Classroom Voting Questions: Statistics
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1 Classroom Voting Questions: Statistics Random Variables 1. Draw the following dart board: A dart board is constructed from three concentric circles with radii 1 inch, 2 inches, and 3 inches, respectively. If a dart lands in the innermost circle, the player receives 4 points. If the dart lands between the innermost circle and the middle circle, the player receives 2 points. If the dart lands between the middle circle and the outermost circle, the player receives 1 point. Assume that the probability of a dart landing in any particular region is proportional to the area of that region. Define the random variable X to be the sum of the player s score on two successive throws. Then X is what type of random variable? Answer: (a). The possible values for X are 2, 3, 4, 5, 6, and 8 a countable number of values. STT CC HZ MA207 F09: 75/25 time 1:30 CC KC MA207 F09: 82/18 time 2:00 AS DH MA3321 Su12: 87/13 time 2:00 AS DH MA F12: 89/11 time 1:50 AS DH MA F12: 95/5 time 2:00 AS DH S13: 100/0 time 3:00 AS DH S14: 97/3 time 2:20, AS DH F14: 100/0 time 3:00, AS DH S15: 96/4 time 2:40, AS DH F15: 97/3 time 2:30, CC KC MA315 F15: 93/7 AS DH F17: 94/6 time 2:30 2. Draw the following dart board: A dart board is constructed from three concentric circles with radii 1 inch, 2 inches, and 3 inches, respectively. If a dart lands in the innermost circle, the player receives 4 points. If the dart lands between the innermost circle and the middle circle, the player receives 2 points. If the dart lands between the 1
2 middle circle and the outermost circle, the player receives 1 point. Assume that the probability of a dart landing in any particular region is proportional to the area of that region. Suppose that a player s score on a single dart throw is defined to be the distance between the dart and the center of the board. Define the random variable X to be the sum of the player s score on two successive throws. Then X is what type of random variable? Answer: (b). The possible values for X are any number between 0 and 6 an uncountable number of values. STT CC HZ MA207 F09: 0/100 time 1:20 AS DH MA3321 Su12: 0/100 time 1:30 AS DH MA F12: 0/100 time 1:30 AS DH MA F12: 19/81 time 1:40 AS DH S13: 12/88 time 1:00 AS DH S14: 5/95 time 1:30, AS DH F14: 8/92 time 1:40, AS DH S15: 4/96 time 1:50, AS DH F15: 0/100 time 2:10, CC KC MA315 F15: 6/94 AS DH F17: 22/78 time 2:10 3. A radioactive mass emits particles at an average rate of 15 particles per minute. Define therandomvariablex tobethenumberofparticlesemittedina10-minutetimeframe. Then X is what type of random variable? Answer: (a). The possibles values for X are all integers between 0 and the number of particles in the mass. Even if there were an infinite number of particles in the mass, this would still be a discrete random variable, since the possible values are countable (1, 2, 3,...). STT
3 CC HZ MA207 F09: 88/12 time 0:50 CC KC MA207 F09: 44/56 time 1:40 AS DH MA3321 Su12: 87/13 time 0:50 AS DH MA F12: 78/22 time 1:00 AS DH MA F12: 81/19 time 1:30 AS DH S13: 81/19 time 2:00 AS DH S14: 38/62 time 2:00, AS DH S15: 85/15 time 2:50, AS DH F15: 85/15 time 2:30, 4. A radioactive mass emits particles at an average rate of 15 particles per minute. A particle is emitted at noon today. Define the random variable X to be the time elapsed between noon and the next emission. Then X is what type of random variable? Answer: (b) X can take on any positive value, which is an uncountable set of values. STT CC HZ MA207 F09: 6/94 time 0:30 AS DH MA3321 Su12: 7/93 time 0:40 AS DH S13: 19/81 time 1:50 AS DH S14: 10/90 time 1:00, AS DH S15: 11/89 time 2:00, AS DH F15: 4/96 time 1:50, 5. A randomly-selected kindergarten class in a large city will get to have a party on Friday of next week. At one point in the party, each child in the class will receive half of a candy bar. Define the random variable X to be the number of candy bars given out in the class next Friday. Then X is what type of random variable? Answer: (a). This questions draws out the fact the discrete-versus-continuous distinction is not an issue of decimals-versus-not-decimals but is more subtle. (a) There are gaps on the real number line between the possible values for X, so X is discrete. (b) The possible values of X do not constitute an interval on the real number line, so X is not continuous. 3
