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Sampling distribution of sample proportion part 2 | AP Statistics | Khan Academy

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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.
Title:
Sampling distribution of sample proportion part 2 | AP Statistics | Khan Academy
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
04:34

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