Question 1132829
mean is 57 hours.
standard deviation is 3.5 hours.


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.


that calculator can be found at <a href = "http://davidmlane.com/hyperstat/z_table.html" target = "_blank">http://davidmlane.com/hyperstat/z_table.html</a>


the results from the use of that calculator, in the order that the the problems were presented, is shown below:


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