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Central Limit Theorem BP Soln

Central Limit Theorem BP Soln - MA 206 Probability&...

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Unformatted text preview: MA 206 Probability & Statistics Lessons 23: The Central Limit Theorem Reflection Questions 1. The Central Limit Theorem (CLT) indicates that if we have a sufi‘ieiently large sample from a random variable, that the distribution of X tends toward the jimmy distribution. 2. What is sufiicz‘ently large when it comes to the (LT? (When can we invoke the CLT to solve problems about the mean?) Is this a “hard and fast” rule? KPSOJ iDIJ'i- no+ loco-t9 6mg} “Quasi. A5 ions as. UV‘ICQQrit-S dish is "‘04" sm’iig skew , Io an ia-e Working with Distributions fimfiiisf‘. l. The lifetime of a certain type of battery is normally distributed with a mean value of 10 hours and astandard deviation of 3 hours. a. If there are four batteries in a pack, what is the probability that the average lifetime of these four batteries lies between 9 and 11? whims) semis: gms may ‘3 ‘i’ “ WWW :- b. 16 battery supefigqiiiai’iqk'? “ii a i y ~N({ot$—fi)t a. .Wssrs Wits. at mm is it f . 1 a “ m it mt Mediates W a ‘ WW . s It 50 batt S I.) swam soil-m W- smarissa 1;; “it 0- ery ca . ibis ‘ mi sapw- {is 'Wi skiff ii 7i” N 00:41:32) ifs-sign) -_ S Qiisgwaiscast: 1/? lass “ d. What does this say about how the sample size affects the distribution of E 7 2. Determine the mean and variance of E of a random sample of size 30 from a distribution havingaPDF: ' f(x)={4x3 05x51 Ax: Scribes“! s 0-? O othenvise U- = x OJCBB Hiram: -—:a (LL-Tl in M/ T N ones: ‘- x~ (m, a. > any . 3° 3. The armor plating for a new Army vehicle has a penetration strength (in pounds per square inch) that follows a Gamma Distribution with parameters or: 20,000 and B = 0.05. a. State the distribution of the average penetration strength for a sample of 40 armor plates. ‘7 a“; — at — moo _ 4 l j ’3 we sour) MK .- (9 y ~N(toao 'i 00/ b. What is the probability ‘ 'u 90 laid}; c. What isthe probab ty l p I. ‘ . ngth is at least 999 pounds per square inch? " ' PCYefiraa) = S Maria .. e r EMS- aas // I: . . . . 4. Sixteen adult nights approach an elevator on the 102'"! floor of the Empirc State Building. The elevator has the sign: Maximum Weight 2900 lbs They enter the elevator. After consulting sorne medical books, you find that the weight of a male is normally distributed with mean 170 pounds and standard deviation -'l 5 poundsl. a. Find the probability that the cable snaps and they plunge to their death; i.e., their combined weight exceeds 2900 pounds _ W tabla. Mu shat? .9 it" :2 2““ =' rat :5 FWD: PO( '3" (“'15) 1 8 do") all»; Hu- ‘5 lBl.25 33 ~ NOWr-mfl — o.oo13,5//, Since our «'5 leesJF CQl‘H\QUS\“-x we me s+‘.l\ ehelsnbtj ha...) 1 Adapted from problem posed by Department of Statistics, Western Michigan University. Retreived from the World Wide Web at: littp:I'iwww.stat.wrnieh,gdp_/~.; lefllQQ. kfnode43.html#n17tl on 27 March, 2007. ...
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