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Unformatted text preview: CEE 304  Uncertainty Analysis In Engineering
Second Exam November 12, 1999 You may use text, your notes and calculators. There are 50 points in total, one per minute. 1. (5 pts) Elizabeth is studying Cornell Lake Source cooling project impacts. Her concern is October
temperatures in Cayuga lake. Using measurements for 8 different years, she found that the average
temperature was 52.1°F with a standard deviation of 3.4°F. (Assume measurements are normally distributed.)
What is a 95% conﬁdence interval for the true averageOctober lake temperature? What is the probability that
the confidence interval Elizabeth computed with these eight years of data contains the true longterm mean
October lake temperature? 2. (9 pts) Sam needs to compute the average weight of football team members. Sam is lazy, so he decides to
report the average he obtains with a sample of 5 players: himself (weight 180 lbs), his friend Fred (weight 220
lbs), and three randomly selected players. Thus he computes W = (180+220+X1+X2+X3)/ 5. If the weight of
footballteam members has mean 190 lbs and standard deviation 20 lbs; what is the mean, variance, and mean
square error of Sam's estimator W of the mean weight of players? Is his estimate biased? Why or why not? Is Sam's estimator better than the sample average of 5 randomly selected players (explain)? 3. (6 pts) If football team player weights were lognormally distributed,
what is the probability that a randomly selected player weights less than 150 lbs? 4. (22 pts) A highway construction inspector needs to demonstrate that the average hardness of asphalt
exceeds 145. Generally, 5 measurements are taken from every road segment.
(a) What are the appropriate null and alternative hypotheses for a road segment?
(b) What is the rejection region for a test with a type I error of 5% ?
For a given road segment, the 5 sample values have a sample mean of 158.9 and a sample standard deviation of 6.8.
(c) What is the t value and the Pvalue for this data set? (d) If the hardness is really 145, what is the probability one will conclude that the asphalt is acceptable?
Assume c = 7, n = 5 and u, = 150. [Devore uses u' rather than u. for the mean when H, is true.) (e) What mean asphalt hardness is needed for the asphalt to be accepted 99% of the time? Assume o = 7. (f) How LARGE a sample is needed need for the type II error to be < 10% ? Assume cs = 7 and u. = 150. (g) Robin in quality engineering wants to determine whether or not the crosssectional area of steel reinforcing in a bridge equals the target of 30 cm2; she tests H0 of u = 30 cm2 versus u at 30 cm2 and obtained a Pvalue = 3%. Her boss never studied statistics and is confused. Explain to him (in S 75 words) what the 3%
should mean to him. 5. (8 pts) Consider the following distribution:
Fx(x)= 1[1—(XIc)]3 Osxsc
where:
E[X] = 0.25 c and Var[X] = 0.0375 c2
Given observations {Xi  i=1,...,n }, an engineer needs to estimate the parameter c which determines the mean
and variance of the distribution. (a) What is the method of moment estimator of c?
(b) What nonlinear equation has as its root the maximum likelihood estimator of c? [Don't solve it!] CEE 304  Uncertainty Analysis In Engineering
Second Exam November 12, 1999 1. A 95% CI is i :t s tomsm / w/ﬁ = 49.3  54.9 where toms]: 2.365 and n = 8. 4 pts.
Elizabeth's interval either does or does not contain the true mean. We do not know which. 1 pts 2. Sam uses as the estimator M = (220 + 180 + X1 + X2 + X3 )/5; EIM] = (400 + u + u + u )/5 = 194. 2 pts The bias is 194190 = 4 lbs. Estimator is biased because its expectation is not true u of 190 lbs. 2 pts
[This occurred because Sam and Joe on average do not weigh 190 lbs each.]
Var[M] =362/51=48= (7)2. MSE=Var+Biasz= 48+42=64. 3pts.
If used sample average of 5, Bias=0; Var = MSE = 02/ 5 = 80 which is LARGER than MSE of Sam's W H!
