The lifetime of a lightbulb in a certain application is normally distributed with mean µ = 1400 hours and standard deviation ? = 200 hours.
a. What is the probability that a lightbulb will last more than 1800 hours?
b. Find the 10th percentile of the lifetimes.
c. A particular lightbulb lasts 1645 hours. What percentile is its lifetime on?
d. What is the probability that the lifetime of a lightbulb is between 1350 and 1550 hours?
e. Eight lightbulbs are chosen at random. What is the probability that exactly two of them have lifetimes between 1350 and 1550 hours?
(a) z = (1800-1400)/200=2.00 . The area to the right of z= 2.00 is 0.0228.
(b) The z-score of the 10th percentile is ? ?1.28.
The 10th percentile is therefore ? 1400 ? 1.28(200) = 1144.
(c) z = (1645-1400)/200=1.23 . The area to the left of z= 1.23 is 0.8907.
Therefore a lifetime of 1645 is on the 89th percentile, approximately.
(d) For 1350, z = (1350-1400)/200=-0.25 . For 1550, z= (1550 ? 1400)/200 = 0.75.
The area between z= ?0.25 and z= 0.75 is 0.7734 ? 0.4013 = 0.3721.
(e) Let X be the number of bulbs with lifetimes between 1350 and 1550 hours. Using part (d), the probability that a bulb has a lifetime between 1350 and 1550 hours is 0.3721. Therefore X~Bin(8, 0.2721).
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