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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+ locot9 6mg} “Quasi. A5 ions as. UV‘ICQQritS dish is "‘04" sm’iig skew , Io an iae
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
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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: '
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. 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? " '
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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
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