WEBVTT

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Let's do another problem from
the normal distribution

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section of ck12.org's
AP statistics book.

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And I'm using theirs because
it's Open Source and it's

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actually quite a good book.

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The problems are, I think,
good practice for us.

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So let's see, number 3.

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You could go to their
site and I think you

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can download the book.

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Assume that the mean eight of 1
year old girls in the U.S. is a

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normally distributed-- or is
normally distributed with the

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mean of about 9.5 grams.

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That's got to be kilograms.

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I have a 10 month old son
and he weighs about 20

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pounds which is about 9
kilograms not 9.5 grams.

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9.5 grams is nothing.

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This would be talking about
like mice or something.

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This has got to be kilograms.

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But anyway, it's about
9.5 kilograms with a

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standard deviation of
approximately 1.1 grams.

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So the mean is equal to 9.5
kilograms I'm assuming and

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the standard deviation
is equal to 1.1 grams.

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Without using a calculator-- so
that's an interesting clue--

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estimate the percentage of 1
year old girls in the U.S. that

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meet the following conditions.

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So when they say that without a
calculator estimate that's a

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big clue or a big giveaway that
we're supposed to use

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the empirical rule.

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Empirical rule sometimes
called the 68-95-99.7 rule.

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And if you remember this
is the name of the rule

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you've essentially
remembered the rule.

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What that tells us that if we
have a normal distribution--

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I'll do a bit of a review
here before we jump

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into this problem.

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If we have a normal
distribution-- let me draw

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a normal distribution.

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It looks like that.

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That's my normal distribution.

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I didn't draw it perfectly
but you get the idea.

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It should be symmetrical.

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This is our mean right there.

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That's our mean.

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If we go one standard deviation
above the mean and one standard

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deviation below the mean,
so this is our mean plus

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one standard deviation.

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This is our mean minus
one standard deviation.

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The probability of finding a
result if we're dealing with a

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perfect normal distribution
that's between one standard

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deviation below the mean and
one standard deviation above

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the mean-- that would be this
area-- and it would be,

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you could guess, 68%.

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68% chance you're going to get
something within one standard

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deviation of the mean.

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Either a standard deviation
below or above or

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anywhere in between.

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Now, if we're talking about two
standard deviations around the

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mean-- so if we go down another
standard deviation, we go down

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another standard deviation in
that direction and another

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standard deviation above the
mean-- and we were to ask

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ourselves what's the
probability of finding

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something within those two or
within that range, then it's,

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you could guess it, 95%.

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And that includes this
middle area right here.

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So the 68% is a
subset of that 95%.

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And I think you know
where this is going.

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If we go three standard
deviations below the mean and

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above the mean, the empirical
rule or the 68-95-99.7 rule

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tells us that there is a 99.7%
chance of finding a result in a

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normal distribution that is
within three standard

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deviations of the mean.

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So above three standard
deviations below the mean

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and below three standard
deviation above the mean.

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That's what the empirical
rule tells us.

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Now let's see if we can
apply it to this problem.

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So they gave us the mean and
the standard deviation.

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Let me draw that out.

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Let me draw my axis
first as best as I can.

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That's my axis.

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Let me draw my bell curve.

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That's about as good as a
bell curve as you can expect

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a freehand drawer to do.

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And the mean here is 9.-- and
this should be symmetric.

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This height should be the
same as that height there.

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I think you get the idea.

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I'm not a computer.

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9.5 is the mean.

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I won't write the units.

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It's all in kilograms.

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One standard deviation above
the mean we should add 1.1 to

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that because they told us the
standard deviation is 1.1.

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That's going to be 10.6.

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If we go-- let me just draw a
little dotted line there-- 1

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standard deviation below the
mean we're going it subtract

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1.1 from 9.5 and so
that would be 8.4.

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If we go two standard
deviations above the mean

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we would add another
standard deviation here.

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Right?

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We went one standard
deviations, two

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standard deviations.

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That would get us to 11.7.

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And if we were to go three
standard deviations

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we'd add 1.1 again.

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That would get us to 12.8.

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Doing it on the other side,
one standard deviation

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below the mean is 8.4.

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Two standard deviations below
the mean-- subtract 1.1

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again-- would be 7.3.

