Question

The manufacturer of a certain type of light bulb believes that the average life of a light bulb is approximately 1000 hours with a standard deviation of 100 hours.

(a) If a light bulb is selected at random from a huge consignment, what is the

probability that it will last for:

Less than 1700 hours?

More than 1780 hours?

Between 1700 hours and 1780 hours?

(b) Above what life time (in hours) would the longest 5% working light bulbs

last? (c) If the manufacturer in (a) above, supplies 3000 light bulbs, how many

would be expected to last between 1700 hours and 1780 hours

Answer 1

$\mu=1000 \\ \sigma = 100$

(a)

$P(X<1700) = P(Z< \frac{1700-1000}{100}) \\ = P(Z< 7) \\ = 0.999968 \\ P(X>1780) = 1 -P(X<1780) \\ = 1 -P(Z< \frac{1780-1000}{100}) \\ = 1 -P(Z< 7.8) \\ = 1-0.999968 \\ = 0.000032 \\ P(1700<X<1780) = P(X< 1780) -P(X<1700) \\ = P(Z< \frac{1780-1000}{100}) -P(Z< \frac{1700 -1000}{100}) \\ = P(Z< 7.8) -P(Z<7) \\ = 0.999968 -0.999968 \\ = 0$

(b) Let the variable A denote life time in hours that would the longest 5% working light bulb last, and calculation for A is given below.

$A_5 = \mu -Z_{0.05} \sigma \\ A = 1000 -(-1.645)(100) \\ A = 1164.5$

(c) N=3000

The required number $= 3000 \times 0 = 0$