The maintenance department at the main campus of a large state university
receives daily requests to replace fluorescent lightbulbs.
The distribution of the number of daily requests
is bell-shaped and has a mean of forty-nine and a standard deviation of eleven.
So because the distribution is bell-shaped, we have a
normal distribution.
Because the mean is equal to forty-nine, we know mu
equals forty-nine. And because the standard deviation is equal to eleven, we
know sigma
equals eleven. We want to use the sixty-eight,
ninety-five, ninety-nine point seven rule, to determine the approximate percent of lightbulb
replacement requests,
numbering between sixteen and forty-nine. So if we let X be equal to the number of requests,
we want to determine the probability or percent
of X, the number of requests,
being greater than sixteen, and less than forty-nine.
So because we have a normal distribution, if we can determine the number of
standard deviations
sixteen and forty-nine is away from the mean, we can determine this percentage.
Well first notice how forty-nine is equal to the mean, or mu.
So we can rewrite this as: the percent
that X is less than
mu, because forty-nine is the mean. And then to finish this,
we need to determine, how many standard deviations below the mean
sixteen is? And because forty-nine
minus sixteen is equal to thirty-three, and thirty-three is equal to three times eleven,
this indicates that sixteen is three standard deviations
away from the mean. And because sixteen is less than the mean,
we know sixteen is three standard deviations below the mean.
So we can express sixteen as mu
minus three sigma. Now that we have the percentage expressed in this way here,
we should be able to use the properties of the normal distribution,
or the empirical rule to determine this percentage.
So for a quick review, mu plus or minus one sigma,
represents approximately sixty-eight percent of the data. Mu plus or minus two sigma
represents
approximately ninety-five percent of the data. And finally, mu plus, or minus three sigma
represents approximately ninety-nine point seven percent of the data.
And because of the symmetry of the bell-shaped curve, we can also break the percentages
down,
as we see here below. So we're trying to find the percent between
mu minus three sigma, which would be here,
and mu which is here in the middle. So we want to find the percent of
data that would be in this region here. To do this, we just need to add the percentages
shown above. So the percent is going to be equal to
two point three, five percent plus thirteen point five percent,
plus thirty-four percent. So this is equal to
two point three, five percent plus thirteen point five percent,
plus thirty-four percent,
which is equal to forty-nine point
eight, five percent. So this tells us that approximately
forty-nine point eight, five percent of the number of requests
is between sixteen and forty-nine. If we have a hard time determining
how many standard deviations a given data value is away from the mean,
another approach to this would be to calculate the Z-score where the Z-score
indicates
how many standard deviations a particular data value is away from the mean.
And the Z-score is equal to X minus mu, divided by sigma.
So let's also show this another way. Again we start with the
percent, or probability that X is greater than sixteen,
and less than forty-nine. Now we want to rewrite this
using their Z-scores. So we'll have the probability
that the z-score is greater than some value,
and less than some value. And now we'll calculate the Z-score for sixteen and forty-nine.
So for X equals sixteen, the Z-score
would be equal to sixteen,
minus forty-nine, divided by
eleven, which equals negative thirty-three,
divided by eleven or negative three. Because the Z-score is negative three,
this indicates that sixteen is
three standard deviations below the mean.
So here we have Z would be greater than negative three, and we should recognize four X equals forty-nine, because
forty-nine is the mean, or mu,
if the Z-scores going to be zero. But we'll go ahead and show that. We'd have forty-nine
minus forty-nine divided by eleven, which equals zero.
Which means we can express the percentage that X is greater than sixteen,
and less than forty-nine, as the percentage of Z being
greater than negative three, and less than zero.
So looking at the graph of the standard normal distribution shown here below,
we're looking for the percent between Z equals negative three,
which is here, and Z equals zero, which is here. Which of course is going to be the same
percentage
that we found above. The difference here is that when we have the standard
normal distribution,
notice how the mean, or mu is equal to zero,
and sigma, the standard deviation is equal to one.
And therefore the horizontal axis can be viewed as Z-scores.
But again, going back to the empirical rule, it's still going to be this sum
of the same percentages shown here.
Which we already know is two point three, five percent,
plus thirteen point five percent,
plus thirty-four percent, which again is equal to
forty-nine point eight, five percent.
I hope you found this helpful.