Question 1163053: The lifetime of a certain light bulb can be standardised to a normal model with a mean of 400 hours and a standard deviation of 65 hours . For a group of 6,400 lights use the empirical rule to determine how many of them are expected to last between 400 hours and 530 hours
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
mu = 400 is the mean
sigma = 65 is the standard deviation
The raw score x = 400 converts to z = 0 as it is exactly the value of the mean. We could show
z = (x-mu)/sigma
z = (400-400)/65
z = 0/65
z = 0
The raw score x = 530 converts to...
z = (x-mu)/sigma
z = (530-400)/65
z = 130/65
z = 2
Showing that this raw score is exactly 2 standard deviations above the mean.
According to the empirical rule, 95% of a normal distribution is within 2 standard deviations of the mean.
If we randomly selected a bulb, then we have approximately a 95% chance of picking a bulb that is within 2 standard deviations of the mean.
In terms of symbols, we would write
P(-2 < Z < 2) = 0.95
which is approximate
But we're only concerned with the interval from z = 0 to z = 2, which is half of the interval -2 < z < 2.
So we simply cut the figure 95% in half to get 47.5%
Meaning, P(0 < Z < 2) = 0.475 approximately
About 47.5% of the distribution has z scores in the interval 0 < z < 2.
The last step is to take 47.5% of the sample size 6400 and we end up with 0.475*6400 = 3,040
We expect about 3040 light bulbs will last between 400 hours and 530 hours.
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