Though it is biased, Sam's estimator is more accurate, when accuracy is measured by MSE. 2 pts
3. First get the variance of logarithms, which is b2 = 1n [1+(omll um]? ] = 0.0110 = (0105):; 2 pts
Mean of logs is a = ln[um,] 0.5 b2 = 5.2415. Pr[ W < 150 ] = @[0n(150) a)/ b = 2.2 ] = 1.4%. 2+2 pts {Note that use of a normal distribution for weights yields: Pr[ W < 150 ] ==<I>[ (150  190)/ 20 =~2.0 ] = 2.2%.} 4) (a) Ho: u = 145; H,: u 2 145 (Thus must prove hardness exceeds 145.) 2 pts
(b) One—sided ttest: reject H0 if «IE (it  145)/ s > ”0.05,; = 2.132 3 pts
(c) for n =5, 7‘; = 158.9, s = 6.8 > t = 4.57 ; p is almost 0.005 = 0.5%. 4 pts.
(d) 5% (just the type I error) 2 pts
(e) To get B = 1% with n = 5 => need d = 2.2. Hence u = 145 + 2.2*7 = 160.4 4 pts (f) d = (150145)/ 7 = 0.7; Table A.13 for lsided test with a = 5%; B = 10% yields 11 2 19+1 =20. 3 pts (g) If it were true that the crosssectional area equals 30 cm2, then there is only a 3% probability that one
would obtain a sample average that deviated from 30 cm2 as much as did the observed sample average, and thus so strongly contradicted the belief in 30 cmz. It is reasonable to conclude that the area is at 30 cm2. 4 pts Most students received 0 points on part (g): the boss has not taken a statistics course, so he does not know
what a null hypothesis or a type I error is. You need to say that these terms mean. Also, it does NOT mean there is a 3% you are wrong when you reject the null hypothesis; that statement is only true it He is true! 5) Method of moments: from EIX] = 0.0.25 c one would get esﬁmator—ofc = 4 i. 3 pts.
MLE: for pdf fx(x) = (3/ c)(1  x/ c)2 , and sample {xi}, obtain likelihood function: 1 pt
L(c) = H fx(xi) ——> In L(r) = n In (3)  n ln(c) + 2 E, ln(1  xi/c) 2 pts 0 = (d/ dc) 1n L(c) =  n/ C + 2 E (1  Xi/ C)'1(Xi)(1/ CZ)
yields n = 2 Z, [ x,/ (c  x,)] ; no analytic equation exists. 2 pts
But a solution can be found between c = xmx and c = on. CEE 304  UNCERTAINTY ANALYSIS IN ENGINEERING
Second Exam November 13, 1998
You may use text, your notes and calculators. There are 50 points in total, one per minute. 1. (8‘ points) Prof. Stedinger is consulting with EPA on estimates of Cryptosporidium oocysts in water. The effective number of oocysts counted by a laboratory depends upon the Recovery Rate R times the real
concentration C. Let W = R * C, where R is lognormal with median = 0.10, and coefficient of variation (CV)
= 0.6, and C is lognormal with median = 0.03 oocysts/ 100 liters, and CV = 1.0. What is the median and CV
of W? Assume R and C are independent. 2. (8‘ points) A structural engineer is concern with the wind load on buildings. The average of the annual
maximum wind speeds is 52 mph with a standard deviation of 17 mph. What design wind should the
engineer use if she wants the probability that the design value is not exceeded to equal 99.9%? Is the
Gumbel distribution a reasonable model for such a random variable? 3. (8‘ points) For a student project, Rachel's group decided to study the weight of cinder blocks used in
construction. They weigh 10 blocks and found the sample mean and sample standard deviation were i = 41.8 and s =35. For lack of anything better, they propose to assume the weights of cinder blocks are
normally distributed. (a) What is a 98% confidence interval for the mean weight of such blocks? (b) If tomorrow six groups of students proposed to each measure 10 different and randomly selected blocks
(60 blocks altogether), what is the probability that all six confidence intervals will contain the true mean?
(c) What is the probability that the confidence interval Rachel’s group calculated with the mean and
variance given above contains the true mean? 4. (18' points) The specifications provided by the manufactor indicate that cinder blocks should weight
40.0 pounds each. This is different from the average value calculated by Rachel’s group. Does their data
indicated thatthe weight of cinder biocks is different from 40.0 pounds?
(a) What are the appropriate null and alternative hypotheses?
(b)What is rejection region for a test with a typel error of 5% for the hypothesis
that the mean weight is really 40 pounds ?
*(c) What is the tvalueand the P—v—alue for this data set?
