ACMS 10140 Section 02 Elements of Statistics October 28, 2010 Midterm Examination II Answers Name DO NOT remove this answer page. DO turn in the entire exam. Make sure that you have all ten (10) pages of the examination this page and 9 pages of the exam. Put your name on this sheet and put your name or initials on each page, in case pages accidentally become separated. You have 75 minutes to complete the examination. You may leave earlier if you are finished. Calculators are allowed, however, please turn off and put away all cell phones, laptops, and other electronic devices (except calculators). Remember that you are taking this examination under the University Honor Code. Record Answers to Multiple Choice Questions Here. Indicate each answer by placing an X in the appropriate circle or by filling it in. a b c d e a b c d e 1. 4 7. 4 2. 4 8. 4 3. 4 9. 4 4. 4 10. 4 5. 4 11. 4 6. 4 12. 4 Please do not write in the table below. Question(s) Maximum Score 1 12 48 13 12 14 14 15 13 16 13 Totals 100
Multiple Choice Section 4 points each Choose the best response to each question. Indicate your answer by placing an X in the appropriate circle on the cover page. Do NOT circle the answer. No partial credit is possible on this part of the examination. 1. In a class of 40 students, 22 are women, 10 are earning an A, and 7 are women that are earning an A. If a student is randomly selected from the class, find the probability that the student is a women or is earning an A. a. 0.25 b. 0.8 c. 0.975 d. 0.625 e. 0.55 Solution. Let W be the event that the student chosen is a woman, and let A be the event that the student chosen is earning an A. By the Additive Rule, P (A W ) = P (A) + p(w ) P (A W ) = 1 4 + 22 40 7 40 = 25 40 = 0.625 2. A one-week study revealed that 60% of the customers of a warehouse store are women and that 30% of women customers spend at least $250 on a single visit to the store. Find the probability that a randomly chosen customer will be a woman who spends at least $250. a. 0.90 b. 0.18 c. 0.36 d. 0.50 e. None of the above 3. Which of the following best characterizes a random sample? 1
a. All members of the sample voluntarily provide all information requested in a timely manner. b. No member of the sample is allowed to communicate with any other member of the sample. c. Methods are employed to guarantee that all subgroups of the population are represented in the sample. d. Every subset of the population with n elements, where n is the sample size, has an equal probability of being chosen. e. No individual in the population is excluded from the sample. 2
Questions 4 7 are about an experiment in which a fair die is rolled twice. We show the sample space below, with the result on the first shown in white and on the second in gray. 4. Let A denote the event that the sum of the numbers showing on the top faces is 3, and let B denote the event that one of the dice shows 1. Which of the following is true of A and B? a. A B = A. b. A B = A. c. A and B are mutually exclusive. d. All the previous statements are true of A and B. e. None of the previous statements is true of A and B. 3
5. Let C denote the event that the first die shows 1, and let D denote the event that one of the dice shows 2. Which of the following is true of C and D? a. C and D are mutually exclusive. b. C and D are independent. c. C D = C. d. C D = D. e. None of the previous statements is true of A and B. 6. Let X denote the sum of the numbers showing on the top faces of the two dice. Which of the the following statements is true of the random variable X? a. X is a discrete random variable with infinitely many values. b. X is a discrete random variable with finitely many values. c. X is a continuous random variable. d. All the previous statements are true of X. e. None of the previous statements is true of X. 7. Let E denote the event that the first die shows 2, and let F denote the event that the sum of the numbers showing on the top faces of the two dice is even. Which of the following is true of the events E and F? a. E F is the event that the second die shows an odd number. b. E F is the event that the second die shows 2. c. E and F are independent events. d. E and F are mutually exclusive events. e. None of the previous statements is true of E and F. 4
