Practice - Simulations with a Random Digit Table Answers 1. A club contains 33 students and 10 faculty members. The students are: Aisen DuFour Kittaka May Rokop Thyen Albrecht Dwivedi Kuhn MacDonald Sommer Wang Bhagava Gartner Lee Neukam Stuy Yerant Bonini Hollensburg Lipp Patel Terrell Chakra Huang Lundberg Pham Thomas Ding Joseph Marshall Ranaweera Thompson The faculty members are: Beck Burse Laumeyer Nesbitt Twining Brown Diamente Mitchener Sohalski Wernke The club can send 4 students and 2 faculty members to a convention and decides to choose those who will go by random selection. a. Use the random digit table to choose a sample of 4 students. Start on line 109. List the numbers and the names below. Answers may vary based on how participants are numbered. Numbering alphabetically, 01 33, first four students are 15, 23, 16, 19 Lee, Pham, Lipp, and May. b. Use the random digit table to choose a sample of 2 faculty members. Start on line 112. List the numbers and the names below Numbering alphabetically, 0-9, first two numbers are 5 and 9 Mitchener and Wernke 2. You are a marketing executive for a clothing company. Choose a SRS of 10 of the 440 retail outlets in New York that sell your company's products. Describe how you would label the retail outlets and select your sample using the random digit table starting on line 102. Label all retail outlets 001-440 and select three digit numbers, ignoring all numbers over 440. First ten numbers are 099, 400, 019, 272, 253, 045, 097, 095, 328, 226. 3. Your school will send a delegation of 35 seniors to a student life convention. 200 girls and 150 boys are eligible to be chosen. If a sample of 20 girls and a separate sample of 15 boys are each selected randomly, it gives each senior the same chance to be chosen to attend the convention. Beginning at line 105 in the random digits table below, select the first four senior girls to be in the sample. Explain your procedures clearly. Number the girls 001 200, ignore all values over 200. First four numbers are 77, 172, 131, 074. 4. Five boxes, each containing 24 cartons of strawberries, are delivered in a shipment to a grocery store. The produce manager always selects a few cartons randomly to inspect. He knows better than to just look at some of the cartons on the top or only in one box, because sometimes the rotten ones are on the bottom. Today he wishes to select a total of 6 cartons to inspect. He has the boxes arranged in order and has a set way to count the cartons inside each box. Explain the process used to make the random selection using a random digit table starting at line 119. Since he has 120 cartons total, he will assign the numbers 001 to 120 to represent the cartons in order. He will use line 119 on a random number table to make the selections. He will need to look at 3-digit numbers every time. He will not want to inspect the same carton twice, so he will ignore any repeats. And, any numbers above 120 will not apply in this case, so he will ignore numbers 121-999 and 000. He will stop once six numbers are selected. He will then find the cartons that the numbers represent and inspect those cartons.
5. Five of the employees at the Stellar Boutique are going to be selected to go to training in Las Vegas for four days. Everyone wants to go of course, so the owner has decided to make the selection randomly. She has decided to send two managers and three sales representatives. The employees' names are listed in the table below. Managers Angela Barbara Elise Gigi Malena Rosie Tammie Veronica Sales Reps Sales Reps Sales Reps Alfie Irma Ray Betty Joe Sandy Carrie Katarina Shirley Cathy Lynn Suzi Darcy Marcie Tawny Fred Nancy Wendy Heidu Orville Zoe Explain the process she can follow to use a random digit table to select the employees who will get to go to the training. Select the managers first, then select the sales representatives. Usie the random digit table starting on line 116. Number managers alphabetically 1 8, starting on line 116, first two are 1 and 4 Angela and Gigi. Numbering the employees 01 21, and continuing on line 116, three sales reps are 03, 10, and 06 Carrie, Katarina, and Fred. 6. The manager at Big-N-Yummy-Burger wishes to know his employees' opinions regarding the work environment. He has 56 employees and plans to select 12 employees at random to complete a survey. Explain the process he can follow to use the random digit table, starting at line 110, to select an SRS of size 12. Which are the first five employees numbers selected? number employees 01 56 and select two digit numbers, starting on line 110: 38, 44, 18, 33, and 46. 7. Use the random digit table, starting on line 112, to select an SRS of five of the fifty U.S. States. Explain your process thoroughly and report the five states that you chose. Repeat this a second time, but begin on a different line on the random digit table. Compare your lists to another classmate's lists. Did you end up with any of the same states in your samples? Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming first 5 states using line 112 and two digit numbers 01 50: 04, 47, 11, 19, 35. Numbering alphabetically, this is Arkansas, Hawaii, Maine, Ohio, and Washington.
