fall 2006 STAT 230 Quiz3Solutions

fall 2006 STAT 230 Quiz3Solutions - STAT 230 Quiz#3 3:40...

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Unformatted text preview: STAT 230 Quiz #3 Oct 26, 2006 3:40 — 4:10 pm Name: S I.D: UWUserid: Mark /20. Circle your instructor and time slot: 1. Metzler(1:30) 2. Drekic(11:30) 3. Springer(10:30) 4. Drekic(12:30) 5. Ramirez(8:30) 1. In a town, E—Coli bacteria is distributed in the water supply according to a Poisson process, with an average of 1 bacteria per 100 ml of water. [3] (a) Give an expression for the probability that a 400 ml bottle of water contains at least three E—coli bacteria. X: Nombcr q. bad-Goa in 400 ml. at water- 7}: z) Xwfijsson (2C400\\=P03880n1(4\ 2 I ‘ JP: PDQ/33 =l—~\°Ex¢3]= \—Z ‘1— 6“1 1:0 7—! _ __ ~‘l r \—ce"‘+<ie“r 55“) =1 — {3-6 [3] (b) Water is considered safe to drink if 400 ml of water contains at most two E—coli bacteria. Give an expression for the probability that ten 400 ml bottles of water will have to be examined to find the second bottle that is safe for drinking. Let ‘1: Number 0(— non—81,9: braille-5 bceoa: 9v WCLWLH‘ZS Then, \iwueca, 1—93 =N3(2,ise“‘) 8+2—v _ 3’ VH0 Enema +0 examsncficPlN=83=< 8 )(la-e‘llfll-i'b'é‘l) : (2 l lei-69c [— as“)? [3] (c) Give an expression for the probability that among one thousand 400 ml bottles of water examined, exactly four bottles are safe for drinking. in: Number cur We boi—HcG am 4900 bellies W ) w/o B‘\ (1000, in?) = g; ( 4090, «BC—“j GHQ PZw=4j: (1:00» “Wit P100041 = (130)18fié—{6(k16.é4) [3] (d) Use an appropriate approximation to evaluate the probability in (c) and justify why your ap- proximation is suitable. Since wa‘itn,q=\—p3 I n h \qnac and (4:0.le is am) we can Lac— he 'Poisson oomimcdfm ’i’o +hc 3imm‘ioklv Than w Olfiwwhmoridq has i’dzsscn draifibd‘ch wi—ih parameter/(Lang: 238.! ‘l - 8.} = N Lain '“9: _ L158. 0“ 7'3 ACMCJ PX“) M 6 — 4! 6 2. Pulses arrive at a Geiger counter in accordance with a Poisson process. It is known that in any one hour time period, there is a 36.78% chance that no pulse arrives to the counter. [2] (a) Show that A, the arrival rate of pulses per hour, is approximately equal to 1 per hour. Lc—i X= Number op 90565 M omwe. in one how am 2 be hm average (Vitae potscs per hem. Than ) 0.391% = sz=oj —_. 23 6% = 6- Then , 2: -\n (0.39%) 2:1.0094 Q51 [3] (b) Assuming that A = 1 /hour, what is the probability that two or more pulses arrive to the counter between 6:00 pm and 8:30 pm of the same day? Lvi \l-‘quxber of putscs mm 6100 308130 (15%} mm” «@01an (2.5). vmazj mam <2] =x— who: “>me - .F‘ — .5' -Q.5 =1—ge"°+2.5-e" l =\—5.5-€ {3] (c) Assuming that A = l/hour, what is the probability that no pulses arrive to the counter between 8:00 am and 10:00 am (of the same day) given that 3 pulses have arrived to the counter by 8:00 am? Lc+ w=mmocr 9; pqscs («am 3100 am in [0:0on Then) u) Nfiiasm (0.3 . PXO potscs in the misc-Von L8, l0\ l 8 page; m (1.3), 1481 = PLO palace in 0340)] ‘0‘ \mrcmsns 0(— home om: rho—mopan -:z .'