Normal conditions for sampling distributions of sample proportions

WEBVTT
Kind: captions
Language: en

00:00:00.876 --> 00:00:02.277
- [Instructor] What we're
going to do in this video

00:00:02.277 --> 00:00:04.496
is think about under which conditions

00:00:04.496 --> 00:00:08.856
does the sampling distribution
of the sample portions,

00:00:08.856 --> 00:00:11.617
in which situations does
it look roughly normal,

00:00:11.617 --> 00:00:16.576
and under which situations
does it look skewed right

00:00:16.576 --> 00:00:18.568
so does it look something like this,

00:00:18.568 --> 00:00:22.335
and under which situations
does it look skewed left

00:00:22.335 --> 00:00:24.688
maybe something like that.

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And the conditions that
we're going to talk about,

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and this is a rough rule of thumb,

00:00:28.811 --> 00:00:31.071
that if we take our sample size

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and we multiply it by
the population proportion

00:00:34.553 --> 00:00:38.720
that we care about and that
is greater than or equal to 10

00:00:39.781 --> 00:00:42.038
and if we take the sample size

00:00:42.038 --> 00:00:45.617
and we multiply it times one
minus the population proportion

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and that also is greater
than or equal to 10,

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if both of these are true,

00:00:49.877 --> 00:00:50.917
the rule of thumb tells us

00:00:50.917 --> 00:00:55.036
that this is going to be
approximately normal in shape,

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the sampling distribution
of the sample proportions.

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So with that in our minds,
let's do some examples here.

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So this first example says,

00:01:06.349 --> 00:01:08.211
Emiliana runs a restaurant

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that receives a shipment
of 50 tangerines every day.

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According to the supplier,

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approximately 12% of the population

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of these tangerines is overripe.

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Suppose that Emiliana calculates

00:01:23.177 --> 00:01:26.282
the daily proportion
of overripe tangerines

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in her sample of 50.

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We can assume the supplier's claim is true

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and that the tangerines each
day represent a random sample.

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What will be the shape of
the sampling distribution,

00:01:40.884 --> 00:01:43.796
what will be the shape of
the sampling distribution

00:01:43.796 --> 00:01:47.734
of the daily proportions
of overripe tangerines?

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Pause this video, think about
what we just talked about

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and see if you can answer this.

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All right, so right over here,

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we're getting daily
samples of 50 tangerines.

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So for this particular example,

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our n is equal to 50

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and our population proportion,

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the proportion that is
overripe is 12% so p is 0.12.

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So if we take n times p, what do we get?

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NP is equal to 50 times 0.12,

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well 100 times this would be 12

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so 50 times this is
going to be equal to six

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and this is less than or equal to 10.

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So this immediately violates
this first condition

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and so we know that we're
not going to be dealing

00:02:33.632 --> 00:02:36.062
with a normal distribution.

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And so the question is, how
is it going to be skewed?

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And the key realization is remember,

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the mean of the sample proportions

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or the sampling distribution
of the sample proportions

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or the mean of the sampling distribution

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of the daily proportions

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that that's going to be the same thing

00:02:51.520 --> 00:02:56.059
as our population proportion
so the mean is going to be 12%.

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So if I were to draw it,

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let me see if I were to
draw it right over here

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where this is 50% and this is 100%,

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our mean is gonna be
right over here at 12%

00:03:06.102 --> 00:03:09.099
and so you're gonna have
it really high over there

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and then it's gonna be
skewed to the right.

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You're gonna have a big long tail.

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So this is going to be
skewed to the right.

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Let's do another example.

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So here we're told, according
to a Nielsen survey,

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radio reaches 88% of children each week.

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Suppose we took weekly random samples

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of n equals 125 children
from this population

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and computed the proportion of children

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in each sample whom radio reaches.

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What will be the shape of
the sampling distribution

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of the proportions of
children the radio reaches?

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Once again, pause this video

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and see if you can figure it out.

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All right, well let's just
figure out what n and p are.

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Our sample size here n is equal to 125

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and our population proportion

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of the proportion of children

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that are reached each week
by radio is 88% so p is 0.88.

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So now let's calculate np
so n is 125 times p is 0.88

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and is this going to be
greater than or equal to 10.

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Well, we don't even have
to calculate this exactly.

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This is almost 90% of 125.

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This is actually going to be over 100

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so it for sure is going
to be greater than 10

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so we meet this first condition.

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But what about the second condition?

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We could take n 125 times one minus p

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so this is times 0.12

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so this is 12% of 125.

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Well, even 10% of 125 would be 12.5

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so 12% is for sure going
to be greater than that

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so this too is going
to be greater than 10.

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I didn't even have to calculate it.

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I could just estimate it

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and so we meet that second condition.

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So even though our population
proportion is quite high,

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it's quite close to one here,

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because our sample size is so large,

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it still will be roughly normal

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and one way to get the
intuition for that is

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so this is a proportion of zero,

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let's say this is 50% and this is 100%,

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so our mean right over
here is gonna be 0.88

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for our sampling distribution
of the sample proportions.

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If we had a low sample size,

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then our standard deviation
would be quite large

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and so then you would end up

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with a left skewed distribution.

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But we saw before the
higher your sample size,

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the smaller your standard deviation

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for the sampling distribution

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and so what that does is it tightens up,

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it tightens up the standard deviation

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and so it's going to look more normal.

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It's gonna look closer to being normal.

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So we'll say approximately normal

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because it met our conditions
for this rule of thumb.

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Is it gonna be perfectly normal?

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No.

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In fact, if we didn't
have this rule of thumb

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to draw the line, some might even argue

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that we still have a
longer tail to the left

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than we do to the right,
maybe it's skewed to the left,

00:06:07.750 --> 00:06:10.622
but using this threshold,
using this rule of thumb

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which is the standard in statistics,

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this would be viewed as
approximately normal.


Introduction to sampling distributions

WEBVTT
Kind: captions
Language: en

00:00:00.242 --> 00:00:01.150
- [Instructor] What we're
gonna do in this video

00:00:01.150 --> 00:00:05.317
is talk about the idea of
a sampling distribution.

00:00:06.716 --> 00:00:09.525
Now, just to make things
a little bit concrete,

00:00:09.525 --> 00:00:13.325
let's imagine that we have
a population of some kind.

00:00:13.325 --> 00:00:15.317
Let's say it's a bunch of balls,

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each of them have a number written on it.

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For that population, we
could calculate parameters.

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So, a parameter you could view as a truth

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about that population.

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We've covered this in other videos.

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So for example, you could
have the population mean,

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the mean of the numbers
written on top of that ball.

00:00:32.833 --> 00:00:35.242
You could have the population
standard deviation.

00:00:35.242 --> 00:00:37.236
You could have the proportion of balls

00:00:37.236 --> 00:00:38.635
that are even, whatever,

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these are all population parameters.

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Now we know from many other videos

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that you might not know
the population parameter

00:00:46.350 --> 00:00:48.308
or might not even be easy to find,

00:00:48.308 --> 00:00:50.408
and so the way that we try to estimate

00:00:50.408 --> 00:00:53.917
a population parameter
is by taking a sample,

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so this right over here is
a sample size of size n.

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Sample of size n.

00:01:01.792 --> 00:01:05.442
And then we can calculate a
statistic from that sample,

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based on that sample, maybe
we picked n balls from there.

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And so from that, we can
calculate a statistic

00:01:13.025 --> 00:01:17.092
that is used to estimate this parameter.

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But we know that this is a
random sample right over here,

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so every time we take a sample,

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the statistic that we
calculate for that sample

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is not necessarily going to be the same

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as the population parameter.

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In fact, if we were to take a
random sample of size n again

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and then we were to calculate
the statistic again,

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we could very well get a different value.

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So, these are all going to be
estimates of this parameter.

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And so an interesting question

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is what is the distribution of the values

00:01:46.608 --> 00:01:48.483
that I could get for the statistics?

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What is the frequency with
which I can get different values

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for the statistic that is trying
to estimate this parameter?

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And that distribution is what
a sampling distribution is.

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So let's make this even a
little bit more concrete.

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Let's imagine where our population,

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I'm gonna make this a very simple example.

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Let's say our population
has three balls in it.

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One, two, three,

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and they're numbered, one, two, and three.

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And it's very easy to calculate.

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Let's say the parameter that
we care about right over here

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is the population mean,

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and that of course is gonna
be one plus two plus three,

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all of that over three,

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which is six divided
by three which is two.

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So, that is our population parameter.

00:02:33.500 --> 00:02:35.858
But let's say that we
wanted to take samples,

00:02:35.858 --> 00:02:38.050
let's say samples of two balls at a time

00:02:38.050 --> 00:02:40.150
and every time we take a
ball, we'll replace it.

00:02:40.150 --> 00:02:43.742
So each ball we take, it
is an independent pick.

00:02:43.742 --> 00:02:46.958
And we're gonna use those
samples of two balls at a time

00:02:46.958 --> 00:02:50.800
in order to estimate the population mean.

00:02:50.800 --> 00:02:54.967
So for example, this could be
our first sample of size two

00:02:55.991 --> 00:02:59.308
and let's say in that
first sample, I pick a one

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and let's say I pick a two.

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Well then I can calculate
the sample statistic here.

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In this case, it would be the sample mean

00:03:06.717 --> 00:03:09.092
which is used to estimate
the population mean.

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And for this sample of
two, it's going to be 1.5.

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Then I can do it again.

00:03:16.041 --> 00:03:19.708
And let's say I get a
one and I get a three.

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Well now, when I
calculate the sample mean,

00:03:23.850 --> 00:03:27.208
the average of one and three
or the mean of one and three

00:03:27.208 --> 00:03:28.892
is going to be equal to two.

