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- [Instructor] This, right over here,
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is a scratch pad on Khan Academy,
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created by Khan Academy
user Charlotte Auen.
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And, what you see here, is
a simulation that allows
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us to keep sampling from
our gumball machine,
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and start approximating
the sampling distribution
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of the sample proportion.
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So, her simulation focuses on
green gumballs, but we talked
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about yellow before, and
the yellow gumballs, we said
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60% were yellow, so let's
make 60% here green.
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And then, let's take samples of ten,
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just like we did before.
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And then, let's just
start with one sample.
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So, we're gonna draw one
sample, and what we wanna show,
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is we wanna show the percentages.
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Which if the proportion of
each sample, that are green.
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So, if we draw that first
sample, notice out of the ten,
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five ended up being green,
and then it plotted that right
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over here, under 50%.
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We have one situation where
50% were green, now let's do
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another sample, so this
sample 60% are green.
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And so, let's keep going.
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Let's draw another sample.
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And now that one, we have,
we have 50% are green,
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and so notice now we see
here on this distribution;
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two of them had 50% green.
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Now, we could keep drawing samples,
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and let's just really increase.
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So, we're gonna do 50
samples of ten at a time.
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And so, here, we can
quickly get to a fairly
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large number of samples.
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So here, we're over a thousand samples.
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And, what's interesting here,
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is we're seeing experimentally,
that our sample;
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the mean of our sample proportion here,
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is zero point six two.
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What we calculated, a few
minutes ago, was that it should
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be zero point six.
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We also see that the standard
deviation of our sample
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proportion, is zero point one six.
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And what we calculated was approximately
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zero point one five.
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And as we draw more and more
samples, we should get even
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closer, and closer to those values.
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And, we see that, for the most
part, we are getting closer,
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and closer, in fact,
now that it's rounded,
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we are at exactly those values,
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that we had calculated before.
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Now, one interesting thing to observe is,
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when your population proportion
is not too close to zero,
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and not too close to one,
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this looks pretty close
to a normal distribution.
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And that makes sense.
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Because, we saw the relation
between the sampling
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distribution of the sample proportion,
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and a binomial random variable.
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But, what if our population
proportion is closer to zero?
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So, let's say our population
proportion is ten percent.
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Zero point one.
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What do you think the
distribution is going
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to look like then?
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Well, we know that the mean
of our sampling distribution
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is going to be ten percent,
and so you could imagine
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that the distribution is
going to be right skewed.
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But, let's actually see that.
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So, here we see that our distribution
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is indeed, right skewed.
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And that makes sense.
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Because, you can only get
values from zero to one,
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and if your mean is closer
to zero, then you're gonna
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see the meat of your distribution
here, and then you're
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gonna see a long tail to the right.
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Which creates that right skew.
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And, if your population
proportion was close to one,
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well, you can imagine the
opposite is going to happen.
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You're going to end up with a left skew.
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And, we indeed, see right
over here, a left skew.
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Now, the other interesting
thing to appreciate is,
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the larger your samples, the
smaller the standard deviation.
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And so, let's do a population proportion
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that is right in-between.
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And so, here, this is similar
to what we saw before,
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this is looking roughly normal.
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But now, and that's when
we had sample size of ten,
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but, what if we have a
sample size of 50 every time?
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Well, notice, now it looks like
a much tighter distribution.
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This isn't even going
all the way to one yet,
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but it is a much tighter distribution.
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And, the reason why that made
sense, the standard deviation
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of your sample proportion,
it is inversely proportional
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to the square root of "n".
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And, so, that makes sense.
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So, hopefully you have a good
intuition now, for the sample
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proportion, it's distribution,
the sampling distribution
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of the sample proportion
that you can calculate it's
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mean, and its' standard deviation.
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And you feel good about it,
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because we saw it in a simulation.