Let's do another problem from the normal distribution
section of ck12.org's AP statistics book.
And I'm using theirs because it's Open Source and it's
actually quite a good book.
The problems are, I think, good practice for us.
So let's see, number 3.
You could go to their site and I think you
can download the book.
Assume that the mean eight of 1 year old girls in the U.S. is a
normally distributed-- or is normally distributed with the
mean of about 9.5 grams.
That's got to be kilograms.
I have a 10 month old son and he weighs about 20
pounds which is about 9 kilograms not 9.5 grams.
9.5 grams is nothing.
This would be talking about like mice or something.
This has got to be kilograms.
But anyway, it's about 9.5 kilograms with a
standard deviation of approximately 1.1 grams.
So the mean is equal to 9.5 kilograms I'm assuming and
the standard deviation is equal to 1.1 grams.
Without using a calculator-- so that's an interesting clue--
estimate the percentage of 1 year old girls in the U.S. that
meet the following conditions.
So when they say that without a calculator estimate that's a
big clue or a big giveaway that we're supposed to use
the empirical rule.
Empirical rule sometimes called the 68-95-99.7 rule.
And if you remember this is the name of the rule
you've essentially remembered the rule.
What that tells us that if we have a normal distribution--
I'll do a bit of a review here before we jump
into this problem.
If we have a normal distribution-- let me draw
a normal distribution.
It looks like that.
That's my normal distribution.
I didn't draw it perfectly but you get the idea.
It should be symmetrical.
This is our mean right there.
That's our mean.
If we go one standard deviation above the mean and one standard
deviation below the mean, so this is our mean plus
one standard deviation.
This is our mean minus one standard deviation.
The probability of finding a result if we're dealing with a
perfect normal distribution that's between one standard
deviation below the mean and one standard deviation above
the mean-- that would be this area-- and it would be,
you could guess, 68%.
68% chance you're going to get something within one standard
deviation of the mean.
Either a standard deviation below or above or
anywhere in between.
Now, if we're talking about two standard deviations around the
mean-- so if we go down another standard deviation, we go down
another standard deviation in that direction and another
standard deviation above the mean-- and we were to ask
ourselves what's the probability of finding
something within those two or within that range, then it's,
you could guess it, 95%.
And that includes this middle area right here.
So the 68% is a subset of that 95%.
And I think you know where this is going.
If we go three standard deviations below the mean and
above the mean, the empirical rule or the 68-95-99.7 rule
tells us that there is a 99.7% chance of finding a result in a
normal distribution that is within three standard
deviations of the mean.
So above three standard deviations below the mean
and below three standard deviation above the mean.
That's what the empirical rule tells us.
Now let's see if we can apply it to this problem.
So they gave us the mean and the standard deviation.
Let me draw that out.
Let me draw my axis first as best as I can.
That's my axis.
Let me draw my bell curve.
That's about as good as a bell curve as you can expect
a freehand drawer to do.
And the mean here is 9.-- and this should be symmetric.
This height should be the same as that height there.
I think you get the idea.
I'm not a computer.
9.5 is the mean.
I won't write the units.
It's all in kilograms.
One standard deviation above the mean we should add 1.1 to
that because they told us the standard deviation is 1.1.
That's going to be 10.6.
If we go-- let me just draw a little dotted line there-- 1
standard deviation below the mean we're going it subtract
1.1 from 9.5 and so that would be 8.4.
If we go two standard deviations above the mean
we would add another standard deviation here.
Right?
We went one standard deviations, two
standard deviations.
That would get us to 11.7.
And if we were to go three standard deviations
we'd add 1.1 again.
That would get us to 12.8.
Doing it on the other side, one standard deviation
below the mean is 8.4.
Two standard deviations below the mean-- subtract 1.1
again-- would be 7.3.
And then three standard deviations below the mean--
which we'd write there-- would be 6.2 kilograms.
So that's our set up for the problem.
So what's the probability that we would find a one year old
girl in the U.S. that weighs less than 8.4 kilograms.
Or maybe I should say whose mass is less
than 8.4 kilograms.
So if we look here, the probability of finding a baby
or a female baby who is one year old with a mass or a
weight of less than 8.4 kilograms, that's this
area right here.
I said mass because kilograms is actually a unit of mass.
Most people use it as weight as well.
So that's that area right there.
So how can we figure out that area under this
normal distribution using the empirical rule?
Well, we know what this area is.
We know what this area between minus one standard deviation
and plus one standard deviation is.
We know that is 68%.
And if that's 68% then that means in the parts that
aren't in that middle region you have 32%.
Because the area under the entire normal distribution is
100 or 100% or 1, depending on how you want to think about it.
Because you can't have-- well, all of the possibilities
combined can only add up to 1.
You can't have it more than 100% there.
So if you add up this leg and this leg-- so this plus that
leg is going to be the remainder.
So 100 minus 68, that's 32.
32%.
32% is if you add up this left leg and this
right leg over here.
And this is a perfect normal distribution.
They told us it's normally distributed.
So it's going to be perfectly symmetrical.
So if this side and that side add up to 32 but they're both
symmetrical, meaning they have the exact same area, then this
side right here-- I'll do it in pink-- this side right here--
it ended up looking more like purple-- would be 16%.
And this side right here would be 16%.
So your probability of getting a result more than one standard
deviation above the mean-- so that's this right hand
side, would be 16%.
Or the probability of having a result less than one standard
deviation below that mean, that's this right here, 16%.
So they want to know the probability of having a
baby at one years old less than 8.4 kilograms.
Less than 8.4 kilograms is this area right here.
And that's 16%.
So that's 16% for part a.
Let's do part b: between 7.3 and 11.7 point seven kilograms.
So between 7.3-- that's right there.
That's two standard deviations below the mean-- and 11.7, one,
two standard deviations above the mean.
So there's essentially asking us what's the probability of
getting a result within two standard deviations
of the mean, right?
This is the mean right here.
This is two standard deviations below.
This is two standard deviations above.
Well that's pretty straightforward.
The empirical rule tells us between two standard deviations
you have a 95% chance of getting a result that is within
two standard deviations.
So the empirical rule just gives us that answer.
And then finally, part c: the probability of having a one
year old U.S. a baby girl more than 12.8 kilograms.
So 12.8 kilograms is three standard deviations
above the mean.
So we want to know the probability of having a result
more than three deviations above the mean.
So that is this area way out there that I drew in orange.
Maybe I should do it in a different color to
really contrast it.
So it's this long tail out here, this little small area.
So what is that probability?
So let's turn back to our empirical rule.
Well we know the probability-- we know this area.
We know the area between minus three standard deviations and
plus three standard deviations.
We know this-- since this is last the last problem I can
color the whole thing in-- we know this area right here
between minus 3 and plus 3, that is it 99.7%.
The bulk of the results fall under there.
I mean, almost all of them.
So what do we have left over for the two tails?
Remember there are two tails.
This is one of them.
Then you have the results that are less than three standard
deviations below the mean.
This tail right there.
So that tells us that this, less than three standard
deviations below the mean and more than three standard
deviations above the mean combined have to be the rest.
Well the rest, there's only 0.3% percent for the rest.
And these two things are symmetrical.
They're going to be equal.
So this right here has to be half of this or 0.15% and
this right here is going to be 0.15%.
So the probability of having a one year old baby girl in the
U.S. that is more than 12.8 kilograms if you assume a
perfectly normal distribution is the area under this curve,
the area that is more than three standard deviations
above the mean.
And that is 0.15%.
Anyway, I hope you found that useful.