00:03:28.892 --> 00:03:30.333
Let's think about all of
the different scenarios

00:03:30.333 --> 00:03:31.317
of samples we can get

00:03:31.317 --> 00:03:33.950
and what the associated
sample means are going to be.

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And then we can see the frequency

00:03:35.375 --> 00:03:37.467
of getting those sample means.

00:03:37.467 --> 00:03:40.725
And so, let me draw a
little bit of a table here.

00:03:40.725 --> 00:03:43.475
So, make a table right over here.

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And let's see, these are
the numbers that we pick

00:03:48.117 --> 00:03:49.408
and remember, when we pick one ball,

00:03:49.408 --> 00:03:51.933
we'll record that number,
then we'll put it back in,

00:03:51.933 --> 00:03:53.158
and then we'll pick another ball.

00:03:53.158 --> 00:03:54.575
So these are going to
be independent events

00:03:54.575 --> 00:03:56.483
and it's gonna be with replacement.

00:03:56.483 --> 00:03:59.766
And so, let's say we could
pick a one and then a one.

00:03:59.766 --> 00:04:02.675
We could pick a one, then
a two, a one and a three.

00:04:02.675 --> 00:04:04.742
We could pick a two and then a one.

00:04:04.742 --> 00:04:07.867
We could pick a two and
a two, a two and a three.

00:04:07.867 --> 00:04:09.758
We could pick a three and a one,

00:04:09.758 --> 00:04:12.692
a three and a two or a three and a three.

00:04:12.692 --> 00:04:14.592
There's three possible
balls for the first pick

00:04:14.592 --> 00:04:16.042
and three possible balls for the second

00:04:16.042 --> 00:04:17.925
'cause we're doing it with replacement.

00:04:17.925 --> 00:04:21.209
And so, what is the sample
mean in each of these

00:04:21.209 --> 00:04:23.312
for all of these combinations?

00:04:23.312 --> 00:04:25.467
So for this one, the sample mean is one.

00:04:25.467 --> 00:04:26.800
Here, it is 1.5.

00:04:28.017 --> 00:04:30.033
Here, it is two.

00:04:30.033 --> 00:04:32.050
Here, it is 1.5.

00:04:32.050 --> 00:04:33.583
Here, it is two.

00:04:33.583 --> 00:04:34.916
Here, it is 2.5.

00:04:35.758 --> 00:04:37.442
Here, it is two.

00:04:37.442 --> 00:04:38.775
Here, it is 2.5.

00:04:40.167 --> 00:04:43.042
And then here, it is three.

00:04:43.042 --> 00:04:45.550
And so, we can now plot the frequencies

00:04:45.550 --> 00:04:49.008
of these possible sample
means that we can get

00:04:49.008 --> 00:04:52.217
and that plot will be
a sampling distribution

00:04:52.217 --> 00:04:53.983
of the sample means.

00:04:53.983 --> 00:04:55.641
So let's do that.

00:04:55.641 --> 00:04:58.700
So, we make a little chart right over,

00:04:58.700 --> 00:05:00.667
a little graph right over here.

00:05:00.667 --> 00:05:03.917
So these are the possible sample means.

00:05:06.533 --> 00:05:09.484
We can get a one, we can get a 1.5,

00:05:09.484 --> 00:05:12.750
we can get a two, we can get a 2.5

00:05:12.750 --> 00:05:14.892
or we can get a three.

00:05:14.892 --> 00:05:17.292
And now let's see the frequency of it.

00:05:17.292 --> 00:05:19.459
I will put that over here.

00:05:21.358 --> 00:05:23.525
And so let's see, how many ones

00:05:23.525 --> 00:05:27.375
out of our nine possibilities
we have, how many were one?

00:05:27.375 --> 00:05:30.358
Well, only one of the
sample means was one,

00:05:30.358 --> 00:05:32.658
and so the relative frequency,
if we just set the number,

00:05:32.658 --> 00:05:35.042
we could make this line go up one

00:05:35.042 --> 00:05:37.375
or we could just say, "Hey,
this is going to be one

00:05:37.375 --> 00:05:39.558
"out of the nine possibilities."

00:05:39.558 --> 00:05:42.117
And so let me just make that.

00:05:42.117 --> 00:05:44.041
I'll call this right over here.

00:05:44.041 --> 00:05:45.958
This is 1/9.

00:05:47.342 --> 00:05:49.675
Now, what about 1.5?

00:05:49.675 --> 00:05:51.975
Let's see, there's one,

00:05:51.975 --> 00:05:54.792
two of these possibilities out of nine.

00:05:54.792 --> 00:05:57.775
So, 1.5, it would look like this.

00:05:57.775 --> 00:06:00.942
This right over here is two over nine.

00:06:02.080 --> 00:06:03.783
And now, what about two?

00:06:03.783 --> 00:06:07.875
Well, we can see there's one, two, three.

00:06:07.875 --> 00:06:11.733
So three out of the nine
possibilities, we got a two.

00:06:11.733 --> 00:06:14.616
So we could say this is two
or we could say this is 3/9,

00:06:14.616 --> 00:06:17.175
which is the same thing of course as 1/3.

00:06:17.175 --> 00:06:20.967
So this right over here
is three over nine.

00:06:20.967 --> 00:06:22.967
And then what about 2.5?

00:06:23.933 --> 00:06:26.108
Well, there's two 2.5's,

00:06:26.108 --> 00:06:28.375
so two out of the nine times.

00:06:28.375 --> 00:06:29.733
Another way you can interpret this

00:06:29.733 --> 00:06:34.050
is when you take a random
sample with replacement

00:06:34.050 --> 00:06:37.850
of two balls, you have a 2/9 chance

00:06:37.850 --> 00:06:40.433
of having a sample mean of 2.5.

00:06:41.883 --> 00:06:45.042
And then last but not
least, right over here,

00:06:45.042 --> 00:06:47.067
there's one scenario out of the nine

00:06:47.067 --> 00:06:50.033
where you get two three's or 1/9.

00:06:50.033 --> 00:06:51.775
And so this right over here,

00:06:51.775 --> 00:06:54.608
this is the sampling distribution,

00:06:55.667 --> 00:06:57.500
sampling distribution,

00:06:58.967 --> 00:07:00.550
for the sample mean

00:07:06.100 --> 00:07:07.433
for n equals two

00:07:10.185 --> 00:07:12.352
or for sample size of two.


Sampling distribution of sample proportion part 1

WEBVTT
Kind: captions
Language: en

00:00:00.308 --> 00:00:03.078
- [Instructor] So I have a
gumball machine right over here.

00:00:03.078 --> 00:00:07.440
It has yellow, and green,
and pink, and blue gumballs.

00:00:07.440 --> 00:00:10.061
Let me throw a few blue ones in there.

00:00:10.061 --> 00:00:12.705
And what we're going to
concern ourselves in this video

00:00:12.705 --> 00:00:14.791
are the yellow gumballs.

00:00:14.791 --> 00:00:17.283
And let's say that we
know that the proportion

00:00:17.283 --> 00:00:20.116
of yellow gumballs over here is P.

00:00:21.422 --> 00:00:24.025
This right over here is a population,

00:00:24.025 --> 00:00:25.775
population parameter.

00:00:29.488 --> 00:00:32.244
And for the sake of argument,
just to make things concrete,

00:00:32.244 --> 00:00:36.894
let's just say that 60% of
the gumballs are yellow,

00:00:36.894 --> 00:00:38.144
or 0.6 of them.

00:00:39.467 --> 00:00:43.210
Now let's review some things
that we have seen before.

00:00:43.210 --> 00:00:45.854
I'm gonna define our
Bernoulli random variable,

00:00:45.854 --> 00:00:49.854
let's call this capital
Y, which is equal to one

00:00:51.027 --> 00:00:54.989
if when we take one random
gumball out of that machine

00:00:54.989 --> 00:00:57.906
we get a, we pick a yellow gumball,

00:00:59.717 --> 00:01:03.930
and it's equal to zero if when
we pick one random gumball

00:01:03.930 --> 00:01:06.812
out of that machine we don't pick yellow,

00:01:06.812 --> 00:01:07.979
so not yellow.

00:01:10.222 --> 00:01:13.099
From previous videos we
know some interesting things

00:01:13.099 --> 00:01:15.712
about this Bernoulli random variable.

00:01:15.712 --> 00:01:17.202
We know its mean.

00:01:17.202 --> 00:01:20.478
We know the mean of our
Bernoulli random variable.

00:01:20.478 --> 00:01:22.837
It's going to be the
proportion of yellow balls

00:01:22.837 --> 00:01:27.105
in this population, so it's
going to be equal to P,

00:01:27.105 --> 00:01:31.197
which in this particular
case we know is 0.6,

00:01:31.197 --> 00:01:32.741
and we know what the standard deviation

00:01:32.741 --> 00:01:35.161
of this Bernoulli random variable is.

00:01:35.161 --> 00:01:39.072
It is going to be P times one minus P,

00:01:39.072 --> 00:01:40.136
actually that's the variance.

00:01:40.136 --> 00:01:41.227
We want to take the square root of that

00:01:41.227 --> 00:01:42.832
to get the standard deviation.

00:01:42.832 --> 00:01:45.099
So in this particular scenario,

00:01:45.099 --> 00:01:49.266
that's going to be the
square root of 0.6 times 0.4.

00:01:52.495 --> 00:01:53.328
Fair enough.

00:01:53.328 --> 00:01:54.826
This is all review so far.

00:01:54.826 --> 00:01:56.899
But now let me define,

00:01:56.899 --> 00:02:00.037
let me define another random variable X,

00:02:00.037 --> 00:02:03.704
which is equal to the
sum of 10 independent,

00:02:07.510 --> 00:02:10.177
independent trials, trials of Y.

00:02:16.210 --> 00:02:18.967
Now, we have seen random
variables like this before.