(d) If the true weight is really 43 lbs, what is the type II error for this test? Assume 0:3.
(e) How LARGE a sample would the engineer need for the typell error to be< 5% ? Assumes: 3.
(f) Using the test in (b), how often will one accept Ho when the mean is REALLY 40 pounds? Assume 0:3. 5. (Slpoints) Consider the following distributionforthe'spacing between cars at a stop light:
Fx(x) = 1 exp( 1:x2 ) 0 _<_ x
and 0 otherwise, wherein addition:
E[X] = 0.8862/105 and Var[X] = 1.5633/1
Theengineer needs to estimate the parameter 1: which determines the mean and variance of the
distribution. Given observations {Xi I i=1,...,n }, what are the method of moment AND the maximum
likelihood estimators of 1? Which is likely to bemore precise? CEE 304 1 UNCERTAINTY ANALYSIS 1N ENGINEERING
SOLUTIONS TO Second Exam
November 13, 1998’ 1. Need to get the variances of logarithms, which are ln[1+(oreallurea1)2 = 0.307 and 0.693 respectively. Then ln(W) ~ Normall 111(medianR) + 111(median—C), 0.307 + 0.693, = 1.006 ] Thus
(median—W) = exp[ln(median—R) + ln(median—C)] = (medianR) * (median—C) = 0.003. Easy
CV{W} = sqrt{ exp(Var‘{an]) — 1 } = sqrﬂ exp(1.006)  1 l = 1.311
The class did very poorly on this problem. LN models are useful. 2. For Gumbel 0x2 = 1.645103; a = 0.0754. [1x = u + 03mm; 11 =44.35; solving
0.999: exp[  exp( ~ «(x0_999u)1 yields x0999 = 136 mph.
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. '3. A 98% CI is i :t s tug11ml Mi 2 38.7 to 44.9 for {0.01’9 '= 2.821 when n = 10.
If they construct 6 random intervals, probability all 6 contain the mean is (0.98)6 = 88.6% Rachel’s interval either does or does not contain the true mean. We do not know which. 4) (a) Ho: u = 40; Ha: u #40 (hence weight does NOT equal specified401bs.)
(b) Twosided ttest; reject Ho if «lﬁ Ii  40! / s > +t0.025’9 = 2.265
(c) for n =10, i = 41.8, s = 3.5 > t = 1.63 ; p is 2*(0.07) = 14% about.
(d) d =(4340)/ 3 =1; Table A.13 for 2—sided test witha = 5% yields {3 ~‘25%
(e) Table A.13 for 2—sided test with on = 5%; B = 5% yields n~20. (f) 95% = 1  a. This test is easy. The professor gives you the answers! 5) Method of moments. From EIX] = 0.8862] ¢0.5 one would gett = I 0.8862 / 5112.
Better to use formula for the mean than the variance because the mean is generally more precise. Use the variance only if you need to estimate a second parameter.
For pdi' fx(x) = 2 x texp( 4x2 ) , and sample {xi}, obtain likelihood function:
Ln) = n, fxm) > ian = n In (2) + z, 1n(xi,)+ n 111(1)  :2; xi2 0=(d/d1:)lnL(1)= n/r  23 xi2 yieldst= n/ 2i x12 MLEs are generally more precise in LARGE samples. 3 pts
2 pts
3 pts 3 pts
3 pts 2 pts 5 pts. 2 pts
1 pts 3 pts
3 pts 4 pts. 3 pts
3 pts
2 pts ‘3 pts. 1 pt
2 pts
1 pts CEE 304  UNCERTAINTY ANALYSIS IN ENGINEERING
Second Exam November 14, 1997 You may use text, your notes and calculators. There are 50 points in total, one per minute. 1. (4 points) Give an example of a realistic physical situation and a corresponding random variable
(of interest in engineering design; but NOT ﬂoods, rain or wind) that should have a Gumbel distribution
because of the way the variable’s value is physically determined. Why is a Gumbel dist. appropriate? 2. (10 points) Ken Hover’s students love to cast and break concrete cylinders. One year the average
breaking strength for the class was 5,140 psi with a standard deviation of 2,840 psi. Assume the strengths
have a lognormal distribution with those moments; what then is the strength that an engineer can be 98% sure that another cylinder would exceed? Might a normal model be reasonable for these data? 3. (8 points) A charcol filter removes chlorinated hydrocarbons from drinking water. An environmental engineer ran 7 samples of water through a filter. The average of the 7 observed removal rates was i = 98.27% with sample standard deviation 5 = 0.31%. Assuming that the measurements are normally
distributed about the true removal rate I, what is a 90% confidence interval for r? What is the probability
that the actual removal rate r, whose value could be obtained by taking many many samples to eliminate
noise from variability in filter performance and laboratory measurement error, is contained in the interval whose end points you calculated above? 4. (18 points) To test the filter described in problem 3, the engineer uses a prepared solution that should have a organic concentration of 20 mg/l. For the 7 samples the average of the initial concentrations was i = 20.440 mg/l with a sample standard deviation 5 = 0.607 mg/l.