8. Washington High School has had some recent problems with students using steroids. The district decides that it will randomly test student athletes for steroids and other drugs. The boy's hockey team is to be tested. There are 13 players on the varsity team and 21 players on the junior varsity team. Use a table of random digits starting at line 122, to choose a stratified random sample of 3 varsity players and 5 junior varsity players to be tested. Remember to clearly describe your process. labeling both groups alphabetically, varsity 01 13, junior varsity 01 21, varsity players selected are 13, 05, and 09 West, Gustaf, Radamacher. JV players selected are 08, 13, 15, 07, 10 James, Peterson, J Randal, Hansen, Linder. 9. Suppose you wish to roll two dice a total of 5 times and keep track of the totals. You don't have any dice, but you do have access to the line of random digits below. Explain how you could simulate the rolls of two dice using the random digit table. Perform the simulation and record your results. The easiest way to roll 2 dice and record the sums is to use 2 single digit numbers 1 6, and ignore 7 9. If you actually represent the sums 02 12, the probabilities will not be represented, since there is a higher probability of getting a sum of 7 than 12. 10. At the start of this season, Major League Baseball fans were asked which American League Central team would be most likely to win the division this year. The table below gives the results of the poll. Most Likely to Win AL Central Team Chicago Cleveland Detroit Kansas City Minnesota Probability 0.14 0.23 0.33 0.02 0.28 Using the random digit table, starting on line 117, simulate the results when asking 10 fans who they think will win the AL Central. Using the probabilities to assign digits 00-99, Chicago: 00 13, Cleveland: 14 36; Detroit: 37 69; KC: 70 71; Minnesota: 72 99. First ten people are 38, 16, 79, 85, 32, 62, 18, 37, 06, 32. Results of these ten give the following probabilities: Chicago: 10%; Cleveland: 40%; Detroit: 30%; KC: 0%; Minnesota: 20% 11. Every person is born on a different day of the month. Some people are born on the 1st and some people are born as late as the 31st. How many people must you go through until you find two that were born on the same day of the month? Simulate this one time using the random digit table starting at line 101. (Ignore the fact that people are not equally likely to be born on all days. It is more likely you were born on the 17th than the 31st since all months have a 17th but not all months have a 31st.) Using line 101 days selected are: 19, 22, 05, 13, 25, 31 28, 19 - so must go through 8 people.