00:02:18.967 --> 00:02:21.840
This is a binomial random variable.

00:02:21.840 --> 00:02:25.238
Now, what do we know about its
mean and standard deviation?

00:02:25.238 --> 00:02:28.416
Well, in previous videos we
know that the mean of this

00:02:28.416 --> 00:02:32.345
binomial random variable is
just going to be equal to

00:02:32.345 --> 00:02:36.345
n times the mean of each
of the Bernoulli trials

00:02:37.493 --> 00:02:39.993
right over here, so n times P,

00:02:40.984 --> 00:02:44.260
which in this particular
situation is going to be,

00:02:44.260 --> 00:02:48.427
n is 10, we're doing 10
trials, and P is zero, 0.6,

00:02:50.479 --> 00:02:51.956
which is equal to six.

00:02:51.956 --> 00:02:53.080
And that makes sense.

00:02:53.080 --> 00:02:55.310
If 60% of the balls here are yellow

00:02:55.310 --> 00:02:58.133
and if you were to take a
sample or if you were to take

00:02:58.133 --> 00:03:01.301
10 trials, so if you were to
take ten balls one at a time,

00:03:01.301 --> 00:03:03.784
they have to be independent,
so you keep looking at them

00:03:03.784 --> 00:03:05.623
and then replacing them,
but if you took 10,

00:03:05.623 --> 00:03:08.186
then you would expect that
six of them would be yellow.

00:03:08.186 --> 00:03:10.273
They're not always going to be six yellow,

00:03:10.273 --> 00:03:12.920
but that would be maybe
what you would expect.

00:03:12.920 --> 00:03:15.350
All right, now what's the
standard deviation here?

00:03:15.350 --> 00:03:18.369
So our standard deviation is equal to,

00:03:18.369 --> 00:03:20.406
and we have proved this in other videos,

00:03:20.406 --> 00:03:21.647
it's equal to the square root

00:03:21.647 --> 00:03:24.314
of n times P, times one minus P.

00:03:26.661 --> 00:03:28.330
Notice you just put an n right over here

00:03:28.330 --> 00:03:29.836
under the radical sign.

00:03:29.836 --> 00:03:33.157
And so this is going to get us
in this particular situation

00:03:33.157 --> 00:03:35.240
to 10 times 0.6 times 0.4

00:03:39.214 --> 00:03:41.795
and then the square root of everything.

00:03:41.795 --> 00:03:43.271
So all of this is review.

00:03:43.271 --> 00:03:46.794
If it's unfamiliar, I
encourage you to review

00:03:46.794 --> 00:03:49.673
some of the videos on
Bernoulli random variables

00:03:49.673 --> 00:03:52.113
and on binomial random variables.

00:03:52.113 --> 00:03:54.011
But what we're going to do in this video

00:03:54.011 --> 00:03:56.672
is think about a sampling distribution

00:03:56.672 --> 00:03:58.598
and it's going to be the
sampling distribution

00:03:58.598 --> 00:04:02.340
for a sample statistic known
as the sample proportion,

00:04:02.340 --> 00:04:04.279
which we actually talked
about when we first introduced

00:04:04.279 --> 00:04:05.951
sampling distributions.

00:04:05.951 --> 00:04:08.608
So let's say, so let's
just park all of this,

00:04:08.608 --> 00:04:11.511
this is background right over here.

00:04:11.511 --> 00:04:14.177
Let's start taking samples of 10.

00:04:14.177 --> 00:04:15.759
And I didn't pick that randomly.

00:04:15.759 --> 00:04:17.685
I want to make it
reconcile with what we do

00:04:17.685 --> 00:04:19.374
with our random variable here.

00:04:19.374 --> 00:04:23.211
And so let's take a sample of 10 gumballs

00:04:23.211 --> 00:04:27.978
and let's calculate the
proportion that are yellow.

00:04:27.978 --> 00:04:31.114
And so we will call that
our sample proportion,

00:04:31.114 --> 00:04:33.494
and I might as well
just do that in yellow.

00:04:33.494 --> 00:04:35.627
So we want to calculate
the sample proportion

00:04:35.627 --> 00:04:36.875
that are yellow.

00:04:36.875 --> 00:04:39.659
And what is this equivalent to?

00:04:39.659 --> 00:04:42.299
Well, you could say, Well
this is equivalent to

00:04:42.299 --> 00:04:43.703
my random variable X.

00:04:43.703 --> 00:04:47.022
I want to count the number
of gumballs that are yellow

00:04:47.022 --> 00:04:50.440
and then I'm gonna divide
it by the sample size.

00:04:50.440 --> 00:04:52.776
So I'm gonna divide it by n.

00:04:52.776 --> 00:04:56.948
And in this case, it
would be X divided by 10.

00:04:56.948 --> 00:04:58.198
I know what some of you are thinking.

00:04:58.198 --> 00:04:59.034
Wait, wait, wait.

00:04:59.034 --> 00:05:00.280
Hold on for a second.

00:05:00.280 --> 00:05:02.530
X is sum of 10 independent,

00:05:03.422 --> 00:05:05.709
independent trials right over here.

00:05:05.709 --> 00:05:08.745
To be independent, you
can't just take 10 gumballs.

00:05:08.745 --> 00:05:10.127
You have to take them one at a time

00:05:10.127 --> 00:05:12.184
and then replace them
back in order for them

00:05:12.184 --> 00:05:14.236
to be truly independent.

00:05:14.236 --> 00:05:17.136
But remember, we have our 10% rule.

00:05:17.136 --> 00:05:21.102
We have our 10% rule, which
tells us that if a sample

00:05:21.102 --> 00:05:24.564
is less than or equal to
10% of the population,

00:05:24.564 --> 00:05:28.138
that you can treat each of
the gumballs in this situation

00:05:28.138 --> 00:05:29.937
as being independent.

00:05:29.937 --> 00:05:31.869
So let's just say for the sake of argument

00:05:31.869 --> 00:05:35.301
that there are 10,000 gumballs in here,

00:05:35.301 --> 00:05:36.832
and so we can feel pretty good

00:05:36.832 --> 00:05:38.803
that these samples,
that each of the things

00:05:38.803 --> 00:05:41.820
in the sample are
independent of each other

00:05:41.820 --> 00:05:43.725
by our 10% rule.

00:05:43.725 --> 00:05:45.808
And so each of these are,

00:05:47.531 --> 00:05:50.125
each of these 10 gumballs what we see,

00:05:50.125 --> 00:05:51.882
they are going to be independent,

00:05:51.882 --> 00:05:54.711
I'm going to put them in
quotes, by the 10% rule.

00:05:54.711 --> 00:05:58.178
And so in that situation,
we can make this claim

00:05:58.178 --> 00:06:02.553
or we can feel good that
this claim is roughly true.

00:06:02.553 --> 00:06:06.106
And so let's say for that
first sample that we do,

00:06:06.106 --> 00:06:09.273
our sample proportion is equal to 0.3,

00:06:11.286 --> 00:06:13.637
so three of our gumballs just happened,

00:06:13.637 --> 00:06:16.657
three of our 10 gumballs
just happened to be yellow.

00:06:16.657 --> 00:06:18.490
Then we'd do it again.

00:06:18.490 --> 00:06:20.002
So we take another sample.

00:06:20.002 --> 00:06:22.974
We calculate our sample
proportion, this statistic again,

00:06:22.974 --> 00:06:25.655
remember, it's trying to estimate
our population parameter,

00:06:25.655 --> 00:06:29.234
and let's say that time it
happens to be seven out of 10,

00:06:29.234 --> 00:06:30.739
and we just keep doing that.

00:06:30.739 --> 00:06:35.505
And if we keep doing that,
and we plot it on a dot chart

00:06:35.505 --> 00:06:37.557
or a dot distribution
I guess we could say,

00:06:37.557 --> 00:06:40.664
where our possible outcomes
you could have zero out of 10,

00:06:40.664 --> 00:06:45.391
one out of 10, two, three,
four, five, so half of them,

00:06:45.391 --> 00:06:49.248
six, seven, eight, nine, 10,
so that would be all of them.

00:06:49.248 --> 00:06:53.716
And so you could plot okay,
0.3 is one, two, three,

00:06:53.716 --> 00:06:56.007
that's one scenario where I got,

00:06:56.007 --> 00:06:59.469
where my sample proportion is 0.3.

00:06:59.469 --> 00:07:03.088
Then 0.7, that's one
situation where I got a 0.7.

00:07:03.088 --> 00:07:06.333
And let's say I were to
take another sample of 10

00:07:06.333 --> 00:07:10.223
and I were to get 0.7, then
you would plot that over here.

00:07:10.223 --> 00:07:13.356
And if you kept taking
samples and kept calculating

00:07:13.356 --> 00:07:16.485
these sample proportions and
you kept plotting it here,

00:07:16.485 --> 00:07:19.315
you would get a better, and
better, and better approximation

00:07:19.315 --> 00:07:23.482
for the sampling distribution
of the sampling proportion.

00:07:25.516 --> 00:07:29.021
But how can we actually
characterize the true sampling

00:07:29.021 --> 00:07:32.071
distribution for the sample proportion?

00:07:32.071 --> 00:07:36.674
What is going to be the mean
of this sampling distribution

00:07:36.674 --> 00:07:39.873
and what is going to be
the standard deviation?

00:07:39.873 --> 00:07:43.717
Well, we can derive that from
what we see right over here.

00:07:43.717 --> 00:07:47.197
The mean of our sampling distribution

00:07:47.197 --> 00:07:51.187
of our sample proportion is
just going to be equal to

00:07:51.187 --> 00:07:55.581
the mean of our random
variable X divided by n.

00:07:55.581 --> 00:07:59.877
It's just going to be the
mean of X divided by n,

00:07:59.877 --> 00:08:01.853
which is equal to what?