(a) What are the appropriate null and alternative hypotheses?
(b) What is rejection region for a test with a type I error of 1% for the hypothesis
that the true concentration is exactly 20 mg / 1 ?
(c) What is the t value and the Pvalue for this data?
(d) If the true concentration of the solution is really 20.75 mg/l (and o = 0.5) what is the type II error for
this test? How LARGE a SAMPLE would the engineer need to collect for the type 11 error to be < 5% ?
(e) Upon computing a Pvalue = 3.4% for this test, an engineer declared: ”This means that there is only a 3.4% probability the null hypothesis is TRUE!” Is this statement justified or not? Explain your answer. 5. (10 points) The errors Vi in a particular analysis have mean zero with a distribution described by the symmetric pdf: fv(v) = (15/16) <p<1+<pv)2(1<pv)2 1/<ps v s +1/cp
and 0 otherwise. Here E[V] = o Var[V] = 0.1524/(p2 The engineer needs to estimate the parameter (p which determines the variance of the errors and their
range. Given observations {Vi I i=1,...,n }, what are the method of moment AND maximum likelihood estimators of (p ? (Get as close to a formula for MLE as you can by deriving equation whose root is the MLE.) CEE 304  UNCEFITAINTY ANALYSIS IN ENGINEERING
SOLUTIONS TO Second Exam November 14, 1997 1. Answers varied. Maximum load on a building or bridge during a year: corresponds to maximum loading
on any day during the year. Maximum wave height during a year: corresponds to maximum height on any day or of any storm during year; max snowfall, max pollutant concentration, max... 4 pts 2. Fitting a lognormal dist, recall the relationship between the realspace and logspace moments.
61085 = sqrt{ h1[1+(orea1/ureal)2] } = 0.516; ”logs = maimed) — 0.5 clogsz = 8.412. 5 pts
Yields 5002 = exp{ 8.412 — 2.054‘0516} = 1558.6 psi where 20.93 = 2.054. 2 pts
IF we used a NORMAL model to the data, then we would obtain 5,140 — 2.054*284O = 690 < 0.
{ OR, if X ~ N(5140, 28402), then Pr[X<0] = Pr[ Z< 5140/2840 = 1.81 ] = 3.5% ! }
Clearly a normal dist. does not work: strengths must be nonnegative, and are likely to be postively skewed away from lower bound of zero with no upper bound, like a lognormal distribution. SD too large. 3 pts. 3. A 90% CI is i i s t0.05,n—1 /1/ﬁ = 98.04 to 98.50 for t0.05,6 = 1.943 when n = 7. 4+2 pts.
This interval either does or does not contain the true mean. We do not know which. 2 pts
4) (a) Ho: u = 20; Ha: u at 20 (hence concentration does NOT equal advertized 20 mg / l.) 3 pts
(b) Twosided ttest; reject Ho if «lﬁ  i  20  /s > +t0.005,6 = 3.707 3 pts
(c) for n :7, i = 20.44, s = 0.608 > t = 1.92 ; p is 2*(0.051) = 10.3% or about 10%. 3 pts.
(d) d = (20.7520)/0.5 =1.5; Table A.13 for 2sided test with on = 1% yields [3 ~ 42% 3 pts
Table A.13 for 2sided test with (X = 1%; B = 5% yields n=13 or 14; 12 possible. 3 pts (e) NOT true. Nothing we have done allows us to assign a probability to the truth of a hypothesis.
