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In this example, we're looking at a manufacturer of soda filling cans, and what we're told
is that the average value in those cans has a mean value of 12.1 ounces. And they know
that the machinery fills the cans with the standard deviation of .15 ounces. If we assume
that the distribution of the liquid in the cans is forming a bell-shaped curve where
we can see approximately a normal distribution, what percentage of those soda cans are under-filled
if there's supposed to be 12 ounces per can? Now in this situation, the way we're going
to answer it is based on this part of the assumption. Otherwise, we wouldn't have enough
information to go on. It has this bell-shaped curve or this normal distribution. So to give
us an idea of what we're working with, our normal distribution for the amount of soda
that is in the cans is given to us by looking at kind of the area under this curve.
And for our distribution, we know that the mean was 12.1 ounces, so that's going to be
where the peak occurs. So the mean here is 12.1. The standard deviation-- so, I'm going
to kind of draw this dash line for my line of symmetry-- the standard deviation is the
distance from the mean. And when you look at the bell-shaped curve, the way you can
figure out where the standard deviation distance is, is by finding what's called an "inflection
point" on that curve. And so this is where it's concave down, and then it kind of turns
so that's being concave up. So the shape is kind of changing at this point, and that's
called our inflection point. So this distance from the mean to that inflection
point is actually the standard deviation. So for us, that's going to be the .15 ounces.
So here, the mean was 12.1 ounces. So what we're trying to identify is that I have cans
being under-filled if they're less than 12 ounces. So I want to find what percentage
of cans are under-filled. So I'm going to look at—okay, here's 12. I want this percentage
or this area that's under that curve. So if I can figure out what this percentage is for
the total area under the curve, I have the percentage of the soda cans that are being
under-filled. So to calculate this, one way that we had
was the Empirical Rule, that 68-95-99.7 rule. But that only works if you're one center deviation
or two standard deviations away or three standard deviations away. I am not exactly one standard
deviation. I'm a little less than one standard deviation away. So I'm not going to be able
to use that Empirical Rule, other than an estimate for my answer to see if I'm in the
ballpark. What I need to look at is using a way to find
this area based on either a calculus technique-- which is beyond what I'm going to do this
video for-- or by using a table of values for a given "Z" score. So I have a table of
values here that I can use. But the first thing I need to do is calculate my actual
Z score. And so a Z score is the observed value minus the mean of the population over
the standard deviation. And so for our situation, the observed value
is going to be the under-filled amount, 12 ounces. So I'm going to subtract off the mean,
12.1, and divide by the standard deviation, .15. When I do that, I get .6 repeating, and
so I'm going to need to approximate this as .67.
Now for some tables, you may have to approximate even more. So we would have to either say
we're going to approximate it to .7 or figure out is it better to round down to .6.
So when I look at my table, I realize that the Z score-- so I'm looking for... that should
have been negative-- 12 minus 12.1 would have been negative.
So I'm going to be looking for something close to -.67. And I come down to the table and
say, well, I've got something that's kind of close. I have these two to kind of choose
from. So there's kind of one or two, a couple ways we can try to get a better approximation
here. One could be deciding which way should I round. Should I round it to -.7 or should
I round it to -.6. If it matters of over-counting or under-counting the percentage, that should
tell you which way to round. What I'm going to look at is because my .67
is in between those two values, I'm just going to average these two. Now if I was really
close to -.7 or really close to -.6, I might just use the value that's given there. But
because I'm kind of towards the middle, I'm just going to average these two percentages
to give me an estimated percentage of soda cans that are being under-filled.
So my percentage that are under-filled is going to be the average of 24.196 and 27.425.
And
this gives me an answer that's roughly 25.8 percent. So based on this graph for our normal
distribution, the percentage of the population of soda cans being filled with less than 12
ounces per can is about 25 percent of those.