00:08:01.853 --> 00:08:04.089
Well, the mean of X is n times P.

00:08:04.089 --> 00:08:06.308
This is n times P.

00:08:06.308 --> 00:08:09.798
You divided it by n,
you're going to get P.

00:08:09.798 --> 00:08:11.396
And that makes sense.

00:08:11.396 --> 00:08:13.261
One way to think about
it, the expected value

00:08:13.261 --> 00:08:15.598
for your sample proportion is going to be

00:08:15.598 --> 00:08:18.256
the proportion of gumballs
that you actually see.

00:08:18.256 --> 00:08:21.068
And so this also a good indicator
that this is going to be

00:08:21.068 --> 00:08:23.501
a reasonably unbiased estimator.

00:08:23.501 --> 00:08:26.084
Now let's think about
the standard deviation

00:08:26.084 --> 00:08:28.828
for our sample proportion.

00:08:28.828 --> 00:08:31.830
Well, we can just view that
as our standard deviation

00:08:31.830 --> 00:08:36.186
of our binomial random
variable X divided by n.

00:08:36.186 --> 00:08:39.809
So this is going to be
equal to the square root of

00:08:39.809 --> 00:08:43.892
n times P, times one minus
P, all of that over n,

00:08:45.031 --> 00:08:47.242
which is the same thing as if we put this,

00:08:47.242 --> 00:08:49.855
if we divide by n inside the radical,

00:08:49.855 --> 00:08:51.468
it'd be the same thing as the square root

00:08:51.468 --> 00:08:54.718
of nP times one minus P over n squared.

00:08:57.276 --> 00:08:59.978
Divide the numerator and denominator by n,

00:08:59.978 --> 00:09:04.145
you will get the square
root of P times one minus P,

00:09:05.770 --> 00:09:07.353
all of that over n.

00:09:08.294 --> 00:09:10.797
And so in this particular situation,

00:09:10.797 --> 00:09:13.611
where our parameter is 0.6,

00:09:13.611 --> 00:09:16.098
where our population parameter is 0.6,

00:09:16.098 --> 00:09:20.350
so it's going to be 0.6,
that's the true proportion

00:09:20.350 --> 00:09:21.835
for our population.

00:09:21.835 --> 00:09:24.410
And then what would out
standard deviation be

00:09:24.410 --> 00:09:26.577
for our sample proportion?

00:09:27.445 --> 00:09:30.879
Well, it's going to be
equal to the square root

00:09:30.879 --> 00:09:34.046
of 0.6 times 0.4, all of that over 10.

00:09:36.006 --> 00:09:38.491
And we can get a calculator
out to calculate that.

00:09:38.491 --> 00:09:42.658
So if we take 0.6 times 0.4
equals, divided by 10, equals,

00:09:45.716 --> 00:09:47.844
and then we take the square root of that,

00:09:47.844 --> 00:09:50.761
and we get it's approximately 0.15.

00:09:52.824 --> 00:09:54.407
Approximately 0.15.


Sampling distribution of sample proportion part 2

WEBVTT
Kind: captions
Language: en

00:00:00.470 --> 00:00:01.557
- [Instructor] This, right over here,

00:00:01.557 --> 00:00:03.823
is a scratch pad on Khan Academy,

00:00:03.823 --> 00:00:07.956
created by Khan Academy
user Charlotte Auen.

00:00:07.956 --> 00:00:11.250
And, what you see here, is
a simulation that allows

00:00:11.250 --> 00:00:13.868
us to keep sampling from
our gumball machine,

00:00:13.868 --> 00:00:16.949
and start approximating
the sampling distribution

00:00:16.949 --> 00:00:18.718
of the sample proportion.

00:00:18.718 --> 00:00:22.628
So, her simulation focuses on
green gumballs, but we talked

00:00:22.628 --> 00:00:25.208
about yellow before, and
the yellow gumballs, we said

00:00:25.208 --> 00:00:29.375
60% were yellow, so let's
make 60% here green.

00:00:31.059 --> 00:00:33.193
And then, let's take samples of ten,

00:00:33.193 --> 00:00:34.931
just like we did before.

00:00:34.931 --> 00:00:37.146
And then, let's just
start with one sample.

00:00:37.146 --> 00:00:39.207
So, we're gonna draw one
sample, and what we wanna show,

00:00:39.207 --> 00:00:40.334
is we wanna show the percentages.

00:00:40.334 --> 00:00:43.453
Which if the proportion of
each sample, that are green.

00:00:43.453 --> 00:00:47.009
So, if we draw that first
sample, notice out of the ten,

00:00:47.009 --> 00:00:50.495
five ended up being green,
and then it plotted that right

00:00:50.495 --> 00:00:52.047
over here, under 50%.

00:00:52.047 --> 00:00:55.461
We have one situation where
50% were green, now let's do

00:00:55.461 --> 00:00:59.321
another sample, so this
sample 60% are green.

00:00:59.321 --> 00:01:01.597
And so, let's keep going.

00:01:01.597 --> 00:01:03.080
Let's draw another sample.

00:01:03.080 --> 00:01:07.135
And now that one, we have,
we have 50% are green,

00:01:07.135 --> 00:01:10.272
and so notice now we see
here on this distribution;

00:01:10.272 --> 00:01:13.474
two of them had 50% green.

00:01:13.474 --> 00:01:15.973
Now, we could keep drawing samples,

00:01:15.973 --> 00:01:18.440
and let's just really increase.

00:01:18.440 --> 00:01:22.555
So, we're gonna do 50
samples of ten at a time.

00:01:22.555 --> 00:01:25.400
And so, here, we can
quickly get to a fairly

00:01:25.400 --> 00:01:26.883
large number of samples.

00:01:26.883 --> 00:01:29.431
So here, we're over a thousand samples.

00:01:29.431 --> 00:01:31.048
And, what's interesting here,

00:01:31.048 --> 00:01:34.551
is we're seeing experimentally,
that our sample;

00:01:34.551 --> 00:01:36.806
the mean of our sample proportion here,

00:01:36.806 --> 00:01:38.688
is zero point six two.

00:01:38.688 --> 00:01:41.925
What we calculated, a few
minutes ago, was that it should

00:01:41.925 --> 00:01:43.442
be zero point six.

00:01:43.442 --> 00:01:46.813
We also see that the standard
deviation of our sample

00:01:46.813 --> 00:01:49.314
proportion, is zero point one six.

00:01:49.314 --> 00:01:51.068
And what we calculated was approximately

00:01:51.068 --> 00:01:52.727
zero point one five.

00:01:52.727 --> 00:01:55.043
And as we draw more and more
samples, we should get even

00:01:55.043 --> 00:01:57.734
closer, and closer to those values.

00:01:57.734 --> 00:02:01.971
And, we see that, for the most
part, we are getting closer,

00:02:01.971 --> 00:02:03.727
and closer, in fact,
now that it's rounded,

00:02:03.727 --> 00:02:05.466
we are at exactly those values,

00:02:05.466 --> 00:02:07.718
that we had calculated before.

00:02:07.718 --> 00:02:09.915
Now, one interesting thing to observe is,

00:02:09.915 --> 00:02:13.602
when your population proportion
is not too close to zero,

00:02:13.602 --> 00:02:15.157
and not too close to one,

00:02:15.157 --> 00:02:18.591
this looks pretty close
to a normal distribution.

00:02:18.591 --> 00:02:19.790
And that makes sense.

00:02:19.790 --> 00:02:22.451
Because, we saw the relation
between the sampling

00:02:22.451 --> 00:02:24.532
distribution of the sample proportion,

00:02:24.532 --> 00:02:27.682
and a binomial random variable.

00:02:27.682 --> 00:02:32.163
But, what if our population
proportion is closer to zero?

00:02:32.163 --> 00:02:36.369
So, let's say our population
proportion is ten percent.

00:02:36.369 --> 00:02:37.698
Zero point one.

00:02:37.698 --> 00:02:38.969
What do you think the
distribution is going

00:02:38.969 --> 00:02:40.540
to look like then?

00:02:40.540 --> 00:02:44.139
Well, we know that the mean
of our sampling distribution

00:02:44.139 --> 00:02:46.566
is going to be ten percent,
and so you could imagine

00:02:46.566 --> 00:02:49.320
that the distribution is
going to be right skewed.

00:02:49.320 --> 00:02:51.363
But, let's actually see that.

00:02:51.363 --> 00:02:54.196
So, here we see that our distribution

00:02:54.196 --> 00:02:56.196
is indeed, right skewed.

00:02:57.244 --> 00:02:58.535
And that makes sense.

00:02:58.535 --> 00:03:02.414
Because, you can only get
values from zero to one,

00:03:02.414 --> 00:03:05.869
and if your mean is closer
to zero, then you're gonna

00:03:05.869 --> 00:03:07.902
see the meat of your distribution
here, and then you're

00:03:07.902 --> 00:03:09.617
gonna see a long tail to the right.

00:03:09.617 --> 00:03:12.441
Which creates that right skew.

00:03:12.441 --> 00:03:16.748
And, if your population
proportion was close to one,

00:03:16.748 --> 00:03:18.487
well, you can imagine the
opposite is going to happen.

00:03:18.487 --> 00:03:21.137
You're going to end up with a left skew.

00:03:21.137 --> 00:03:25.220
And, we indeed, see right
over here, a left skew.

00:03:27.162 --> 00:03:30.196
Now, the other interesting
thing to appreciate is,

00:03:30.196 --> 00:03:34.599
the larger your samples, the
smaller the standard deviation.

00:03:34.599 --> 00:03:38.317
And so, let's do a population proportion

00:03:38.317 --> 00:03:40.105
that is right in-between.

00:03:40.105 --> 00:03:43.972
And so, here, this is similar
to what we saw before,

00:03:43.972 --> 00:03:46.688
this is looking roughly normal.

