WEBVTT

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Where we left off in the
last video I kind

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of gave you a question.

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Find an interval so that we're
reasonably confident-- we'll

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talk a little bit more about why
I have to give this kind

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of vague wording right here--
reasonably confident that

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there's a 95% chance that the
true population mean, which is

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p, which is the same thing as
the mean of the sampling

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distribution of the
sampling mean.

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So there's a 95% chance that
the true mean-- and

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let me put this here.

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This is also the same thing as
the mean of the sampling

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distribution of the sampling
mean is in that interval.

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And to do that let me just
throw out a few ideas.

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What is the probability that if
I take a sample and I were

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to take a mean of that sample,
so the probability that a

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random sample mean is within two
standard deviations of the

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sampling mean, of
our sample mean?

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So what is this probability
right over here?

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Let's just look at our
actual distribution.

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So this is our distribution,
this right here is our

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sampling mean.

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Maybe I should do it in
blue because that's

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the color up here.

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This is our sampling mean.

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And so what is the probability
that a random sampling mean is

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going to be two standard
deviations?

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Well a random sampling is a
sample from this distribution.

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It is a sample from the sampling
distribution of the

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sample mean.

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So it's literally what is the
probability of finding a

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sample within two standard
deviations of the mean?

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That's one standard deviation,
that's another standard

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deviation right over there.

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In general, if you haven't
committed this to memory

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already, it's not a bad thing
to commit to memory, is that

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if you have a normal
distribution the probability

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of taking a sample within two
standard deviations is 95--

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and if you want to get
a little bit more

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accurate it's 95.4%.

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But you could say it's roughly--
or maybe I could

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write it like this--
it's roughly 95%.

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And really that's all that
matters because we have this

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little funny language here
called reasonably confident,

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and we have to estimate the
standard deviation anyway.

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In fact, we could say if we
want, I could say that it's

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going to be exactly
equal to 95.4%.

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But in general, two standard
deviations, 95%, that's what

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people equate with each other.

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Now this statement is the
exact same thing as the

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probability that the sample
mean, that the sampling mean--

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not the sample mean, the
probability of the mean of the

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sampling distribution is within
two standard deviations

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of the sampling distribution of
x is also going to be the

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same number, is also going
to be equal to 95.4%.

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These are the exact
same statements.

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If x is within two standard
deviations of this, then this,

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then the mean, is within two
standard deviations of x.

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These are just two ways of
phrasing the same thing.

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Now we know that the mean of the
sampling distribution, the

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same thing as a mean of the
population distribution, which

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is the same thing as the
parameter p-- the proportion

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of people or the proportion of
the population that is a 1.

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So this right here is the same
thing as the population mean.

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So this statement right here
we can switch this with p.

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So the probability that p is
within two standard deviations

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of the sampling distribution
of x is 95.4%.

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Now we don't know what this
number right here is.

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But we have estimated it.

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Remember, our best estimate of
this is the true standard, or

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it is the true standard
deviation of the population

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divided by 10.

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We can estimate the true
standard deviation of the

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population with our sampling
standard deviation, which was

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0.5, 0.5 divided by 10.

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Our best estimate of the
standard deviation of the

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sampling distribution of the
sample mean is 0.05.

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So now we can say-- and I'll
switch colors-- the

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probability that the parameter
p, the proportion of the

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population saying 1, is within
two times-- remember, our best

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estimate of this right here is
0.05 of a sample mean that we

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take is equal to 95.4%.

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And so we could say the
probability that p is within 2

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times 0.05 is going to be equal
to-- 2.0 is going to be

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0.10 of our mean is equal to
95-- and actually let me be a

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little careful here.

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I can't say the equal now,
because over here if we knew

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this, if we knew this parameter
of the sampling

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distribution of the sample
mean, we could

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say that it is 95.4%.

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We don't know it.

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We are just trying to find our
best estimator for it.

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So actually what I'm going to
do here is actually just say

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is roughly-- and just to show
that we don't even have that

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level of accuracy, I'm going
to say roughly 95%.

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We're reasonably confident that
it's about 95% because

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we're using this estimator that
came out of our sample,

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and if the sample is really
skewed this is going to be a

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really weird number.

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So this is why we just have to
be a little bit more exact

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about what we're doing.

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But this is the tool
for at least saying

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how good is our result.

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So this is going to
be about 95%.

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Or we could say that the
probability that p is within

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0.10 of our sample mean
that we actually got.

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So what was the sample mean
that we actually got?

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It was 0.43.

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So if we're within 0.1 of 0.43,
that means we are within

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0.43 plus or minus 0.1 is
also, roughly, we're

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reasonably confident
it's about 95%.

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And I want to be very clear.

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Everything that I started all
the way from up here in brown

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to yellow and all this magenta,
I'm just restating

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the same thing inside of this.

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It became a little bit more
loosey-goosey once I went from

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the exact standard deviation of
the sampling distribution

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to an estimator for it.

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And that's why this is just
becoming-- I kind of put the

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squiggly equal signs there
to say we're reasonably

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confident-- and I even got rid
of some of the precision.

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But we just found
our interval.

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An interval that we can be
reasonably confident that

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there's a 95% probability that
p is within that, is going to

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be 0.43 plus or minus 0.1.

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Or an interval of-- we have
a confidence interval.

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We have a 95% confidence
interval of, and we could say,

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0.43 minus 0.1 is 0.33.

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If we write that as a percent
we could say 33% to-- and if

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we add the 0.1, 0.43 plus
0.1 we get 53%-- to 53%.

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So we are 95% confident.

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So we're not saying kind of
precisely that the probability

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of the actual proportion is 95%,
but we're 95% confident

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that the true proportion
is between 33% and 55%.

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That p is in this
range over here.

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Or another way, and you'll see
this in a lot of surveys that

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have been done, people will say
we did a survey and we got

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43% will vote for number one,
and number one in this case is

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candidate B.

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And then the other side, since
everyone else voted for

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candidate A, 57% will
vote for A.

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And then they're going to
put on margin of error.

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And you'll see this in any
survey that you see on TV.

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They'll put a margin of error.

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And the margin of error is just
another way of describing

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this confidence interval.

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And they'll say that the margin
of error in this case

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is 10%, which means that there's
a 95% confidence

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interval, if you go plus or
minus 10% from that value

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right over there.

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And I really want to emphasize,
you can't say with

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certainty that there is a 95%
chance that the true result

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will be within 10% of this,
because we had to estimate the

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standard deviation of
the sampling mean.

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But this is the best measure
we can with the information

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you have. If you're going to
do a survey of 100 people,

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this is the best kind of
confidence that we can get.

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And this number is actually
fairly big.

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So if you were to look at this
you would say, roughly there's

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a 95% chance that the true
value of this number is

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between 33% and 53%.

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So there's actually still a
chance that candidate B can

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win, even though only
43% of your 100 are

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going to vote for him.

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If you wanted to make it a
little bit more precise you

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would want to take
more samples.

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You can imagine.

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Instead of taking 100 samples,
instead of n being 100, if you

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made n equal 1,000, then you
would take this number over

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here, you would take this number
here and divide by the

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square root of 1,000 instead
of the square root of 100.

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So you'd be dividing
by 33 or whatever.

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And so then the size of the
standard deviation of your

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sampling distribution
will go down.

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And so the distance of two
standard deviations will be a

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smaller number, and so
then you will have a

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smaller margin of error.

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And maybe you want to get the
margin of error small enough

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so that you can figure out
decisively who's going to win

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the election.

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