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And then three standard
deviations below the mean--

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which we'd write there--
would be 6.2 kilograms.

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So that's our set up
for the problem.

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So what's the probability that
we would find a one year old

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girl in the U.S. that weighs
less than 8.4 kilograms.

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Or maybe I should say
whose mass is less

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than 8.4 kilograms.

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So if we look here, the
probability of finding a baby

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or a female baby who is one
year old with a mass or a

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weight of less than 8.4
kilograms, that's this

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area right here.

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I said mass because kilograms
is actually a unit of mass.

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Most people use it
as weight as well.

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So that's that
area right there.

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So how can we figure out
that area under this

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normal distribution using
the empirical rule?

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Well, we know what
this area is.

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We know what this area between
minus one standard deviation

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and plus one standard
deviation is.

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We know that is 68%.

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And if that's 68% then that
means in the parts that

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aren't in that middle
region you have 32%.

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Because the area under the
entire normal distribution is

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100 or 100% or 1, depending on
how you want to think about it.

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Because you can't have-- well,
all of the possibilities

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combined can only add up to 1.

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You can't have it more
than 100% there.

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So if you add up this leg and
this leg-- so this plus that

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leg is going to be
the remainder.

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So 100 minus 68, that's 32.

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32%.

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32% is if you add up
this left leg and this

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right leg over here.

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And this is a perfect
normal distribution.

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They told us it's
normally distributed.

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So it's going to be
perfectly symmetrical.

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So if this side and that side
add up to 32 but they're both

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symmetrical, meaning they have
the exact same area, then this

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side right here-- I'll do it in
pink-- this side right here--

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it ended up looking more
like purple-- would be 16%.

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And this side right
here would be 16%.

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So your probability of getting
a result more than one standard

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deviation above the mean-- so
that's this right hand

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side, would be 16%.

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Or the probability of having a
result less than one standard

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deviation below that mean,
that's this right here, 16%.

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So they want to know the
probability of having a

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baby at one years old
less than 8.4 kilograms.

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Less than 8.4 kilograms
is this area right here.

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And that's 16%.

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So that's 16% for part a.

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Let's do part b: between 7.3
and 11.7 point seven kilograms.

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So between 7.3--
that's right there.

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That's two standard deviations
below the mean-- and 11.7, one,

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two standard deviations
above the mean.

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So there's essentially asking
us what's the probability of

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getting a result within two
standard deviations

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of the mean, right?

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This is the mean right here.

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This is two standard
deviations below.

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This is two standard
deviations above.

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Well that's pretty
straightforward.

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The empirical rule tells us
between two standard deviations

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you have a 95% chance of
getting a result that is within

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two standard deviations.

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So the empirical rule just
gives us that answer.

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And then finally, part c: the
probability of having a one

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year old U.S. a baby girl
more than 12.8 kilograms.

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So 12.8 kilograms is three
standard deviations

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above the mean.

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So we want to know the
probability of having a result

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more than three deviations
above the mean.

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So that is this area way out
there that I drew in orange.

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Maybe I should do it in
a different color to

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really contrast it.

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So it's this long tail out
here, this little small area.

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So what is that probability?

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So let's turn back to
our empirical rule.

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Well we know the probability--
we know this area.

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We know the area between minus
three standard deviations and

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plus three standard deviations.

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We know this-- since this is
last the last problem I can

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color the whole thing in-- we
know this area right here

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between minus 3 and plus
3, that is it 99.7%.

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The bulk of the results
fall under there.

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I mean, almost all of them.

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So what do we have left
over for the two tails?

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Remember there are two tails.

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This is one of them.

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Then you have the results that
are less than three standard

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deviations below the mean.

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This tail right there.

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So that tells us that this,
less than three standard

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deviations below the mean and
more than three standard

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deviations above the mean
combined have to be the rest.

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Well the rest, there's only
0.3% percent for the rest.

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And these two things
are symmetrical.

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They're going to be equal.

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So this right here has to be
half of this or 0.15% and

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this right here is
going to be 0.15%.

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So the probability of having a
one year old baby girl in the

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U.S. that is more than 12.8
kilograms if you assume a

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perfectly normal distribution
is the area under this curve,

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the area that is more than
three standard deviations

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above the mean.

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And that is 0.15%.

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Anyway, I hope you
found that useful.