The 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, and has a mean of
fifty-one months, and a standard deviation of nine months.
Using the sixty-eight, ninety-five, and ninety-nine point seven rule,
also known at the empirical rule, what is the approximate percentage of cars that
remain in service between
sixty and sixty-nine months? We use the variable mu to present the mean, so here
mu is equal to
fifty-one months. And we use the variable sigma to represent the standard deviation
which is given as nine months. Now before we answer this question, let's review
the sixty-eight, ninety-five, ninety-nine point seven rule.
When we have a normal distribution, approximately sixty-eight
percent of the data falls between mu
plus or minus one sigma. Which means sixty-eight percent of the date is between
the mean and plus or minus one standard deviation.
And because of the symmetry and the fact that sixty-eight percent of divided by two equals
thirty-four percent, we can also say approximately thirty-four percent of the data
falls between the mean, and one standard deviation above the mean.
And thirty-four percent the data falls between the mean and one standard deviation
below the mean. Next approximately ninety-five percent the data
falls between mu plus or minus two sigma,
or the mean plus or minus two standard deviations.
And because ninety-five percent divided by two is equal to forty-seven point five percent, because of the
symmetry of the bell shaped curve,
we can say that thirteen point five percent of the data
falls between one and two standard deviations above the mean.
And thirteen point five percent of the data falls between
one or two standard deviations below the mean. Notice that thirty-four percent plus
thirteen point five percent does equal forty-seven point five percent.
And then finally, approximately ninety-nine point seven percent of the data falls between
mu plus or minus three sigma, .or the mean plus or minus three standard
deviations
And again because a ninety-nine point seven percent divided by two
is equal to forty-nine point eight, five percent. Because of our symmetry and the bell shaped curve,
we can also say approximately two point three, five percent of the data
falls between two and three standard deviations above the mean,
and also approximately two point three, five percetn of the data falls between
two and three standard deviations below the mean. Keep in mind if we had all these
percentages,
it's not one hundred percent there would still be some data
outside plus or minus three standard deviations.
Now going back to our problem, we're trying to determine the number of cars that remain in service
between sixty
and sixty-nine months. Well because mu is equal to fifty-one months,
and sigma is equal to nine months. Notice that mu
plus one standard deviation would be equal to fifty-one
plus nine or sixty months.
And also notice that mu
plus two standard deviations, or two sigma,
would equal to fifty-one plus two times nine,
which equals sixty-nine months. So to answer this question,
we want to determine what person of the data falls between
one and two standard deviations above the mean.
Looking at a graph here on the right,
we know mu is here in the middle, and therefore, the data between one and two
standard deviations above the mean
would be in this region here. And because we have a normal distribution we know
approximately
thirteen point five percent of the data falls between
one and two standard deviations above the mean.
Notice here we have plus one sigma,
or one standard deviation. Here we have plus two
sigma, or two standard deviations. So to answer the question,
approximately thirteen point five percent
of the cars remain in service
between
sixty and sixty-nine months.
I hope you found this helpful.