4 STT AS DH F16: 100/0 time 1:40 AS DH F17: 94/6 time 2:30 6. Consider the continuous random variable X = the weight in pounds of a randomly selected newborn baby born in the United States during Let f be the probability density function for X. It is probably safe to say that P(X < 0) = 0 and P(X < 20) = 1. Which of the following is not a justifiable conclusion about f given this information? (a) No portion of the graph of f can lie below the x-axis. (b) The area under the entire graph of f equals 1. (c) The area under the graph of f between x = 0 and x = 20 is 1. (d) The nonzero portion of the graph of f lies entirely between x = 0 and x = 19. Answer: (d). Since X is a continuous random variable, it can take on values between 19 and 20. There may be some nonzero portion of the graph of f that lies between x = 19 and x = 20. (In fact, the Guinness Book of World Records lists as the heaviest baby born to a healthy mother a boy weighing 22 pounds, 8 ounces, born in Aversa, Italy, in September 1955.) Draw a graph for f and illustrate these three properties. STT AS DH MA3321 Su12: 0/0/8/92 time 3:00 7. Two standard, six-sided dice are rolled. What is the probability that the sum of the dice is 6? (a) 1 6 (b) 5 6 (c) 1 12 (d) 5 12 (e) 1 36 (f) 5 36 Answer: (f). STT DH 20 4
5 AS DH MA F12: 18/0/6/12/65/0 time 2:30 AS DH MA F12: 6/11/0/11/72/ time 3:00 AS DH S13: 11/0/5/0/16/68 time 2:00 AS DH S14: 10/3/20/0/3/63 time 2:20, AS DH S14: 25/0/8/0/0/67 time 3:10, AS DH F14: 9/3/9/12/0/67 time 3:10, AS DH S15: 0/0/0/23/5/73 time 2:20, AS DH F15: 3/0/6/3/0/87 time 2:30, CC KC MA315 F15: 0/0/12/0/0/81 AS DH F17: 3/0/51/6/0/40 time 3:30 8. Two standard, six-sided dice are rolled. What is the most probable sum? (a) 2 (b) 6 (c) 7 (d) 12 Answer: (c). STT DH 30 AS DH MA F12: 0/0/88/12 time 2:00 AS DH MA F12: 0/0/100/0 time 2:20 AS DH S13: 0/11/63/26 time 1:00 AS DH S14: 0/23/77/0 time 2:30, AS DH F14: 0/17/83/0 time 3:10, AS DH F15: 0/16/84/0 time 2:00, CC KC MA315 F15: 0/7/93/0 AS DH F17: 0/26/71/3 time 3:30 9. Consider rolling a standard, six-sided die. Let A be the event that the number rolled is even. Let B be the event that the number rolled is a multiple of 3. The event (notb) consists of (a) 1,3,5 (b) 1,2,4,5 (c) 2,4,6 (d) 1,3,5 5
6 Answer: (b). STT DH 40 AS DH MA F12: 6/88/6/0 time 1:30 AS DH MA F12: 0/71/18/12 time 1:40 AS DH S13: 0/86/14/0 time 3:00 AS DH S14: 0/100/0/0 time 2:20, AS DH F14: 0/97/3/0 time 2:50, AS DH S15: 0/95/5/0 time 2:30, AS DH F15: 0/84/16/0 time 1:40, CC KC MA315 F15: 0/93/7/0 AS DH F17: 0/83/17/0 time 2: Consider rolling a standard, six-sided die. Let A be the event that the number rolled is even. Let B be the event that the number rolled is a multiple of 3. The event (A and B) consists of (a) 2,3,4,6 (b) 2,3,4,6,6 (c) 6 Answer: (c). STT DH 50 AS DH MA F12: 71/0/29 time 1:00 AS DH MA F12: 17/22/61 time 1:00 AS DH S13: 33/0/67 time 2:00 AS DH S14: 36/4/60 time 1:50, AS DH S14: 4/9/87 time 2:00, AS DH F14: 0/26/74 time 2:30, AS DH F15: 16/0/84 time 1:20, CC KC MA315 F15: 11/0/89 time 1: Consider rolling a standard, six-sided die. Let A be the event that the number rolled is even. Let B be the event that the number rolled is a multiple of 3. The event (AorB) consists of (a) 2,3,4,6 (b) 2,3,4,6,6 6
7 (c) 6 Answer: (a). STT DH 60 AS DH MA F12: 100/0/0 time 0:45 AS DH MA F12: 83/17/0 time 0:50 AS DH S13: 100/0/0 time 1:00 AS DH S14: 96/0/4 time 0:50, AS DH F14: 97/3/0 time 2:10, AS DH F14: 83/17/0 time 2:10, AS DH S15: 85/0/15 time 1:20, AS DH F15: 100/0/0 time 0:40, CC KC MA315 F15: 100/0/0 AS DH F17: 100/0/0 time 1: A standard, six-sided die is rolled. What is the probability of rolling an even number or a number divisible by 3? (a) 2 3 (b) 5 6 (c) 4 (d) 5 Answer: (a). STT DH 70 AS DH MA F12: 65/35/0/0 time 1:30 AS DH MA F12: 100/0/0/0 time 2:00 AS DH S13: 81/19/0/0 time 3:10 AS DH S14: 80/20/0/0 time 2:30, AS DH F14: 94/6/0/0 time 3:10, AS DH F15: 88/12/0/0 time 3:00, CC KC MA315 F15: 96/4/0/0 time 1:10 AS DH F17: 86/8/6/0 time 2: A card is drawn at random from a standard deck of 52 playing cards. What is the probability that the card is a red card or a jack? (a) 28 7
8 (b) 30 (c) 7 13 (d) Answer: (c). STT DH 80 AS DH MA F12: 0/0/71/29 time 1:40 AS DH MA F12: 0/0/94/6 time 2:30 AS DH S13: 0/0/71/29 time 2:40 AS DH F14: 0/0/85/15 time 3:50, AS DH S15: 0/0/83/17 time 2:50, AS DH F15: 0/0/89/11 time 2:50, AS DH F15: 6/11/44/39 time 3:30, CC KC MA315 F15: 0/0/82/18 time 1:30 8
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