00:03:46.688 --> 00:03:49.459
But now, and that's when
we had sample size of ten,

00:03:49.459 --> 00:03:54.368
but, what if we have a
sample size of 50 every time?

00:03:54.368 --> 00:03:58.535
Well, notice, now it looks like
a much tighter distribution.

00:04:01.519 --> 00:04:03.632
This isn't even going
all the way to one yet,

00:04:03.632 --> 00:04:06.718
but it is a much tighter distribution.

00:04:06.718 --> 00:04:09.858
And, the reason why that made
sense, the standard deviation

00:04:09.858 --> 00:04:13.568
of your sample proportion,
it is inversely proportional

00:04:13.568 --> 00:04:15.736
to the square root of "n".

00:04:15.736 --> 00:04:17.309
And, so, that makes sense.

00:04:17.309 --> 00:04:20.337
So, hopefully you have a good
intuition now, for the sample

00:04:20.337 --> 00:04:23.486
proportion, it's distribution,
the sampling distribution

00:04:23.486 --> 00:04:26.685
of the sample proportion
that you can calculate it's

00:04:26.685 --> 00:04:30.188
mean, and its' standard deviation.

00:04:30.188 --> 00:04:31.360
And you feel good about it,

00:04:31.360 --> 00:04:33.341
because we saw it in a simulation.


Biased and unbiased estimators from sampling distributions examples

WEBVTT
Kind: captions
Language: en

00:00:00.415 --> 00:00:02.599
- [Instructor] Alejandro
was curious if sample median

00:00:02.599 --> 00:00:06.352
was an unbiased estimator
of population median.

00:00:06.352 --> 00:00:10.237
He placed ping pong balls
numbered from zero to 32

00:00:10.237 --> 00:00:12.185
in a drum and mixed them well.

00:00:12.185 --> 00:00:16.753
Note that the median of
the population is 16.

00:00:16.753 --> 00:00:20.565
He then took a random
sample of five balls,

00:00:20.565 --> 00:00:23.157
and calculated the median of the sample.

00:00:23.157 --> 00:00:25.333
He replaced the balls,
and repeated this process

00:00:25.333 --> 00:00:28.073
for a total of 50 trials.

00:00:28.073 --> 00:00:31.974
His results are summarized
in the dotplot below

00:00:31.974 --> 00:00:34.806
where each dot represents
the sample median

00:00:34.806 --> 00:00:37.889
from a sample of five balls.

00:00:37.889 --> 00:00:40.471
So this right over here
is a sampling distribution

00:00:40.471 --> 00:00:42.610
of the sample median.

00:00:42.610 --> 00:00:44.973
You have a population of balls.

00:00:44.973 --> 00:00:46.280
We know a parameter.

00:00:46.280 --> 00:00:50.498
We know that the median
of the population is 16.

00:00:50.498 --> 00:00:53.888
This right over here is the population.

00:00:53.888 --> 00:00:57.369
And then he keeps taking
samples of five balls,

00:00:57.369 --> 00:01:00.942
so that is five-ball
sample right over there.

00:01:00.942 --> 00:01:02.859
And then he calculates the statistic,

00:01:02.859 --> 00:01:04.362
the median of the sample,

00:01:04.362 --> 00:01:07.998
so he calculates the median
there, and he does it again.

00:01:07.998 --> 00:01:10.270
He calculates the median,
and he does it again.

00:01:10.270 --> 00:01:11.548
He calculates the median.

00:01:11.548 --> 00:01:13.860
And then he does this 50 times.

00:01:13.860 --> 00:01:15.079
He does this 50 times.

00:01:15.079 --> 00:01:19.387
And so, what you see plotted
over here, do you see 50 dots?

00:01:19.387 --> 00:01:21.807
Every time he took a sample of five balls,

00:01:21.807 --> 00:01:23.168
and calculated the median,

00:01:23.168 --> 00:01:25.514
we see that as one of the dots here.

00:01:25.514 --> 00:01:29.773
So for example, four times,
he calculated a median of 20.

00:01:29.773 --> 00:01:32.554
Two times, he calculated a median of 19.

00:01:32.554 --> 00:01:35.896
One time, he calculated a median of four.

00:01:35.896 --> 00:01:38.608
Now they ask us, based on these results,

00:01:38.608 --> 00:01:42.182
does the sample median
appear to be a biased

00:01:42.182 --> 00:01:46.591
or unbiased estimator of
the population median?

00:01:46.591 --> 00:01:48.656
So pause this video, and
see if you can come up

00:01:48.656 --> 00:01:50.855
with an answer to that.

00:01:50.855 --> 00:01:53.023
In order to be an unbiased estimator,

00:01:53.023 --> 00:01:55.262
and that is what you want
your estimator to be,

00:01:55.262 --> 00:01:56.995
you want it to be unbiased,

00:01:56.995 --> 00:02:00.353
the sampling distribution
for that statistic

00:02:00.353 --> 00:02:02.576
has to be evenly distributed

00:02:02.576 --> 00:02:06.142
about the true parameter
for the population.

00:02:06.142 --> 00:02:08.052
So for example, in this situation,

00:02:08.052 --> 00:02:09.960
the true parameter for the population

00:02:09.960 --> 00:02:14.081
is a note that the median
of the population is 16,

00:02:14.081 --> 00:02:15.978
so that's the thing that
you're trying to estimate.

00:02:15.978 --> 00:02:19.582
So this is where the median
of the population is.

00:02:19.582 --> 00:02:22.657
And so, if we look at the
sampling distribution,

00:02:22.657 --> 00:02:25.236
we see that it is roughly balanced

00:02:25.236 --> 00:02:28.862
to the left and to the right
of that true parameter.

00:02:28.862 --> 00:02:32.224
So I would say that this
is looking pretty unbiased

00:02:32.224 --> 00:02:35.210
because the sampling distribution

00:02:35.210 --> 00:02:38.245
is evenly balanced to
the left and the right

00:02:38.245 --> 00:02:42.558
of the true median of the
population, that true parameter.

00:02:42.558 --> 00:02:43.503
If, for some reason,

00:02:43.503 --> 00:02:46.501
the distribution looked
something like this,

00:02:46.501 --> 00:02:49.440
then it would not be unbiased.

00:02:49.440 --> 00:02:53.555
Or if the distribution
looked something like this,

00:02:53.555 --> 00:02:55.485
if the distribution looked like this,

00:02:55.485 --> 00:02:57.787
if our sampling
distribution was like this,

00:02:57.787 --> 00:03:00.739
this would mean that we're
consistently underestimating

00:03:00.739 --> 00:03:02.216
the true parameter.

00:03:02.216 --> 00:03:04.763
And if our distribution looked like this,

00:03:04.763 --> 00:03:08.558
this means that our
statistic is overestimating

00:03:08.558 --> 00:03:10.557
the true parameter.

00:03:10.557 --> 00:03:13.221
Let's do another example.

00:03:13.221 --> 00:03:15.211
The dotplots below show an approximation

00:03:15.211 --> 00:03:17.119
to the sampling distribution

00:03:17.119 --> 00:03:19.035
for three different estimators

00:03:19.035 --> 00:03:22.317
of the same population parameter.

00:03:22.317 --> 00:03:26.975
If the actual value of the
population parameter is five,

00:03:26.975 --> 00:03:31.760
which dotplot displays the
estimator with both low bias

00:03:31.760 --> 00:03:33.427
and low variability?

00:03:34.828 --> 00:03:35.790
Pause this video,

00:03:35.790 --> 00:03:38.448
and see if you can come
up with the answer.

00:03:38.448 --> 00:03:40.181
So the population parameter

00:03:40.181 --> 00:03:42.501
that we're trying to estimate is five.

00:03:42.501 --> 00:03:46.349
So let's show that on each
of these distributions,

00:03:46.349 --> 00:03:50.730
these approximations of
the sampling distributions.

00:03:50.730 --> 00:03:51.728
Over here, we have our scale.

00:03:51.728 --> 00:03:56.357
This should be one, two,
three, four, five, six, seven,

00:03:56.357 --> 00:04:00.712
and one, two, three,
four, five, six, seven.

00:04:00.712 --> 00:04:04.281
And so, our population parameter is five,

00:04:04.281 --> 00:04:06.747
so that is right over there,

00:04:06.747 --> 00:04:09.104
that is right over there,

00:04:09.104 --> 00:04:11.854
and that this is right over here.

00:04:15.188 --> 00:04:18.178
Which of these statistics
seem to be biased?

00:04:18.178 --> 00:04:19.196
Let's just start there.

00:04:19.196 --> 00:04:20.029
Pause this video.

00:04:20.029 --> 00:04:22.732
Which of these look biased to you?

00:04:22.732 --> 00:04:26.317
The one that looks clearly
biased is statistic C.

00:04:26.317 --> 00:04:29.069
When you look at its
sampling distribution,

00:04:29.069 --> 00:04:32.616
it is consistently to the
left of our true parameter,

00:04:32.616 --> 00:04:36.264
so it's consistently
underestimating our parameter,

00:04:36.264 --> 00:04:39.489
so this one right over here is biased.

00:04:39.489 --> 00:04:41.648
Now statistic A and statistic B

00:04:41.648 --> 00:04:44.324
both look reasonably unbiased.

00:04:44.324 --> 00:04:45.894
Remember, these are approximations

00:04:45.894 --> 00:04:47.385
of the sampling distribution.

00:04:47.385 --> 00:04:48.763
If you want to get closer and closer

00:04:48.763 --> 00:04:49.985
to the sampling distribution,

00:04:49.985 --> 00:04:51.501
the true sampling distribution,

00:04:51.501 --> 00:04:52.995
you would keep taking samples,

00:04:52.995 --> 00:04:55.045
and keep calculating the statistic,

00:04:55.045 --> 00:04:58.461
and keep plotting them on this
distribution right over here.

