SOLUTION: 1000 light bulbs with an average life of 120 days are installed in streets of Mumbai. Their length of life is normally distributed with variance 400 days. (i) How many will expir

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Question 1193377: 1000 light bulbs with an average life of 120 days are
installed in streets of Mumbai. Their length of life is normally
distributed with variance 400 days. (i) How many will expire in
less than 90 days? (ii) If it is decided to replace all the bulbs
together what interval should be allowed between
replacements if not more than 10% should expire before
replacement?
This question is related to normal distribution and standard normal variate
My answer for(i) 668 but didn't get the answer for(ii)
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
mean is 120 days
variance is 400 days
standard deviation is sqrt(400) = 20 days
z = (x - m) / s
z is the z-score
x if the raw score
m is the mean
s is the standard deviation

formula becomes:
z = (90 - 120) / 20 = -1.5
area under normal distribution curve less than z-score of -1.5 = .0668072287.
.0668072287 * 1000 = 66.8072287

if you don't want more than 10% to expire before replacement, then the z-score for for an area of 10% to the left of it is equal to -1.281551567.
use the z-score formula to find the raw score.
-1.281551567 = (x - 120) / 20 = 94.36898867
solve for x to get x = -1.281551567 * 20 + 120 = 94.36896867.
is you don't want more than 10% to fail before replacing, then you should replace bulbs when no more than 94 days have elapsed before replacing.

my solutions to your questions are:

(i) How many will expire in
less than 90 days?

66.8072287 are expected to expire in under 90 days.

(ii) If it is decided to replace all the bulbs together what interval should be allowed between replacements if not more than 10% should expire before replacement?
no more than 94.36896867 days should be allowed before replacing defective light bubs.

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round as required.

let me know if you have any questions and whether or not these numbers were good for you.
theo