Question

Video projector light bulbs are known to have a mean lifetime of µ = 100 hours and standard deviation σ = 75 hours. The university uses the projectors for 9000 hours per semester. (a) (15 points) Use the central limit theorem to estimate the probability that 100 light bulbs will last the whole semester? (b) (10 points) Explain how to estimate the number of light bulbs necessary to have a 1% chance of running out of light bulbs before the semester ends. Don’t actually do the whole computation.

Answer 1

a)

expected time 100 light bulbs will work =100*100 =10000

and standard deviaiton =75*√100 =750

therefore from central limit theorem , normal approximation of sampling distribution of sample mean:

P( 100 light bulbs will last the whole semester ):

P(X>9000)=1-P(X<9000)=1-P(Z<(9000-10000)/750)=1-P(Z<-1.33)=1-0.0918 =0.9082

b)

let number of light bulbs required =n

expected life of n bulbs =100n

standard deviation =75*√n

for 1st percentile critical value of z=-2.33

for 1% chance of running out of light bulbs before the semester ends

therefore mean +z*standard deviation >=9000

100n-2.33*75*√n >=9000

100n-174.75√n-9000 >=0

solving above quadratic equation: n>=108.2

or n>=109