[Instructor] A company has a policy of retiring company cars
this policy looks at the number of miles driven, purpose
of trips, style of car, and other features.
The distribution of the number of months in service
for the fleet cars is bell-shaped which means we have
a normal distribution and has a mean of 39 months
and a standard deviation of five months.
So we're given the mean equals 39 months which means mu
equals 39.
And we're given the standard deviation is five months
so sigma equals five.
We're asked to use the 68-95-99.7 rule, the empirical rule
to determine the approximate percent of cars that remain
in service between 44 and 54 months.
So if you let x be equal to the number of months,
we want to determine the percent or probability that x
is greater than 44 and less than 54.
So because we have a normal distribution if we can determine
how many standard deviations 44 is away from the mean of 39
and how many standard deviations 54 is away
from the mean of 39, we can determine this percentage.
So notice how 44 minus the mean of 39 is equal to five
this indicates that 44 is one standard deviation
above the mean and because 54 minus the mean of 39
is equal to 15, then 15 is equal to three times five
or three times the standard deviation this indicates
that 54 is three standard deviations above the mean.
So we can rewrite this percentage as the percent
or probability that x is greater than, again 44
is one standard deviation above the mean which we can
express as mu plus one sigma and x is less than 54
which would be mu plus three sigma.
So now we can use the empirical rule
to determine this percent.
We'll also come back and solve this a second way
using z scores.
But for a quick review when we have a normal distribution,
mu plus or minus one sigma represents 68% of the data,
mu plus or minus two sigma represents approximately 95%
of the data and finally mu plus or minus three sigma
represents approximately 99.7% of the data.
And then because of the symmetry of the bell-shaped curve
we can break these percentages down as shown here below.
So notice how the percent between mu plus one sigma
which would be here and mu plus three sigma which would
be here, we want to determine the percent
of this region here.
Notice how this would be approximately 13.5% plus 2.35%.
So this is equal to again, 13.5% or approximately
plus 2.35% which is equal to 15.85%.
Which means approximately 15.85% of the cars remain
in service between 44 and 54 months.
Now let's solve the same problem using z scores.
Remember z scores indicate how many standard deviations
a particular data value is away from the mean.
And the z score is equal to x, the data value,
minus the mean, divided by sigma, the standard deviation.
So starting with the same probability and percent
where we want to find the percent that x is greater than 44
and less than 54.
We'll now calculate the z score for 44 and 54
and write this percent using z scores.
So for x equals 44, the z score is going to be equal to 44
minus the mean of 39 divided by five which would five
divided by five which equals one,
which indicates 44 is one standard deviation above the mean
which we found above as well and then for x equals 54,
the z score would be equal to 54 minus 39 divided by five,
which is 15 divided by five which equals three,
which means 54 is three standard deviations above the mean.
So we can write the same percentage as the percent
or probability that z is greater than one
and less than three.
So looking at the graph of the standard normal distribution
shown here below, we know this is a standard normal
distribution because notice how the mean or mu equals zero,
and the standard deviation or sigma is equal to one.
And therefore the horizontal axis can also be viewed
as the z scores.
So here's where z equals one and here's where z equals
three so we're looking for the percent of data in this
region here.
Which as we already found, would be 13.5% plus 2.35%,
which equals 15.85%.
I hope you found this helpful.