12. Suppose that 80% of a school s student population is in favor of eliminating final exams. a) Explain how could you assign digits from a random digit table to simulate this situation? b) Suppose you ask 10 students if they would like to eliminate final exams. Simulate a random selection of 20 students and record how many of the 20 are in support of eliminating final exams. Use the random digit table starting at line 110. a) Assign number 00 79 to students in favor, and 80 99 to students against. b) Numbers selected are: 38, 44, 84, 87, 89, 18, 33, 82, 46, 97, 39, 36, 44, 20, 06, 76, 68, 80, 87, 08 Of these 20 numbers 13 are from 00 79, so 13 in support of eliminating final exams. 13. Suppose that students at a particular college are asked about their class rank when they were in high school. The table below shows what they said. Class Rank Top 10% Top 10% to 25% Top 25% to 50% Bottom 50% Prob. 0.2 0.4 0.3 a) What must the probability be for the bottom 50%? 10% b) Explain how you could assign digits to carry out a simulation for this situation. Let 00 19 = top 10%, 20 59 = top 10-25%; 60 89 = top 25 50%; 90 99 = bottom 10% c) Using your set up, perform a simulation using the random digit table starting at line 118. Use 20 students in your simulation and record your results. Numbers selected are: 73, 19, 03, 25, 33, 04, 47, 02, 96, 69, 84, 40, 79, 07, 85, 65, 95, 68, 63, 82 5 students in top 10%, 4 students top 10% to 25%, 9 students top 25% to 50%, 2 students bottom 50% 14. Suppose the grades for students in your math course were distributed as shown in the table. Grade A B C D or F Prob. 0.20 0.29 0.35 0.16 a) Explain how you could assign digits to simulate the grades of randomly chosen students. Let 00 19 = A; 20 49 = B; 50 84 = C; 85-99 = D or F b) Simulate the grades for 30 students using the random digit table starting at line 113. Record your results. Numbers selected are: 62, 56, 87, 02, 06, 40, 32, 50, 36, 99, 71, 08, 02, 25, 53, 11, 48, 61, 17, 76, 45, 14, 93, 29, 92, 75, 73, 06, 62, 80 c) How closely did your simulation match the actual distribution? A 8 out of 30 = 27%, Prob =.27 (actual.20) B 7 out of 30 = 23%, Prob =.23 (actual.29) C 11 out of 30 = 37%, Prob =.37 (actual.35) D or F 4 out of 30 = 13%, Probl =.13 (actual.16) 15. How many five card poker hands must you be dealt in order to get a hand with two cards that have matching values? (For example, the 7 of hearts and 7 of diamonds have matching values.) a) Explain how you will assign digits for this situation. Since all the cards are distinguishable, you need to label every card individually. Let 01-52 represent the cards in the deck, ordered by card number (A,2,3,4,5,6,7,8,9,10,K,Q,J), and then by suit: clubs, diamonds, hearts, and spades. So clubs are 01-13, diamonds are 14 26, hearts are 27-39, spades are 40-52. b) Perform the simulation one time using the random digit table starting at line 105, and state how many five-card hands it took for you to get your first hand of two cards with matching values. Note, cards that match will differ by 13 (ex. A clubs =1, A diamonds = 14, A hearts = 26, A spades = 40)
Starting at line 105: 95, 59, 29, 40, 07, 69, 97, 19, 14, 81, 60, 77, 95, 37, 91, 17, 29, 75, 93, 35, 68, 41, 73, 50, 13, 15, 52, 97, 27, 65, 85, 08, 95, 70, 67, 50, 21, 14, 74, 87 1 st 5 card hand: 29, 40, 07, 19, 14-40 is an Ace of diamonds, 14 is an Ace of spades 2 nd 5 card hand: 37, 17, 29, 35, 41 3 rd 5 card hand: 50, 13, 14, 52, 27 It only took one 5 card hand to get a hand that had two cards with mathcing values. 16. Suppose we have a class of 30 students and you are wondering what the chances are that there is at least one pair of students who have the same birthday. Assume that there are 365 days in a year. a) Explain how could you assign digits from a random digit table to simulate this situation? Assign every day of the year a different number, 001 365, in order. b) Perform this simulation one time using the random digit table starting at line 111, and record whether or not there was a match in the class of 30 students. Starting at line 11: 814, 866, 948, 760, 513, 092, 970, 041, 271, 238, 276, 493, 995, 059, 636, 888, 040, 463, 471, 197, 193, 527, 308, 984, 898, 457, 856, 256, 870, 206, 403, 250, 369, 971, 080, 225, 531, 148, 611, 776, 451, 493, 299, 275, 730, 662, 800, 381, 956, 202, 029, 387, 091, 561, 041, 776, 849, 432, 224, 709, 736, 981, 452, 631, 893, 325, 921, 445, 926, 056, 314, 248, 037, 165, 103 Using only numbers from 001-365 the 30 students are: 092, 041, 271, 238, 276, 059, 040, 197, 193, 308, 256, 206, 250, 080, 225, 148, 299, 275, 202, 029, 091, 041, 224, 325, 056, 314, 248, 037, 165, 103 There is a match in the class of 30 students.