Assignment 0 Due: 9:00 am January 25, 1996 SIMPLE RANDOM SAMPLING AND THE SURVEY SIMULATIONS THE SIMULATED SURVEY ENVIRONMENT: Stephens County Stephens County is a fictitious county in the midwestern part of the United States with a population of approximately 103,000. It has two main cities: Lockhart City, population 57,500, and Eavesville, population 11,700. Both cities are commercial and transportation centers and boast a variety of light industries. Among the county's industrial products are farm chemicals, pet foods, cable and wire, aircraft radios, greeting cards, corrugated paper boxes, industrial gases, and pipe organs. The county has three smaller municipalities: Villegas, Weldon, and Routledge with populations between one and two thousand. These cities are local commercial centers. The surrounding areas are agricultural, although a sizeable number of persons commute to the larger cities. The county's main agricultural products are beef cattle, wheat, sorghum and soybeans. Stephens County has been organized into 75 districts with the houses within a district numbered consecutively starting with 1. For the purposes of these exercises, you may assume that houses in the same district with close numbers are physically close. The Stephens County Cablevision Company has been formed to provide cable TV service to Stephens County. It has commissioned this survey to help it with its pricing and programming decisions. THE INTERVIEW QUESTIONNAIRE: The Stephens County Cablevision Company has supplied an interview questionnaire for your use: Stephens County Cablevision Inc. Lockhart City 1. How many persons aged 12 and older live at this address? 2. How many persons aged 11 and younger live at this address? 3. How many television sets do you have? 4. (Interviewer: Ask questions 4a,b,c,d,e and record the highest price-- $0, $5, $10, $15, $20, or $25--respondent is willing to pay.) a. If Cable TV cost $5 per month, would you subscribe? b. If Cable TV cost $10 per month, would you subscribe? c. If Cable TV cost $15 per month, would you subscribe? d. If Cable TV cost $20 per month, would you subscribe? e. If Cable TV cost $25 per month, would you subscribe? 5. How many hours per week does your household watch TV? Estimate the number of hours in question 5 that are spent watching each of the following types of programming: 6. News and "public affairs" 7. Sports 8. Children's programming 9. Movies In addition, for each surveyed household, the Company has obtained from the county tax assessor the assessed valuation of that household's living quarters. This information is meant to provide a measure of family income (without having to ask about it). The SURVEY program was written in 1983. Clearly it has been dated by inflation and the addition of other types of optional cable channels (e.g. MTV). DISTRICT MAP OF STEPHENS COUNTY SCALE I---------I 5 miles I---------I---------I---------I---------I---------I---------I I I I I I I I I 1 I 2 I 3 I 4 I 5 I 6 I I I I I I I I I 44 I I I I I I I---------I---------I---------I---------I---------I---------I I I I I I I I I 7 I 8 I 9 I 10 I 11 I 12 I I I I I I I I I I I I I I I I---------I---------I---------I---------I---------I---------I I I I 51 52 53 54 55 I I 45 I I 13 I 14 I I 15 I I I I I 56 57 58 59 60 I I 16 I I I I I I I I---------I---------I 61 62 63 64 65 I---------I---------I I I I I I I I 17 I 18 I 66 67 68 69 70 I 19 I 20 I I I I I I I I I I 71 72 73 74 75 I I I I---------I---------I---------I---------I---------I---------I I I I I I I I I 21 I 22 I 23 I 24 I 25 I 26 I I I I I I I I I I I I I I I I---------I---------I---------I---------I---------I---------I I I I I I I I I 27 I 28 I 29 I 30 I 31 I 32 I I I I I I I I I I I I I I I I---------I---------I---------I---------I---------I---------I I 46 I I I I I I I I 34 I 35 I 36 I 37 I 38 I I 33 I I I I I I I I I I I I I I---------I---------I---------I---------I---------I---------I I I I I 47 48 I I I I 39 I 40 I 41 I I 42 I 43 I I I I I 49 50 I I I I I I I I I I I---------I---------I---------I---------I---------I---------I