A pvalue describes the probability for a random T, that T > tobserved when H 0 is true! 3 pts
{ QUESTION: (f) The engineer was concerned that the solution was too strong, perhaps 20.75 mg/l. If the actual concentration is 20 mg/l, what is the probability the test in part (b) will make the wrong decision?
ANSWER: (f) The value of on = 1% was specified. } 5) For fV(v) = (15/ 16) cp(1+(pv)2(1(pv)2 , and sample {vi}, obtain likelihood function:
L((p) = Hi fV(vi) —> In L(r) = n In (15/ 16) + n ]n(<p) + 2 Xi ln(1+<pvi) + 2 Xi 1n(1<pvi) 3 pts 0 = (d/dtp) hi L(r) = n/q) + 2 2i vi/(1+cpvi)  2 2i vi/(1(pvi) (can stop here) 3 pts > solve numerically for (B mle; Continuing: 0 = n/(p  4(p Xi viz/ [1((pvi)2] so MLE has the form 1 / (2 (T) mle)2 = 2i vi2/ [1(q3 mlevi)2] / n, which is like a variance.
However, MLE will yield Ivil < 1/ ([5 for all i; moment estimator may violate this constraint.
Method of moments. V is symmetric with a ﬁxed mean of zero, so average values of Vi will not help.
Instead set sample variance 52 = 0.1524 / (13 2 yielding (I) m =_ sqrt[ 0.1524/s2 ]. 4 pts. Estimate variance 52 by regular sample variance dividing by (n1). Even better use 2i v12 / n to estimate variance because we know the true mean is zero! CEE 304  UNCERTAINTY ANALYSIS IN ENGINEERING Second Exam UNUSIIHI. VEHB
November 15, 1996 You may use the text and your notes. There are 50 points in total, one per minute. 1. (4 pts) A highway engineer needs to estimate the maximum 24hour rainfall R to size a culvert. The
mean 24hour maximum is E{R} = 2.5 inches, so that the mean and median of ln(R) are both 0.9163. The standard deviation of ln(R) is 0.3. What is the probability the max 24hour rainfall R exceeds 4.6 inches? 2. (8 pts) Joan and Jeff are in an athletic contest. Their score 8 will be their average time, plus 10 seconds
due to an earlier penalty: S = (Joan + Jeff)/2 +10.
E{Joan] = 105; EUeff] = 95; Var[Joan] = 70; VarUefﬂ = 80; with Correlation[ Joan, Jeff ] = 0.4. If the
distribution of Joan and Jeff’s scores are normally distributed, but not independent, what is the mean and
standard deviation of their joint score S? What value will their score exceed with probability 90% ? 3. (20 pts) An environmental engineer is monitoring zebra mussel densities in a fresh water lake.
Ten lfootsquare plates left suspended in the lake yielded n = 10 samples with i = 232 /ft2 and a sample
standard deviation s = 57 m2. (a) Construct a 95% confidence interval for the true but unknown mean mussel density.
Last year mussel densities were determined to be 190 /ft2 with a standard deviation 0 = 50 /ft2. Our
engineer fears that mussel concentrations will increase again this year. In fact a computer model has
projected that they will increase to 240 mussels/ftz. (b) What are the appropriate null and alternative hypotheses to test for a population increase ? (c) What is the appropriate ttest 8: THE rejection region if on = 1% ? What value of t do you obtain ? (d) If concentrations have not increased from last year, what is the probability the test in part (c)
will incorrectly conclude that they have increased ? (e) If the computer model is correct and the concentration is 240 mussels / ftz, what is the probability
the test in (c) will incorrectly conclude that they have stayed the same ? (f) What sample size is needed to obtain a type II error 3 10% for the test in (c) ?
UNUSUHL VEHH  Neat problem normally on prelim #l . 4. (12 pts) A structural engineer is concerned about the load on a pier due to large ocean waves. Assume that
large wavegenerating storms occur in time according to a Possion process, with an average of 2 such storms
per decade. The impact of waves W when they occur is described by Pr{ W s w } = 1exp[ (w/300)2 ], w 2 0.
(a) Last month one such storm occurred. What are the mean and variance of the time until the next large
wavegenerating storm ? (b) What is the probability that there will be exactly one large wavegenerating storm in the next 10 years ?
(c) The engineer wants to design the pier so that s...
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