SOLUTION: Suppose that the lifetimes of light bulbs are approximately normally distributed, with a mean of 57 hours and a standard deviation of 3.5 hours. With this information, answer
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Question 1132829: Suppose that the lifetimes of light bulbs are approximately normally distributed, with a mean of 57 hours and a standard deviation of 3.5 hours. With this information, answer the following questions.What proportion of light bulbs will last more than 61 hours?
(b) What proportion of light bulbs will last 52 hours or less?
(c) What proportion of light bulbs will last between 57 and 62 hours?
(d) What is the probability that a randomly selected light bulb lasts less than 46 hours Answer by Theo(13342) (Show Source):
use the z-score to find the answer you need.
formula for z-score is z = (x - m) / s
z is the z-score
x is the raw score
m is the raw score mean
s is the standard deviation.
(a) what proportion of light bulbs will last more than 61 hours.
z = (61 - 57) / 3.5 = 1.142857143.
area to the right of that z-score is equal to .1265490059.
that's the probability of the life of the light bulb to be greater than 61.
(b) What proportion of light bulbs will last 52 hours or less?
z = (52 - 57) / 3.5 = -1.428571429
area to the left of that z-score is equal to .0765637714.
that's the probability of the life of the light bulb to be less than 52.
(c) What proportion of light bulbs will last between 57 and 62 hours?
the z-score for 57 is equal to (57 - 57) / 3.5 = 0
the z-score for 62 is equal to (62 - 57) / 3.5 = 1.428571429
the area to the left of a z-score of 0 is equal to .5
the area to the left of a z-score of 1.428571429 is equal to .9234362286.
the area between a z-score of 0 and a z-score of 1.428571429 is equal to .9234362286 minus .5 = .4234362286.
that's the probability of the life of the light bulb to be between 57 and 62 hours.
(d) What is the probability that a randomly selected light bulb lasts less than 46 hours
z = (46 - 57) / 3.5 = -3.142857143.
the area to the left of that z-score is equal to .0008366040026.
round to 2 decimal places and it becomes 0.
i used the TI-84 Plus calculator to get the answers.
you can also use the z-score tables, but your z-score would need to be rounded to 2 decimal digits.
the z-score table gives you the area to the left of the z-score.
to get the area to the right of the z-score, you would need to take 1 minus the area to the left of the z-score.
there is an online calculator that can also do this directly from the mean and the standard deviation of the raw score as well as from the z-score.