1993_Final_Exam_Soln - CEE 304 UNCERTAINTY ANALYSIS IN...

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CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING FINAL EXAM Wednesday, December 8,1993 Exam is open notes and open-book. The exam lasts 150 minutes and there are 150 poin is. SHOW WORK! 1. (10 points) An engineer must use the cumulative distribution function FX(x) = (da f o r O ~ x 1 1 to describe a particular phenomona. Given a random sample of n independent observations {XI, X2, ... , Xn), what is the maximum likelihood estimator of the unknown shape parameter a? 2. (10 points) Please explain in 25 words or less why method of moment estimators make sense. 3. (10 points) An engineer is concerned with the loading on a bridge over its lifetime. Suppose that major dynamic shock loadings can be described by an exponential distribution with a mean of 4,000. Assume that the bridge would need to absorb 1000 such shocks over its operational lifetime. Using the appropriate asymptotic distribution, what is the mean and variance of the largest of those 1000 shocks? (Which is the load for which bridge should be designed.) 4. (5 points) Of course, large shocks really arrive randomly in time. If on average the bridge receives 30 shocks per year, and they occur independently over time, what is the mean and variance of the time until 1000 shocks occur? 5. (10 points) Rainfall is often well described by a lognormal distribution. Assume that the mean and standard deviation of maximum annual 24-hour rainfall depths are 3.1 inches and 1.8 inches. What is the rainfall depth exceeded with a 1% probability? 6. (10 points) Consider 8 individual weights submitted by class members 117 150 162 145 170 125 175 160 Might these be described by a normal distribution? Use the attached probability paper to construct a visual test of normality. Indicate the numerical values of the quantities you are plotting. What is your assessment?
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GEE 304 - UNCERTAINTY ANALYSlS IN ENGINEERING FINAL EXAM Wednesday, December 8,1993 7. (10 points) Twelve women provided their heights; they had a sample mean of 65.15" and a standard deviation of 2.44". Let us consider these women to be a random sample of Cornell women engineering students. What is a 90% confidence interval for the true mean height of Cornell women engineering students? (~ssume that heights are normally distributed.) What is the probability that the interval you just constructed actually contains the mean height of Cornell women engineering students, assuming the 12 who responsed are a representative random sample? 8. (10 points) Now the big question. Are men taller than women, as is often believed? Look at the data that was turned in on heights: N - M MEDIAN STDEV Women 1 2 65.1 46 64.500 2.441 Men 3 1 71.258 71 .OOO 2.889 Use a pooled t test because Nwomen only equals 12.
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