00:04:58.461 --> 00:05:01.129
But for statistic A, it
looks reasonably balanced

00:05:01.129 --> 00:05:03.548
to the left and the right.

00:05:03.548 --> 00:05:05.842
It's not perfectly balanced
but it's reasonable.

00:05:05.842 --> 00:05:07.316
Remember, these are approximations.

00:05:07.316 --> 00:05:09.722
And statistic B is also
reasonably balanced

00:05:09.722 --> 00:05:11.360
to the left and the right.

00:05:11.360 --> 00:05:14.116
Now they don't just say
which one has a low bias

00:05:14.116 --> 00:05:16.375
but they also say low variability.

00:05:16.375 --> 00:05:19.088
I would say both statistic
A and statistic B

00:05:19.088 --> 00:05:21.494
have relatively low bias

00:05:21.494 --> 00:05:23.712
but if we talk about low variability,

00:05:23.712 --> 00:05:26.166
that's how much spread do they have,

00:05:26.166 --> 00:05:28.131
especially around the true parameter.

00:05:28.131 --> 00:05:31.824
And it looks like statistic
A has much lower spread

00:05:31.824 --> 00:05:34.279
than statistic B right over here.

00:05:34.279 --> 00:05:36.367
In statistic B, you've
calculated statistics

00:05:36.367 --> 00:05:39.588
that are too way from the true parameter

00:05:39.588 --> 00:05:42.463
while in statistic A, you've
calculated a statistic

00:05:42.463 --> 00:05:46.174
of certain samples that
are roughly half away

00:05:46.174 --> 00:05:48.147
from the true parameter.

00:05:48.147 --> 00:05:52.098
So the statistic that has both
low bias and low variability,

00:05:52.098 --> 00:05:54.265
that would be statistic A.


Give to Khan Academy today!

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.420
Hi, I'm Sal Khan, founder of Khan Academy,

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Introduction to sampling distributions

WEBVTT
Kind: captions
Language: en

00:00:00.484 --> 00:00:03.283
- [Instructor] Let's say I have
a bag of colored balls here

00:00:03.283 --> 00:00:07.829
and we know that 40% of
the balls are orange.

00:00:07.829 --> 00:00:11.329
Now, imagine defining a random variable x.

00:00:12.192 --> 00:00:14.350
And x is based on a trial

00:00:14.350 --> 00:00:16.459
where we stick our hand in this bag,

00:00:16.459 --> 00:00:19.417
we don't look around, and
we randomly pick a ball,

00:00:19.417 --> 00:00:21.267
look at its color, we record it,

00:00:21.267 --> 00:00:22.467
and then we're gonna put it back.

00:00:22.467 --> 00:00:24.833
So we're gonna assume that
we have our placement here,

00:00:24.833 --> 00:00:27.300
and we're gonna say that
our random variable x

00:00:27.300 --> 00:00:31.467
is going to be equal to one
if we pick an orange ball

00:00:34.325 --> 00:00:38.075
and it's going to be
equal to zero otherwise.

00:00:38.950 --> 00:00:40.683
You might already recognize this

00:00:40.683 --> 00:00:43.317
as a Bernoulli random variable.

00:00:43.317 --> 00:00:47.158
And we can construct a
probability distribution for x.

00:00:47.158 --> 00:00:49.433
In fact, let's do this.

00:00:49.433 --> 00:00:52.450
So, x is going to be discrete.

00:00:52.450 --> 00:00:55.942
It can only take on two different values.

00:00:55.942 --> 00:00:59.859
So x can take on zero
or it can take on one,

00:01:00.925 --> 00:01:03.325
and if 40% of the balls are orange,

00:01:03.325 --> 00:01:06.267
the one has a 40% chance of happening,

00:01:06.267 --> 00:01:07.883
so let me do that.

00:01:07.883 --> 00:01:12.050
So there's gonna be 40%
chance of getting a one.

00:01:13.517 --> 00:01:16.184
So, that's 0.4 right over there.

00:01:17.133 --> 00:01:20.983
And there would be a 60%
chance or 0.6 probability

00:01:20.983 --> 00:01:23.083
of getting a zero.

00:01:23.083 --> 00:01:25.150
So this right over here,

00:01:25.150 --> 00:01:27.225
just trying to hand-draw it.

00:01:27.225 --> 00:01:31.392
So this would be 0.6
probability of getting a zero.

00:01:32.342 --> 00:01:36.017
So we can call this the
probability distribution,

00:01:36.017 --> 00:01:38.600
probability distribution for x.

00:01:41.633 --> 00:01:44.958
This is all review so far,
but the reason why I did this

00:01:44.958 --> 00:01:47.983
is because we're now
going to introduce ourself

00:01:47.983 --> 00:01:51.583
to the notion of a sampling distribution.

00:01:51.583 --> 00:01:54.257
And it can be a little bit
confusing because in our brain,

00:01:54.257 --> 00:01:56.858
we tend to think in terms
of probability distributions

00:01:56.858 --> 00:02:01.258
and not as much in terms
of sampling distributions.

00:02:01.258 --> 00:02:03.392
So what you do in a sampling distribution

00:02:03.392 --> 00:02:07.642
is you still start with a population here,

00:02:07.642 --> 00:02:11.192
but then you take a
sample of that population.

00:02:11.192 --> 00:02:12.817
Let me label things.

00:02:12.817 --> 00:02:15.758
So this is our population.

00:02:15.758 --> 00:02:17.867
This is our population here.

00:02:17.867 --> 00:02:19.200
We take a sample

00:02:22.975 --> 00:02:24.900
from the population,

00:02:24.900 --> 00:02:26.592
and it could have a certain sample size,

00:02:26.592 --> 00:02:27.759
sample size n.

00:02:30.300 --> 00:02:35.017
And then we'll calculate some
statistic for that sample,

00:02:35.017 --> 00:02:37.934
so we will calculate the statistic.

00:02:42.517 --> 00:02:44.733
And then we're gonna think
about the distributions

00:02:44.733 --> 00:02:48.450
of the statistics that we
can get from these samples.

00:02:48.450 --> 00:02:50.950
So one way to think about
this is keep doing this.

00:02:50.950 --> 00:02:53.417
So this is our first
sample of sample size n,

00:02:53.417 --> 00:02:55.158
we calculate the statistics.

00:02:55.158 --> 00:02:58.217
Then we take another
sample of sample size n

00:02:58.217 --> 00:03:01.025
and then we calculate the statistic again.

00:03:01.025 --> 00:03:02.792
And then we take another sample

00:03:02.792 --> 00:03:04.517
and we just keep doing this.

00:03:04.517 --> 00:03:07.092
We take another sample of sample size n

00:03:07.092 --> 00:03:10.792
and we calculate the statistic again.

00:03:10.792 --> 00:03:14.125
And let's say we were to do
this an infinite number of times

00:03:14.125 --> 00:03:17.700
and we were to plot the
distribution of the statistic

00:03:17.700 --> 00:03:18.917
that we're calculating.

00:03:18.917 --> 00:03:22.042
Well then, we have our
sampling distribution.

00:03:22.042 --> 00:03:23.850
Let's try to make this a
little bit more tangible

00:03:23.850 --> 00:03:28.333
by going back to our colored
balls example, and calculate

00:03:28.333 --> 00:03:31.400
or think about a sampling
distribution for that.

00:03:31.400 --> 00:03:34.733
So let's say we have our population here

00:03:37.400 --> 00:03:41.483
and we know that a parameter
for this population,

00:03:42.408 --> 00:03:45.225
we know that the proportion
of balls that are orange.

00:03:45.225 --> 00:03:46.892
40% are orange.

00:03:47.908 --> 00:03:49.533
We don't always know the parameters.

00:03:49.533 --> 00:03:51.758
Oftentimes, we're
estimating the parameters

00:03:51.758 --> 00:03:53.792
by looking at samples.

00:03:53.792 --> 00:03:57.475
But let's say we then
take sample sizes of 10.

00:03:57.475 --> 00:03:59.058
So, sample size 10,

00:04:02.725 --> 00:04:06.892
and every time we calculate
the statistic for our sample

00:04:07.800 --> 00:04:09.892
of what percentage are orange.

00:04:09.892 --> 00:04:13.508
So let's say the first
time we take a sample,

00:04:13.508 --> 00:04:17.008
this time over here, we get three oranges.

00:04:19.875 --> 00:04:24.100
Let's say the next time
we get two oranges.

00:04:24.100 --> 00:04:25.375
Actually, let me do these as a proportion.

00:04:25.375 --> 00:04:29.825
So if my sample size is
10, I get three oranges,

00:04:29.825 --> 00:04:31.325
which is 30%.

00:04:32.700 --> 00:04:36.450
And then if I do it
again, I get two oranges,

00:04:37.291 --> 00:04:39.041
and that is 20%.

00:04:40.392 --> 00:04:42.258
And I just keep doing this.

00:04:42.258 --> 00:04:46.092
And eventually, I can plot a distribution

00:04:46.092 --> 00:04:49.200
of these sample proportions.

00:04:49.200 --> 00:04:53.700
You would end up with some type
of a discrete distribution.

00:04:53.700 --> 00:04:57.350
And the way to read this
discrete distribution

00:04:57.350 --> 00:04:59.567
is let's say this right over here ends up,

00:04:59.567 --> 00:05:00.742
and I'm just gonna make up a number,

00:05:00.742 --> 00:05:02.741
this isn't going to be the actual number,

00:05:02.741 --> 00:05:05.408
but let's say that this is 0.15.