Questions 8 and 9 are about an analysis of four hundred accidents that occurred on a Saturday night. The number of vehicles involved and whether alcohol played a role in the accident were recorded. The results are shown in the table below. Number of Vehicles Involved Totals Did Alcohol Play a Role? 1 2 3 or more Totals Yes 53 100 17 170 No 24 176 30 230 Totals 77 276 47 400 8. Suppose that one of the 400 accidents is chosen at random. What is the probability that the accident involved more than a single vehicle? 81 21 11 323 276 a. b. c. d. e. 400 400 80 400 400 9. Given that an accident involved multiple vehicles, what is the probability that it involved alcohol? 17 117 17 117 276 a. b. c. d. e. 47 323 400 400 400 10. Which of the following is a valid probability distribution for a random variable? a. x 0 1 2 3 4 p(x) 0.1 0.1 0.2 0.1 0.2 b. x 0 1 2 3 4 p(x) 0.1 0.1 0.2 1 0.2 c. x 0 1 2 3 4 p(x) 0.1 0.2 0.3 0.3 0.1 d. x 0 1 2 3 4 p(x) -0.1 0.1 0.2 0.3 0.5 e. x 0 1 2 3 4 p(x) 0.3 0.2 0.2 0.2 0.2 5
Questions 11 and 12 are about the random variable X, which is equal to the number of lemon meringue pies sold in a day by a local bakery, The Baker s Circle. The bakery has determined a probability distribution for X, which is given by the table below. x 1 2 3 4 5 6 7 8 9 10 p(x) 0.001 0.021 0.135 0.15 0.203 0.211 0.119 0.11 0.025 0.025 11. What is the probability that the bakery will sell 7 or more lemon meringue pies in a day? a. 0.16 b. 0.119 c. 0.721 d. 0.84 e. 0.279 12. What are the expected value E(X) and the standard deviation σ(x) of X? a. E(X) = 1.825, and σ(x) = 5.517. b. E(X) = 5.517, and σ(x) = 3.330. c. E(X) = 5.5, and σ(x) = 2.0. d. E(X) = 5.517, and σ(x) = 1.825. e. None of the above 6
Free Response Section Partial credit is possible. Show your work. Unsupported answers might not receive full credit. 13. (12 points) A life insurance company sells a term life insurance policy to a 21-year-old male that pays $100,000 if the insured dies within the next 5 years. The probability that a randomly chosen male will die each year can be found in mortality tables. The company collects a premium of $250 each year as payment for the insurance. The amount X that the company earns on this policy is $250 for each year the policy is in force, less the $100,000 that it must pay if the insured dies. Here is the distribution of X. Age at death 21 22 23 24 25 26 X -$99,750 -$99,500 -$99,250 -$99,000 -$98,750 $1250 Probability 0.00183 0.00186 0.00189 0.00191 0.00193? a. Fill in the missing value in the table. 1 (0.00183 + 0.00186 = 0.00189 + 0.00191 + 0.001930 = 0.99058 b. Find the expected value E(X) of X. Using a calculator, we find E(X) = 303.3525. c. Find the standard deviation σ(x) of X. Using a calculator we find σ(x) = 9708. d. How many standard deviations from E(X) is the value 0? (The standard deviation is often used as a measure of the risk associated with a financial instrument.) 303.3525/9708 = 0.0312 7
14. (14 points) In his novel Bomber, Len Deighton argues that a World War II pilot had a 2% chance of being shot down on each mission. So in 50 missions the pilot is mathematically certain to be shot down, because 50 2% = 100%. Is this a good argument? (Hint: Can you model this situation as a coin tossing experiment? It is allright to make the simplifying assumption that the outcomes of the missions are independent.) It is not a good argument. Think of a biased coin with 2% probability of turning up heads. We identify tossing the coin with flying a mission. The coin turning up heads represents the airplane being shot down. It is certainly possible to toss the coin 50 times and have it come up tails all 50 times. Indeed the probability of this event is 0.98 50 0.36. 8
15. (13 points) Urn A contains 3 red marbles and 2 green marbles, and Urn B contains 2 red marbles and 3 green marbles, A fair coin is tossed. If it turns up heads, a marble is drawn from Urn A, and if it turns up tails, a marble is drawn from Urn B, a. What is the probability that the coin turns up heads? The probability that the coin turns up heads is 1/2. b. Find the probability that a red marble is drawn. Let R denote the event that a red marble is drawn, and let H denote the event that the coin turns up heads. P (R) = P (R H)P (H) + P (R H c )P (H c ) = 3 5 1 2 + 2 5 1 2 = 1 2 c. Find the conditional probability that the coin turns up heads, given that a red marble is drawn. P (H R) = P (H R P (R) P (R H)P (H) = P (R) = = 3 5 3 1 5 2 1 2 9
16. (13 points) An airline has requests for standby flights at half the usual one-way air fare. Past experience has shown that these passengers have about a 1 in 5 chance of getting on the standby flight. When they fail to get on a flight as a standby, the only other choice is to fly first class on the next flight out. Suppose the usual one-way air fare to a certain city is $148 and the cost of flying first class is $480. Should a passenger who wishes to fly to this city opt to fly as a standby? Let s calculate the expected cost of the flight for someone who tries to fly standby. This person pays $74 with probability 1/5, and with probability 4/5 the person pays $480. Thus the person expects to pay 1 5 $74 + 4 $480 = $398.80 5 It doesn t appear to be a good deal. 10