e) For the Poisson, gives probability of successes in an interval in which the average number of successes is . For the exponential, , where the mean time to a success is , implies that . Thus (i) for the Poisson with and (ii) for the exponential . Note that251y0431d 5/03/04 ECO251 QBA1 THIRD HOUR EXAMApr 20, 2004 Name: ___KEY______________ Student Number: _____________________ Class Time (Circle) 1pm 2pmPart I: 10 points.IQs are supposedly Normally distributed with a population mean of 100 and a standard deviation of 16 16,100~ Nx. Find the following. Make diagrams! Add a vertical line at zero to the diagrams below.1. . 161004.784.78 zPxP 35.1zP 9115.5.4115.0035.1 zPzPColors in diagram are reversed.2. 35.135.1161006.121161004.786.1214.78zPzPxP 8230.4115.235.102 zP3. 161002.841610052.472.8452.47zPxP 1606.3389.4995.099.0028.399.028.3zPzPzP(The value for 3.28 comes from the lower part of the Normal table.)43210-1-2-3-40.40.30.20.10.0xf43210-1-2-3-40.40.30.20.10.0xf43210-1-2-3-40.40.30.20.10.0xf251y0431d 4/20/044. 6.12104.111 xP161006.1211610004.111zP 35.169.0 zP 1566.2459.4155.69.0035.10 zPzP43210-1-2-3- 40.40.30.20.10.0xf5. To get into Mensa you must be in the top 2% 0f the population. What IQ do you need? (Hint: find 02.x) the 98th percentile. Make a diagram: Show a Normal curve with a mean at zero. Label the area below zero ‘50%.’ By definition 02.z is the point with a probability of 2% above it, which means it must have 98% below it and 98% - 50% = 48% between it and zero. We can try to find a point of the Normal table with .4800.002. zzP The closest we can come to .4800 is 4798.05.20 zP or .4803.06.20 zP Either is acceptable, but a good compromise might be 054.202.z. So, ifyou use the formula, ,zx your answer could be 8.1321605.210002.x, 16054.210002.x 9.132 or 0.1331606.210002.x. Check: 161009.1329.132 zPxP 06.2zP 02.0197.4803.5.06.200 zPzP2251y0431 4/20/04Part II: (20+ points) Do all the following: All questions are 2 points each except as marked. Exam is normed on 50 points including take-home. (Showing your work can give partial credit on some problems! In open-ended questions it is expected. Please indicate clearly what sections of the problem you are answering and what formulas you are using. Neatness counts!)1. Thirty-six of the staff of 80 teachers at a local intermediate school are certified in Cardio-Pulmonary Resuscitation (CPR). In 180 days of school, about how many days can we expect that the teacher on bus duty will likely be certified in CPR?a) 5 daysb) 45 daysc) 65 daysd) *81 daysExplanation: If you want to get technical this is a Binomial distribution with 45.8036pand .180n So .8118045. np2. What type of probability distribution will most likely be used to analyze the number of cars with defective radios in the following problem?From an inventory of 48 new cars being shipped to local dealerships, corporate reports indicate that 12 have defective radios installed. The sales manager of one dealership wants to predict the probability that out of the 8 new cars it just received, when each is tested, no more than 2 of the cars have defective radios.a) binomial distribution.b) Poisson distribution.c) *hypergeometric distribution.d) none of the above.Explanation: This is hypergeometric and not binomial because there are only 12 cars with defective radios. On the first pick the probability of getting a defective radio is 4812p , but on the second pick 4711p or 4712. Without a constant value for p, we cannot have a binomial distribution.3. A company has 125 personal computers. The probability that any one of them will require repair on a given day is 0.025. To find the probability that exactly 20 of the computers will require repairon a given day, one will use what type of probability distribution?a) *binomial distribution.b) Poisson distribution.c) hypergeometric distribution.d) none of the above.Explanation: This is binomial and not hypergeometric because the probability that any pc needs repair is a constant .025. Though ‘binomial’ is the correct answer , the binomial distribution can be approximated by the Poisson because .5005000025.125pn4. The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8. You have 2 such alarms in your home and they operate independently. The probability that both sound an alarm in the presence of smoke is __.64____.Explanation: If you want to get technical this is a Binomial distribution with 8.p and.2n So xnxnxqpCxP and .64.2.8.20222CP. Alternatively, let 1A be‘First alarm works’ and 2A be ‘Second alarm works.’ If the events are independent, 21AAP .64.8.221 APAP3251y0431 4/20/045. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. The probability that there will be at least 1 power outage in a year is _0.9975___.Explanation: 99752.00248.1011 PxP.6. An Undergraduate Study Committee of 6 members at a major university is to be formed from a pool of faculty of 18 men and 6 women. If the committee members are chosen randomly, what is the probability that all of the members will be men? (3)Solution: This is hypergeometric because we are taking a sample of 6 that is more than 5% of a population of 18 + 6 = 24. The formula is NnMxMNxnCCCxP, which gives theprobability of 6x successes in a sample of 6n taken from a population of24N in which there are 18M successes. 1379.192021222324131415161718!6!18!24!6!12!18624618660CCCP, but you could also argue that the probability of a male on the first try is 2418, on the
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