00:05:06.800 --> 00:05:11.317
The way to read that is
you have a 15% chance

00:05:11.317 --> 00:05:15.484
of getting a sample where
50% of your balls are orange

00:05:17.158 --> 00:05:20.075
or if this right over here is 0.07,

00:05:21.217 --> 00:05:24.075
that would mean that you have a 7% chance

00:05:24.075 --> 00:05:27.408
where 20% of your balls are orange.

00:05:28.708 --> 00:05:31.025
Now, to make this even a
little bit more tangible,

00:05:31.025 --> 00:05:34.842
let's run a simulation
that actually does this.

00:05:34.842 --> 00:05:36.400
This right over here is a simulation

00:05:36.400 --> 00:05:37.666
created on Khan Academy,

00:05:37.666 --> 00:05:41.483
on our computer programming
scratchpads by Charlotte Auen.

00:05:41.483 --> 00:05:46.408
It's a simulation to construct
a sampling distribution.

00:05:46.408 --> 00:05:51.308
So here, she's using candies
instead of just colored balls

00:05:51.308 --> 00:05:53.542
but these candies are
essentially colored balls.

00:05:53.542 --> 00:05:56.783
And so here, we can set
the population proportion.

00:05:56.783 --> 00:05:58.925
So let's say that the actual proportion

00:05:58.925 --> 00:06:02.192
as we saw on our example
of, let's say it's green

00:06:02.192 --> 00:06:05.700
as opposed to orange here, is 40%.

00:06:05.700 --> 00:06:07.308
And so let's say in each sample,

00:06:07.308 --> 00:06:09.517
just as we said, our sample size is 10,

00:06:09.517 --> 00:06:11.650
so we'll just take a sample size of 10.

00:06:11.650 --> 00:06:14.441
And let's just do one sample first,

00:06:14.441 --> 00:06:17.258
so let's just draw a sample.

00:06:17.258 --> 00:06:21.558
And so what we did is we
took 10 of these gumballs out

00:06:21.558 --> 00:06:25.058
and we are counting how
many of them are green.

00:06:25.058 --> 00:06:27.033
So in this first sample of 10,

00:06:27.033 --> 00:06:29.249
we see that one, two, three,

00:06:29.249 --> 00:06:32.517
four, five, six of them are green.

00:06:32.517 --> 00:06:36.008
So out of the possible outcomes,

00:06:36.008 --> 00:06:39.092
we're now gonna tally one of our outcomes

00:06:39.092 --> 00:06:42.233
having hey, we got six
of our 10 to be green.

00:06:42.233 --> 00:06:44.083
And if we wanna show the proportion

00:06:44.083 --> 00:06:45.792
instead of just the count,

00:06:45.792 --> 00:06:48.233
we can just pick percentage here.

00:06:48.233 --> 00:06:51.525
And so here, we've had
one scenario already

00:06:51.525 --> 00:06:54.167
where 60% were green.

00:06:54.167 --> 00:06:55.383
But we don't wanna just do one sample.

00:06:55.383 --> 00:06:56.775
We just wanna keep drawing sample.

00:06:56.775 --> 00:06:58.633
So let's draw another sample.

00:06:58.633 --> 00:07:02.917
So in this last sample, 50% are green.

00:07:02.917 --> 00:07:05.242
And so now that we have one was 50% green,

00:07:05.242 --> 00:07:07.358
one was 60% green.

00:07:07.358 --> 00:07:08.783
Let's draw another sample.

00:07:08.783 --> 00:07:12.492
Now we have another sample
where we got 60% green,

00:07:12.492 --> 00:07:16.300
so there are two situations
where we had 60% green.

00:07:16.300 --> 00:07:21.250
And so I can keep doing
this over and over and over,

00:07:21.250 --> 00:07:23.392
and so what we're creating right over here

00:07:23.392 --> 00:07:26.267
is a sampling distribution.

00:07:26.267 --> 00:07:28.408
And if we were to do this
an infinite number of times,

00:07:28.408 --> 00:07:31.608
we would get the true
sampling distribution

00:07:31.608 --> 00:07:34.250
of the sample proportion

00:07:34.250 --> 00:07:38.717
given the actual population
proportion that is green.

00:07:38.717 --> 00:07:41.750
And so, this is after 77 samples.

00:07:41.750 --> 00:07:45.683
Notice, this is saying that
out of 77 of our samples,

00:07:45.683 --> 00:07:47.266
22 of those samples

00:07:48.233 --> 00:07:52.092
resulted in 40% of our
gumballs being green.

00:07:52.092 --> 00:07:53.666
Only one of our samples

00:07:53.666 --> 00:07:57.026
had 80% of our gumballs being green.

00:07:57.026 --> 00:07:58.758
And if we just wanna
do a ton more samples,

00:07:58.758 --> 00:08:02.225
I'll go all the way to
drawing 50 samples at a time.

00:08:02.225 --> 00:08:05.225
So let me just keep increasing this.

00:08:07.133 --> 00:08:09.800
Notice, we have 17 samples now

00:08:09.800 --> 00:08:12.650
where we had zero percent that are green.

00:08:12.650 --> 00:08:15.150
We have 91 of the 2200 samples

00:08:16.475 --> 00:08:18.783
where 10% were green,

00:08:18.783 --> 00:08:21.883
where one out of the 10
in our sample were green.

00:08:21.883 --> 00:08:26.233
Now we could convert any of
these number, 17 and 91, 256,

00:08:26.233 --> 00:08:28.783
we could turn these into percentages

00:08:28.783 --> 00:08:31.558
by just dividing by the number of samples.

00:08:31.558 --> 00:08:33.192
But this is fun, we could just keep going

00:08:33.192 --> 00:08:36.217
and making this larger
and larger and larger.

00:08:36.217 --> 00:08:37.359
I encourage you to play with this.

00:08:37.359 --> 00:08:40.358
I'll provide a link for it in
the description on this video

00:08:40.358 --> 00:08:42.050
and on Khan Academy,

00:08:42.050 --> 00:08:45.133
but the main idea is to get an intuition

00:08:45.133 --> 00:08:48.125
for how a sampling
distribution is different

00:08:48.125 --> 00:08:51.875
from just a traditional
probability distribution.

00:08:51.875 --> 00:08:53.825
That in a sampling distribution,

00:08:53.825 --> 00:08:56.033
you're taking samples from a population,

00:08:56.033 --> 00:08:58.784
calculating some
statistic for that sample,

00:08:58.784 --> 00:09:01.308
and what you're plotting in
the sampling distribution

00:09:01.308 --> 00:09:04.838
are the various probabilities,
the various likelihoods,

00:09:04.838 --> 00:09:09.005
of the outcomes for those
statistics in those samples.


10% Rule of assuming "independence" between trials

WEBVTT
Kind: captions
Language: en

00:00:00.404 --> 00:00:03.097
- [Instructor] As we go further
in our statistical careers,

00:00:03.097 --> 00:00:05.088
it's going to be valuable to assume

00:00:05.088 --> 00:00:08.603
that certain distributors
are normal distributions

00:00:08.603 --> 00:00:11.818
or sometimes to assume that
they are binomial distributions

00:00:11.818 --> 00:00:13.596
because if we can do that,

00:00:13.596 --> 00:00:17.293
we can make all sorts of
interesting inferences about them

00:00:17.293 --> 00:00:19.447
when we make that assumption.

00:00:19.447 --> 00:00:22.169
But one of the key things
about normal distributions

00:00:22.169 --> 00:00:25.866
or binomial distributions is
we assume that they're the sum

00:00:25.866 --> 00:00:26.790
or they can be viewed

00:00:26.790 --> 00:00:29.920
as the sum of a bunch
of independent trials

00:00:29.920 --> 00:00:32.003
so we have to assume that

00:00:33.107 --> 00:00:35.024
trials are independent.

00:00:36.975 --> 00:00:39.756
Now that is reasonable
in a lot of situations,

00:00:39.756 --> 00:00:41.373
but sometimes let's say you're conducting

00:00:41.373 --> 00:00:44.461
a survey of people exiting a mall

00:00:44.461 --> 00:00:46.489
and in that case and
let's say you're saying

00:00:46.489 --> 00:00:48.788
whether they have done
their taxes already.

00:00:48.788 --> 00:00:50.451
If they're exiting the mall,

00:00:50.451 --> 00:00:53.514
it's hard to do these
samples with replacement.

00:00:53.514 --> 00:00:54.355
They're leaving the mall.

00:00:54.355 --> 00:00:55.325
You can't say, "Hey, hey, wait.

00:00:55.325 --> 00:00:56.580
"I just asked you a question.

00:00:56.580 --> 00:00:58.008
"Now you've answered it.

00:00:58.008 --> 00:00:59.417
"Now go back into the mall

00:00:59.417 --> 00:01:02.860
"because I want each trial
to be truly independent."

00:01:02.860 --> 00:01:04.897
But we all know it feels intuitive

00:01:04.897 --> 00:01:07.728
that hey if there are
10,000 people in the mall

00:01:07.728 --> 00:01:10.215
and I'm going to sample 10 of them,

00:01:10.215 --> 00:01:13.396
does it really matter that
it's truly independent?

00:01:13.396 --> 00:01:15.139
Doesn't it matter that we're just

00:01:15.139 --> 00:01:17.603
close to being independent?

00:01:17.603 --> 00:01:19.116
And because of that idea

00:01:19.116 --> 00:01:21.385
and because we do wanna make inferences

00:01:21.385 --> 00:01:24.425
based on things being close
to a binomial distribution

00:01:24.425 --> 00:01:25.955
or a normal distribution,

00:01:25.955 --> 00:01:29.455
we have something called the 10% rule

00:01:30.834 --> 00:01:35.354
and the 10% rule says that if our sample,

00:01:35.354 --> 00:01:39.521
if our sample is less than or
equal to 10% of the population

00:01:42.320 --> 00:01:46.487
then it is okay to assume
approximate independence

00:01:51.595 --> 00:01:54.442
and there are some fairly
sophisticated ways of coming up

00:01:54.442 --> 00:01:56.410
with this 10% threshold.