INCORPORATED MUNICIPALITIES: LOCKHART CITY - 51 TO 75 EAVESVILLE - 47 TO 50 WELDON - 45 VILLEGAS - 44 ROUTLEDGE - 46 STEPHENS COUNTY DISTRICT INFORMATION Column 1: District number Column 2: Number of houses Column 3: Cumulative house count Column 4: Population Column 5: Mean assessed house valuation (1) (2) (3) (4) (5) 1 142 142 549 66448 2 153 295 609 60122 3 135 430 514 65797 4 128 558 477 56084 5 110 668 431 56592 6 103 771 418 58795 7 105 876 374 69629 8 385 1261 1386 77155 9 296 1557 1059 73605 10 287 1844 1002 69039 11 253 2097 905 58635 12 172 2269 660 55189 13 198 2467 759 64406 14 432 2899 1548 76384 15 248 3147 964 69636 16 251 3398 922 53881 17 221 3619 858 67928 18 297 3916 1059 78802 19 235 4151 860 71418 20 171 4322 641 53214 21 135 4457 517 67860 22 254 4711 893 64890 23 203 4914 763 77586 24 244 5158 1013 82727 25 202 5360 741 68232 26 103 5463 401 55069 27 102 5565 407 61346 28 115 5680 438 60559 29 180 5860 694 67770 30 190 6050 817 70615 31 152 6202 548 66993 32 141 6343 555 58621 33 143 6486 588 57266 34 135 6621 481 57311 35 178 6799 661 59424 36 221 7020 849 62233 37 174 7194 654 53194 38 101 7295 317 49778 39 95 7390 355 57749 40 130 7520 477 57899 41 152 7672 613 56030 42 169 7841 712 58756 43 91 7932 352 50803 44 283 8215 1036 60249 45 562 8777 1979 57333 46 312 9089 1108 52056 47 897 9986 3203 62436 48 734 10720 2617 61524 49 963 11683 3462 60091 50 642 12325 2280 55114 51 525 12850 1869 95235 52 726 13576 2513 68411 53 674 14250 1931 54494 54 585 14835 1203 48185 55 553 15388 1111 42886 56 583 15971 2084 95676 57 911 16882 2717 84813 58 1051 17933 2450 56963 59 918 18851 1832 36610 60 799 19650 1610 44400 61 545 20195 1891 101460 62 895 21090 2768 75407 63 1313 22403 2772 56079 64 968 23371 2418 62998 65 717 24088 2239 69795 66 651 24739 1692 91677 67 886 25625 2846 82857 68 912 26537 2911 77141 69 898 27435 2647 72167 70 759 28194 2569 80097 71 722 28916 2512 86883 72 753 29669 2701 79953 73 793 30462 2737 79166 74 725 31187 2139 82016 75 802 31989 2845 80879 (1) (2) (4) (5) 1-43 7932 29841 65520 RURAL 44-46 1157 4123 56623 VILLEGAS, WELDON, ROUTLEDGE 1-46 9089 33964 64388 RURAL 47-50 3236 11562 60079 EAVESVILLE 51-75 19664 57007 71034 LOCKHART CITY 1-75 31989 102533 68037 STEPHENS COUNTY EXERCISES: 1. Why is the following procedure not suitable for drawing a simple random sample of addresses in Lockhart City? a. Randomly select a district between 51 and 75. b. Randomly select a house from those in the chosen district. c. Reject both district and house selection if the house is already in the sample. d. Repeat a - c until the desired sample size is achieved. 2. Notice that no district in Lockhart City has more than 1313 houses. Prove that the following procedure does produce a simple random sample of houses in Lockhart City: a. Randomly select a district between 51 and 75. b. Randomly select a random number (the potential house selection) between 1 and 1313. c. Reject both district and house selection if the house number exceeds the number of houses in the district or if the house is already in the sample. d. Repeat a-c until the desired sample size is achieved. 3. Let N=6, n=3. For purposes of studying sampling distributions, we assume that all population values are known: y1 = 98 y2 = 102 y3 = 154 y4 = 133 y5 = 190 y6 = 175 We are interested in mu, the population mean. What is the value of mu? For each of the following sampling plans, find (i) E[ybar]; (ii) V[ybar]; (iii) Bias(ybar); (iv) MSE(ybar). Which sampling plan do you think is the best? Why? Plan 1. 8 possible samples sample Units Prob. 1 (1,3,5) 1/8 2 (1,3,6) 1/8 3 (1,4,5) 1/8 4 (1,4,6) 1/8 5 (2,3,5) 1/8 6 (2,3,6) 1/8 7 (2,4,5) 1/8 8 (2,4,6) 1/8 Plan 2. Three possible samples sample Units Prob 1 (1,4,6) 1/4 2 (2,3,6) 1/2 3 (1,3,5) 1/4 4. Select a simple random sample of size 10 from Lockhart City. Use any random number table. Hand in a list of the random numbers you selected and the addresses to which they correspond. Describe exactly how you converted a random number to an address.