1989_Final_Exam_Soln

# 1989_Final_Exam_Soln - CEE 304 UNCERTAINTY ANALYSIS IN...

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CEE 304 - UNCERTAINTY ANALYSIS IN ENGlhlEERlNG Final Exam Saturday, December 15,1989 This exam is open notes and open-book. The exam lasts 150 minutes and there are 150 points. SHOW WORK! 1. (15 points) An engineer believes that a random variable X has a cumulative distribution function FX(x) = 1 - exp( - kx3 } for x 2 0 Given a random sample of n independent observations ( Xi, X2, ... , Xn}, what is the maximum likelihood estimator of k? Why do statisticians like maximum likelihood estimators? 2. (10 points) In a region, annual maximum rainfall depths are modeled by a lognormal distribution with a coefficient of variation of 0.20 and a median value of 3 inches. What is the depth that would be exceeded with a probability of only 2%? he CV is o,/p, where X is the real-space random variable.) 3. (10 points) Consider the seven observations 127 106 115 60 87 78 114 which are thought to be drawn from a normal distribution. By hand, or using the attached probability paper, construct a probability plot of the data as a visual test of normality. Provide a table of the two coordinates of the points you plotted. How should one construct a probability plot to examine whether a sample was drawn from an exponential distribution? What would one plot against what? 4. (15 points) Global climate change is a big issue now. A meterologist is trying to track the concentration of C02 in the atmosphere. Last week, 14 measurements of atmospheric C02 were made at his station; the observations have a sample average of 334.12 ppm with a sample variance of 0.14 (ppm)2. What is a 95% confidence interval for the true atmospheric concentration of C02 that week? (Assume that the variations are normally distributed about the true weekly mean concentration.) If the meterologist repeats his analysis every week, what is the probability that every one of the confidence interval he constructs over the next 52 week contains the true atmospheric C02 concentration for its week? What is the probability the particular confidence interval you just constructed with the given data contains the true atmospheric C02 concentration for last week?

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CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING Final Exam Saturday, December 15,1989 5. (20 points) Two fraternities got into an argument about which is better athletically. They decided to resolve the issue by randomly pairing off members of the two houses (denoted r and A) for 50-yard races. They conducted 25 races and saw who won.
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