kmarks Window People Help d Learn x D x httpswebcoursesniued

kmarks Window People Help d Learn x D x https://webcourses.niu.edu dav/pid-3831611-dt-content-rid-23001326 2/courses/20152-ISYE-335-----1/Hw%234.pdf ISYE335 Homework #4 (Due March 4th, 2015) Spring 2015 8. Among homeowners on a metropolitan area, 75% recycle plastic bottles each week. A waste management company services 1500 homeowners (assumed independent). Approximate the following probabilities: (a) At least 1150 recycle plastic bottles in a week (b) Between 1075 and 1175 recycle plasticbottles in a week 9. An Pad contains 500 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.99 and assume that the components fail independently. Approximate the probability that 10 or more of the original 500 components fail during the useful life of the product

Solution

(8)(a)

mean=n*p=1500*0.75=1125

standard deviation= sqrt(n*p*(1-p))=sqrt(1500*0.75*0.25)=16.77051

So the probability is

P(X>=1150) = P((X-mean)/s >(1150-1125)/16.77051)

=P(Z>1.49) =0.0681 (from standard normal table)

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(b)P(1075<X<1175) = P((1075-1125)/16.77051 <Z< (1175-1125)/16.77051)

=P(-2.98<Z<2.98)

=0.9971(from standard normal table)

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(9) mean=n*p=500*0.01=5

standard deviation =sqrt(500*0.01*0.99)=2.22486

So the probability is

P(X>=10) = P(Z>(10-5)/2.22486)

=P(Z>2.25) = 0.0122 (from standard normal table)