Solutions to Final Exam, Ma441, Spr06 - Math 441...

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Unformatted text preview: Math 441 Engineering Statistics I Name: fowl-70113.5 Final Exam Spring 2006 RWJ Instructions: Read questions carefully Help me award you (partial) credit by carefully showing your work/reasoning Note point values on problems (102 listed, 100 maximum) Good luck! Miscellaneous: A random variable having the density le'b‘ x 2 O f(x)={ 0 x<0 has mean ,u =1/xi and standard deviation 0 =1/x1. 1. / 12 2. /25 3. /20 4. /10 5. /20 6. / 15 / 102 Fflomf. l: l. (12 pts.) Consider the density function ___...——————o J— J— 1 m2 > I“ c. ¢ ¢ L _ _ 0 ’A 1- f(x) = 2 e x C '4 0 x < 0 ’L ’L O" 3 a s 2. a a) a. One 0 - - ' to ; ams below displays 1,000 values generated from this density. Which on — histogram 1 histogram 2, or histogram 3 (circle one)? b. One hundred observations were generated from the above density and then added. This was repeated 1 HI -- - 7 to give 1,000 sums. Which histogram — histogram 1, histogram 2, or displays these 1,000 sums (circle one)? c. Four hundred observations were generated from the above density and then added. Thi as re eated 1,000 times to give 1,000 sums. Which histogram - histogram 1, or histogram 3 — displays these 1,000 sums (circle one)? Histogram 1 Frequency N O O continued Histogram 2 Frequency Histogram 3 (D 0 Frequency 2. (25 pts.) A short cable is made of 40 strands of a fixed length. Individually, the strands have breaking strengths with mean 35 1b. and standard deviation 5 lb. Suppose further that the breaking strength of a cable is roughly the sum of the strengths of the strands that ak . 1. Q- ‘— m “up 3C5: gf'fifl'klnb swam CWL€_I_ L'I,2,.../ $0 a. Estimate the mean breaking strength for such cables. 7. 35)- -»-4-35 : I, fan [9, £44: W 4am... b. Estimate the standard deviation of the breaking strength for such cables (assume independence as necessary). ,. ,q flXfiwd— 21+, ’ 2:4. k A ’L ’2. L fill-"+24, ” 5?. *”"' “717:” " 5 *""r 57- : Moo (6; W 417 M5 0' 7. .._ c. Estimate the chance that such a (randomly selected) cable will support a load of 1,450 lbs. (Note: To support a load of at least 1,450 lbs. the breaking strength must be above 1,450 lbs.) /(:c.+.-~4— 3:... 2. hrs.) : f XVFHM’ Zea; (" (\kJ-s : ,,_ I (I919) » row) JpTr = /— $0.59) 1 This is roughly true for “short” cables. For an entertaining discussion of this as a key element of a story plot read Eyes of the Dragon by Stephen King (a good read!). 3. (20 pts.) Shown below is a histogram of the knee diameter2 for a sample of a few hundred U.S. adults3. For what follows assume this sample is representative of US. adults. For this sample the average is nearly 18.80 cm with a standard deviation of nearly 1.35 cm. Histogram of Knee Diameter 18.0 21.0 . 24.0 Knee Diameter a. Using the fact that this histogram is well approximated by a normal, estimate the proportion of US. adults with knee diameters between 18.00 cm and 19.70 cm. Wit.» < K < We) (770’0’10 /é,op»/?.Eo = — a‘ §< /. 35 ) < /- 3)“ ) : €(0.£7)—§(—0v77) 37474-1774 = b. Using the fact that this histogram is well approximated by a normal, estimate the 30th percentile of knee diameter values. That is, estimate the value such that 30% of all knee diameters fall at or below it. _ adage 0.3»: flKéx) ~ ’5 ['35) 0.301;: £50.51), K_/8,’90- L35 49.57— ’9 'K: {87.09300 - d dimensions”, 2 Sum of diameters for the two knees, in cm. - 3 From Heinz, G., Peterson, L., Johnson, R., and Kerk, C. (2003), “Exploring relationships ' Journal of Statistics Education, vol. 11, no. 2. 4. (10 pts.) Suppose runners in a race are wearing numbers ranging from 1 through N, with N unknown. For such a population of numbers it can be readily shown that ,u = You glimpse a portion this race seeing a random sample of runners with the 2 following numbers: f #025443 + “‘3 126, 33, 243, 172, 279 I + m r 17 '7)/5 Give the method of moments estimate of the number of runners, N. : [’20,£ 361’ fizz/u - ’\ “6" (70.4: ’93 MM;31’O"L a: 24.0 5. (20 pts.) Transmissions are relayed to a communications center from two different sources, A and B. The chance a transmission from source A is corrupted when it arrives at the center is .08. The chance a transmission from source B is corrupted when it arrives at the center is .05. Suppose 70% of all transmissions are relayed from source A and 30% of all transmissions are relayed from source B. If a message at the center is known to have been corrupted, what is the chance it originated from source A? Paw) : Flew) WM) / WélfiWU’r) + We! a) PC a) ((08) (~79) _____________________________.. Qog) (.75) 4— (.05)C3o> IR , WW7} 6. (15 pts.) In a test for extrasensory perception (ESP), an experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle, a wave, a square, or a cross. As the experimenter looks at each of the 20 cards in turn, the subject names the shape on the card. If the subject is simply guessing, then (s)he will have a l in 5 chance of guessing correctly each trial. Write an expression for (but don’t evaluate) the chance of matching at least 15 out of 20. M: ##4##) [/(MZIS) ; Z %>Zo-7c Ks]; NM Vefwvlfl all ywblfin (bwl’u; (45¢ yaul’K Cut/70(3)): V/Mz/E) 9 (.5’03xlzf7 ...
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