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It never hurts to get
a bit more practice.

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So this is problem number five
from the normal distribution

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chapter from ck12.org's
AP statistics FlexBook.

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So they're saying, the 2007 AP
statistics examination scores

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were not normally distributed
with a mean of 2.8

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and a standard
deviation of 1.34.

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They cite some College
Board stuff here.

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I didn't copy and paste that.

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What is the approximate z-score?

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Remember, z-score
is just how many

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standard deviations you
are away from the mean.

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What is the approximate
z-score that

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corresponds to an
exam score of 5?

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So we really just
have to figure out--

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this is a pretty
straightforward problem.

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We just need to figure out how
many standard deviations is

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5 from the mean?

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Well, you just take
5 minus 2.8, right?

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The mean is 2.8.

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Let me be very
clear, mean is 2.8.

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They give us that.

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Didn't even have
to calculate it.

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So the mean is 2.8.

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So 5 minus 2.8 is equal to 2.2.

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So we're 2.2 above the mean.

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And if we want that in terms
of standard deviations,

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we just divide by our
standard deviation.

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You divide by 1.34.

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Divide by 1.34.

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I'll take out the
calculator for this.

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So we have 2.2 divided
by 1.34 is equal to 1.64.

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So this is equal to 1.64.

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And that's choice C. So this was
actually very straightforward.

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We just have to see how far
away we are from the mean

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if we get a score of
5-- which hopefully you

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will get if you're
taking the AP statistics

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exam after watching
these videos.

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And then you divide by the
standard deviation to say,

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how many standard deviations
away from the mean

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is the score of 5?

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It's 1.64.

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I think the only tricky
thing here might have been,

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you might have been tempted
to pick choice E, which says,

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the z-score cannot be calculated
because the distribution is not

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normal.

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And I think the reason why you
might have had that temptation

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is because we've
been using z-scores

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within the context of
a normal distribution.

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But a z-score literally
just means how many

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standard deviations you
are away from the mean.

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It could apply to
any distribution

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that you could calculate a mean
and a standard deviation for.

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So E is not the correct answer.

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A z-score can apply to a
non-normal distribution.

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So the answer is C. And I
guess that's a good point

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of clarification to
get out of the way.

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And I thought I would do
two problems in this video,

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just because that
one was pretty short.

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So problem number six.

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The height of fifth grade
boys in the United States

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is approximately
normally distributed--

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that's good to know-- with
a mean height of 143.5

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centimeters.

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So it's a mean of
143.5 centimeters

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and a standard deviation
of about 7.1 centimeters.

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What is the probability that
a randomly chosen fifth grade

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boy would be taller
than 157.7 centimeters?

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So let's just draw
out this distribution

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like we've done in a
bunch of problems so far.

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They're just asking
us one question,

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so we can mark this
distribution up a good bit.

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Let's say that's
our distribution.

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And the mean here, the
mean they told us is 143.5.

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They're asking us
taller than 157.7.

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So we're going in the
upwards direction.

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So one standard
deviation above the mean

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will take us right there.

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And we just have to add 7.1
to this number right here.

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We're going up by 7.1.

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So 143.5 plus 7.1 is what?

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150.6.

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That's one standard deviation.

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If we were to go another
standard deviation,

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we'd go 7.1 more.

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What's 7.1 plus 150.6?

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It's 157.7, which
just happens to be

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the exact number they ask for.

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They're asking for
the probability

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of getting a height
higher than that.

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So they want to know, what's
the probability that we

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fall under this area right here?

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Or essentially more than
two standard deviations

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from the mean.

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Or above two
standard deviations.

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We can't count this
left tail right there.

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So we can use the
empirical rule.

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If we do our standard
deviations to the left,

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that's one standard deviation,
two standard deviations.

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We know what this whole area is.

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Let me pick a different
color so that I don't.

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So we know what this
area is, the area

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within two standard deviations.

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The empirical rule tells us.

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Or even better, the
68, 95, 99.7 rule

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tells us that this
area-- because it's

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within two standard
deviations-- is 95%, or 0.95.

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Or it's 95% of the area under
the normal distribution.

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Which tells us that what's
left over-- this tail

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that we care about and
this left tail right here--

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has to make up the
rest of it, or 5%.

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So those two combined
have to be 5%.

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And these are symmetrical.

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We've done this before.

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This is actually a little
redundant from other problems

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we've done.

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But if these are added, combined
5%, and they're the same,

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then each of these are 2.5%.

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Each of these are 2.5%.

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So the answer to
the question, what

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is the probability that a
randomly chosen fifth grade boy

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would be taller then
157.7 centimeters.

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Well, that's literally
just the area

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under this right green part.

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Maybe I'll do it in
a different color.

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This magenta part that
I'm coloring right now.

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That's just that area.

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We just figured out it's 2.5%.

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So there's a 2.5% chance we'd
randomly find a fifth grade

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boy who's taller than
157.7 centimeters,

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assuming this is the mean,
the standard deviation,

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and we are dealing with
a normal distribution.