SOLUTION: A certain brand of light bulb has a mean life of 600 hours and a standard deviation of 53 hours.
Assuming the data are bell-shaped, what percentage of these light bulbs will hav
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Question 1196426: A certain brand of light bulb has a mean life of 600 hours and a standard deviation of 53 hours.
Assuming the data are bell-shaped, what percentage of these light bulbs will have a life between 547 and 706 hours?
Group of answer choices
95%
18.5%
68%
81.5%
13.5%
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
"bell shaped" is another way of saying we have a normal distribution. Some textbooks will call it a Gaussian curve.
mu = 600 = mean
sigma = 53 = standard deviation
Let's compute the z score for x = 547
z = (x - mu)/sigma
z = (547-600)/53
z = -53/53
z = -1
This score is exactly one standard deviation below the mean.
Repeat for x = 706
z = (x - mu)/sigma
z = (706-600)/53
z = 106/53
z = 2
This score is exactly two standard deviations above the mean
The task of finding P(547 < x < 706) is the same as P(-1 < z < 2) for this problem's context.
Let's turn to the Empirical Rule
We'll be adding the two pink 34% regions, and the right-hand region of 13.5 in blue
These three regions span from z = -1 to z = 2
0.34+0.34+0.135 = 0.815
According to the Empirical Rule
P(-1 < z < 2) = 0.815 approximately
which translates back to
P(547 < x < 706) = 0.815 approximately when mu = 600 and sigma = 53
This means roughly 81.5% of the light bulbs have a lifespan between 547 hours and 706 hours.
Answer: 81.5% (choice C)
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