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304_Exam2_98_Solution

304_Exam2_98_Solution - CEE 304 UNCERTAINTY ANALYSIS IN...

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CEE 304 - UNCERTAINTY ANALYSIS IN ENGINEERING Second Exam Solution November 13, 1998 1. Need to get the variances of logarithms, which are ln[1+( σ real / μ real ) 2 = 0.307 and 0.693 respectively. Then ln(W) ~ Normal[ ln(median-R) + ln(median-C), 0.307 + 0.693 = 1.006 ]. Thus 3 pts (median-W) = exp[ln(median-R) + ln(median-C)] = (median-R) * (median-C) = 0.003. Easy 2 pts CV[W] = sqrt{ exp(Var[lnW]) - 1 } = sqrt{ exp(1.006) - 1 } = 1.311 3 pts The class did very poorly on this problem. LN models are useful. 2. For Gumbel σ x 2 = 1.645/ α 2 ; α = 0.0754. μ x = u + 0.5772/ α ; u = 44.35 ; solving 3 pts 0.999 = exp[ - exp( - α (x 0.999 -u)] yields x 0.999 = 136 mph. 3 pts We are describing annual maxima (the largest in each year), and the Gumbel distribution is the asymptotic distribution (limit as n -> infinity) for the largest of n events unbounded above, as would be appropriate in a model of wind speeds. 2 pts 3. A 98% CI is x ¯ ± s t 0.01,n-1 / n = 38.7 to 44.9 for t 0.01,9 = 2.821 when n = 10. 5 pts. If they construct 6 random intervals, probability all 6 contain the mean is (0.98) 6 = 88.6% 2 pts Rachel’s interval either does or does not contain the true mean. We do not know which.
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