 50 specialized clinics with bedspace, but detailed numbers are not available. They need a quick estimate, and decide to use a sample of n = 10 clinics which they will task you to survey. To get started, use the random number table in the appendix, applying the acceptance-rejection technique, and generate a sequence of 10 random numbers in the range 1, . . . , 50. This is your subject ID set S. Next use the ID set S to extract the set of measurements {y1, y2, . . . , y10} from the sampling frame; this is your sample. 1Back in the 1970's, a UC Riverside student accidentally collided with Dr. F. N. David, causing her to spill a deck of computer punch cards. Horrified, the student offered to help sort the cards back into order. Dr David laughed, "Not to worry, it's a random number table. Any order will do." 1

3. CALCULATE SAMPLE STATISTICS AND ESTIMATES Calculate the sample total, mean, and variance for your random sample: t = X iS yi = y = t n 2 = s 2 = P iS y 2 i ny 2 n 1 Then use these sufficient statistics to calculate interval estimates[1] for the population mean and total: = 2 s s 2 n N n N T = N 2N s s 2 n N n N If you're curious as to how good your estimates are, the summary parameters for the sampling frame (n = N = 50) are T = 3004 = 60.08 2 = 1077.914 So if your interval estimates include the values for T and , you've generated a 95% "confident" interval

4. YOUR REPORT Your report should include your random number protocol, to include - which number was your random start, e.g. 4963 - which direction you scanned the table, e.g. down and right, left and up, etc. - which digits you chose, e.g. first two, middle two, last two, etc. your list of "accepted" two-digit random numbers from the table, the set S your list of sample values, the set {y1, y2, . . . , yn} your summary sufficient statistics, t, y, and s 2 . your interval estimates for and T, in the form ME. 2 If you compare your work with a classmate, don't freak out if you get somewhat different estimates. This is after all random sampling.

2 APPENDIX: TABLES TABLE 1. Four-Digit Random Numbers 3334 4952 0902 5786 0364 2302 1934 8567 3632 4963 9668 0744 8661 6150 8920 8641 9526 0976 4817 7482 4741 1234 7232 4777 6099 8234 9392 7815 6352 7808 9696 3950 5804 2678 6117 3237 3091 6107 2328 3238 0151 5533 3666 9879 2890 1463 1035 0230 9159 2800 7878 3213 3912 4166 4009 5453 9468 9263 2537 4943 6507 8289 0713 0106 2346 3296 9351 9591 5125 3558 5589 5564 2992 2892 8549 7629 8875 5585 7300 1109 4877 0458 3209 3981 5247 3243 0095 4269 3400 1940 6246 3339 2684 7602 7610 1082 3339 0446 7608 2039 0577 2795 0430 0284 5457 6297 2462 2380 9232 2478 2988 7018 8396 4508 7001 8626 1135 8302 4536 5931 4414 9135 2016 2492 2378 6510 5192 0134 8720 5122 5922 4982 0015 1386 8193 5227 4646 4137 9053 7139 1568 9688 7961 9440 5415 8789 6351 5156 6168 6663 7098 6744 5540 7612 7046 0880 2925 2557 2619 3254

TABLE 2. Private Clinic Beds (the sampling frame) 1 2 3 4 5 6 7 8 9 10 27 93 55 108 70 84 98 34 113 62 11 12 13 14 15 16 17 18 19 20 115 12 52 27 106 81 30 18 45 20 21 22 23 24 25 26 27 28 29 30 73 74 75 115 77 91 66 15 94 69 31 32 33 34 35 36 37 38 39 40 57 71 20 73 48 36 50 51 78 1 41 42 43 44 45 46 47 48 49 50 28 107 56 31 6 111 86 61 36 102 Each cell contains two numbers. The number in the upper left-hand corner is the subject ID, used in random sampling. The number in the lower right-hand corner is the value associated with the subject, the number used in calculating sample statistics