. Pb pokes m L8, 103 is puis<§ m 0%)]: lezoj :e STAT 230 Quiz #3 Oct 26, 2006 3:40 — 4:10 pm Name: S I.D: UWUserid: Mark / 20 Circle your instructor and time slot: 1. Metzler(1:30) 2. Drekic(11:30) 3. Springer(10:30) 4. Drekic(12:30) 5. Ramirez(8:30) 1. In a town, E—coli bacteria is distributed in the water supply according to a Poisson process, with an average of 2 bacteria per 100 ml of water. [3] (a) Give an expression for the probability that a 500 ml bottle of water contains at least three E—ooli bacteria. X: Number 09 boum‘q in 500 m\ q. WC:- 7s : W :5 X ~?oisson (a (500\] :"fia‘issm (10) 2 2 00)‘ —-l0 {2: PDQ/33: l-Plx<33= lr 2' Pixaj: l-Z :1 6 1:0 ~ :9 —o _ ~10 —Lo r\—Lel ‘i‘IO-e‘o-VSO‘C )= l—él-C [3] (b)' Water is considered safe to drink if 500 ml of water contains at most two E—coli bacteria. Give an expression for the probability that ten 500 ml bottles of water will have to be examined to find the third bottle that is safe for drinking. L-c—i V: Number 0+ fbn-SQCC bo’rHcs bean? 354$e'b‘i'Hes Thenflu NBCB, i—JP) = 145(3) We“) q+3-\ 4 -lo 3 —\o P Lso Mb) >m m3m\=?L\l=:fl:{ q )(eI-c News ] = (2)6?6500—61‘6—‘0? [3] (c) Give an expression for the probability that Among one thousand 500 ml bottles of water examined, exactly five bottles are safe for drinking. W =N0mber- at Sate: boll-ts am 4000 bcl-ch Then) w M’Bi £1000, l-f): Ba (1000, elem) CH5. Pzw=5]: (19500 ) WP? PlOOO-S = (U920) Hie—5% Pekc-io) [3] (d) Use an appropriate approximation to evaluate the probability in (c) and justify why your ap- proximation is suitable. S‘mcc wNB\Ln,q=l—?7 ,n 1; MC w 3:23? mo— sW“: NC can 03‘- ‘H‘t 1’on approximation Jro —ihc Emmi. Thcn UL) qpyronnwcu, ms Porsaon dismbuhon im—lh .pqmmc’rcwng : 2:}? Hence) 9):“):53 M 3 \S 5 5 L —r\ -2.T-'l __ “2? c: g e zoossla 2. Pulses arrive at a Geiger counter in accordance with a Poisson process. It is known that in any one hour time period, there is a 13.53% chance that no pulse arrives to the counter. [2} (a) Show that A, the arrival rate of pulses per hour, is approximately equal to 2 per hour. Let X: Number or, pulses rim-v orfive in one hoof and 7i be +hc Gueran we: of. poses per- mm. Than) mafia = PZ%:01: g '7‘ = e‘“ 'lhcn ) zs-m @4353) .'. a ': 2.000260 9&9- [3] (b) Assuming that /\ = 2/ hour, what is the probability that two or more pulses arrive to the counter between 6:00 pm and 8:30 pm of the same day? La ‘1 =N0mber 0+ pom From 6:00 io 8160 (25%} 1hcn ) \{ N'RJTSSUHCZCQSD :i’omson L5) - 911223 = l—vn <13 .— r- @1qu +?L‘1 :0) ~ = \- e-e’ : 035% [3] (c) Assuming that /\ = 2/ hour, what is the probability that no pulses arrive to the counter between 8:00 am and 11:00 am (of the same day) given that 5 pulses have arrived to the counter by 8:00 am? 145+ U0=Numbcr Cl. potsCs in 3mm (from film toH'vOOGm) W0 7 W N?Oisson (.6) Flower: in +hc mm (8»0I5puscs m (1187,7“83 =PZopu5¢5 m (8,111)] “9‘ \nmats 0+ +mc out mn—o/ermopmcd "(a PLO paces m mils in (7,3)] 2 le=03 :e ...
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