SOLUTION: For a certain type of fluorescent light in a large building, the cost per bulb of replacing bulbs all at once is much less than if they are replaced individually as they burn out.

Question 1194396: For a certain type of fluorescent light in a large building, the cost per bulb of replacing bulbs all at once is much less than if they are replaced individually as they burn out. It is known that the lifetime of these bulbs is normally distributed, and that 60% last longer than 2500 hours, while 30% last longer than 3000 hours.
a)What are the approximate mean and standard deviation of the lifetimes of the bulbs?
b)If the light bulbs are completely replaced when more than 20% have burned out, what is the time between complete replacements?

Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
the distribution of the lifespan has a mean and a sd
we know that 2500 hrs. is at the 40th percentile, and that 3000 hours is at the 70th percentile
z(0.40)=-0.253 and z(0.70)=0.524
we also know that z=(x-mean)/sd
we have two equations
-0.253=(2500-mean)/sd or 2500-mean=-0.253 sd
+0.524=(3000-mean)/sd or 3000-mean=0.524 sd
subtract the 2nd from the first: -500=-0.777sd
sd=643.50 hours
mean=2663 hours when one substitutes into either of the equations above
-
z (0.20)=-0.842
so -0.842=(x-mean)/sd
-0.842=(x-2663)/643.50
-542=x-2663
x=2121 hours

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