00:01:56.410 --> 00:01:57.954
People could have picked 9%.

00:01:57.954 --> 00:02:00.234
They could have picked 10.1%,

00:02:00.234 --> 00:02:02.523
but 10% is a nice round number.

00:02:02.523 --> 00:02:04.907
And if we look at some tangible examples,

00:02:04.907 --> 00:02:07.657
it seems to do a pretty good job.

00:02:08.725 --> 00:02:11.390
So for example right over here,

00:02:11.390 --> 00:02:15.557
let's let x be the number
of boys from three trials

00:02:16.406 --> 00:02:19.583
selecting from a classroom of n students

00:02:19.583 --> 00:02:21.688
where 50% of the class is a boy

00:02:21.688 --> 00:02:24.485
and 50% of the glass is a girl

00:02:24.485 --> 00:02:26.485
and so what we have over here is

00:02:26.485 --> 00:02:27.954
we have a bunch of different n's.

00:02:27.954 --> 00:02:29.426
What if we have 20 students in the class?

00:02:29.426 --> 00:02:30.259
What if we have 30?

00:02:30.259 --> 00:02:31.166
What if we have 100?

00:02:31.166 --> 00:02:33.122
What if we have 10,000?

00:02:33.122 --> 00:02:34.548
And so we could find the probability

00:02:34.548 --> 00:02:37.677
that we select three boys with replacement

00:02:37.677 --> 00:02:39.123
in each of these scenarios

00:02:39.123 --> 00:02:40.393
and we could also find the probability

00:02:40.393 --> 00:02:42.735
that we select three
boys without replacement

00:02:42.735 --> 00:02:43.740
and then we could think about

00:02:43.740 --> 00:02:48.557
what proportion is our sample
size of the entire population

00:02:48.557 --> 00:02:49.390
and then we could say,

00:02:49.390 --> 00:02:52.540
"Hey, does the 10% rule
actually make sense?"

00:02:52.540 --> 00:02:54.304
So this first column

00:02:54.304 --> 00:02:57.360
where we are picking three
boys with replacement,

00:02:57.360 --> 00:02:59.498
in this case because we are replacing,

00:02:59.498 --> 00:03:02.862
each of these trials are independent,

00:03:02.862 --> 00:03:04.695
are truly independent.

00:03:05.804 --> 00:03:07.989
And if our trials are independent,

00:03:07.989 --> 00:03:11.841
then x would be truly a binomial variable.

00:03:11.841 --> 00:03:16.146
Here, we aren't independent
because we are not replacing,

00:03:16.146 --> 00:03:17.729
so not independent,

00:03:19.964 --> 00:03:22.160
and so officially in this
column right over here

00:03:22.160 --> 00:03:23.370
when we're not replacing,

00:03:23.370 --> 00:03:26.236
x would not be considered
a binomial random variable.

00:03:26.236 --> 00:03:27.271
Let's see if there's a threshold

00:03:27.271 --> 00:03:30.909
where if our sample size is
a small enough percentage

00:03:30.909 --> 00:03:34.291
of our entire population
where we would feel not so bad

00:03:34.291 --> 00:03:38.161
about assuming x is
close to being binomial.

00:03:38.161 --> 00:03:42.377
So in all of the cases where
you have independent trials

00:03:42.377 --> 00:03:46.099
and 50% of the population
is boys, 50% is girls,

00:03:46.099 --> 00:03:49.957
you're going to amount to
1/2 times 1/2 times 1/2

00:03:49.957 --> 00:03:53.718
so in all of those situations
we have a 12.5% chance

00:03:53.718 --> 00:03:56.870
that x is going to be equal to three

00:03:56.870 --> 00:04:00.919
and in this case x would
be a binomial variable.

00:04:00.919 --> 00:04:02.254
But look over here.

00:04:02.254 --> 00:04:06.421
When three is a fairly large
percentage of our population,

00:04:07.490 --> 00:04:09.628
in this case it is 15%,

00:04:09.628 --> 00:04:14.505
the percent chance of getting
three boys without replacement

00:04:14.505 --> 00:04:18.672
is 10.5% which is reasonably
different from 12.5%.

00:04:19.547 --> 00:04:23.766
It is 2% different but
2% relative to 12.5%

00:04:23.766 --> 00:04:27.413
so that's some place in
between 10 and 20% difference

00:04:27.413 --> 00:04:29.533
in terms of the probability.

00:04:29.533 --> 00:04:31.960
So this is a reasonably big difference.

00:04:31.960 --> 00:04:34.532
But as we increase the population size

00:04:34.532 --> 00:04:37.034
without increasing the sample size,

00:04:37.034 --> 00:04:38.612
we see that these numbers

00:04:38.612 --> 00:04:40.856
get closer and closer to each other

00:04:40.856 --> 00:04:42.505
all the way so that if you have

00:04:42.505 --> 00:04:44.688
10,000 people in your population

00:04:44.688 --> 00:04:46.932
and you're only doing three trials

00:04:46.932 --> 00:04:49.293
that the numbers get very, very close.

00:04:49.293 --> 00:04:52.426
This is actually 12.49 something percent,

00:04:52.426 --> 00:04:54.589
but if you round to the
nearest tenth of a percent,

00:04:54.589 --> 00:04:56.342
you see that they are close.

00:04:56.342 --> 00:04:58.329
So I think most people would say,

00:04:58.329 --> 00:05:00.510
"All right, if your sample

00:05:00.510 --> 00:05:03.933
"is three ten-thousandths
of the population

00:05:03.933 --> 00:05:05.702
"that you'd feel pretty good

00:05:05.702 --> 00:05:08.003
"treating this column without replacement

00:05:08.003 --> 00:05:11.762
"as being pretty close to
being a binomial variable."

00:05:11.762 --> 00:05:12.695
And most people would say,

00:05:12.695 --> 00:05:14.593
"All right, this first scenario

00:05:14.593 --> 00:05:18.509
"where your sample size
is 15% of your population,

00:05:18.509 --> 00:05:20.629
"you wouldn't feel so good treating this

00:05:20.629 --> 00:05:24.766
"without replacement column as
a binomial random variable."

00:05:24.766 --> 00:05:26.916
But where do you draw the line?

00:05:26.916 --> 00:05:29.955
And as we alluded to earlier in the video,

00:05:29.955 --> 00:05:33.316
the line is typically drawn at 10%.

00:05:33.316 --> 00:05:34.870
That if your sample size

00:05:34.870 --> 00:05:38.569
is less than or equal to
10% of your population,

00:05:38.569 --> 00:05:42.561
it's not unreasonable to
treat your random variable,

00:05:42.561 --> 00:05:45.179
even though it's not
officially binomial to say,

00:05:45.179 --> 00:05:46.695
"Okay, maybe it is.

00:05:46.695 --> 00:05:49.095
"Maybe I can functionally
treat it as binomial

00:05:49.095 --> 00:05:49.955
"and then from there

00:05:49.955 --> 00:05:52.236
"I can make all of the
powerful interferences

00:05:52.236 --> 00:05:54.529
"that we tend to do in statistics."

00:05:54.529 --> 00:05:57.194
With that said, the lower the percentage

00:05:57.194 --> 00:06:00.692
the sample is of the
population the better.

00:06:00.692 --> 00:06:01.733
Now to be clear,

00:06:01.733 --> 00:06:04.971
that's not saying that small
sample sizes are better

00:06:04.971 --> 00:06:06.702
than large sample sizes.

00:06:06.702 --> 00:06:10.206
In statistics, large sample
sizes tend to be a lot better

00:06:10.206 --> 00:06:11.717
than small sample sizes.

00:06:11.717 --> 00:06:15.774
But if you wanna make this
independence assumption,

00:06:15.774 --> 00:06:18.485
so to speak, even when
it's not exactly true,

00:06:18.485 --> 00:06:19.923
you want your sample

00:06:19.923 --> 00:06:22.874
to be a small percentage
of the population.

00:06:22.874 --> 00:06:25.737
So the ideal, let's say you're
doing a survey at the mall,

00:06:25.737 --> 00:06:27.725
you might wanna survey 100 people

00:06:27.725 --> 00:06:29.606
but you would hope that there's at least

00:06:29.606 --> 00:06:33.338
1,000 people in the mall
in order for you to feel

00:06:33.338 --> 00:06:35.739
like your trials are
reasonably independent.

00:06:35.739 --> 00:06:37.231
If there's 10,000 people in the mall

00:06:37.231 --> 00:06:38.872
or somehow 50,000 people in the mall,

00:06:38.872 --> 00:06:42.166
which would be a very large
mall, well that's even better.


Identifying individuals, variables and categorical variables in a data set

WEBVTT
Kind: captions
Language: en

00:00:05.280 --> 00:00:07.186
- [Narrator] We're told
that millions of Americans

00:00:07.186 --> 00:00:10.181
rely on caffeine to get
them up in the morning.

00:00:10.181 --> 00:00:13.350
Which is true, although, if I
drank caffeine in the morning,

00:00:13.350 --> 00:00:14.267
I've very sensitive.

00:00:14.267 --> 00:00:15.895
I wouldn't be able to sleep at night.

00:00:15.895 --> 00:00:18.595
Here's nutritional data
on some popular drinks

00:00:18.595 --> 00:00:20.845
at Ben's Beans coffee shop.

00:00:22.347 --> 00:00:25.126
All right, so here we
have the different names

00:00:25.126 --> 00:00:26.379
of the drinks.

00:00:26.379 --> 00:00:29.070
And then here we have
the type of the drink,

00:00:29.070 --> 00:00:31.498
and it looks like they're
either hot or cold.

00:00:31.498 --> 00:00:34.296
Here we have the calories
for each of those drink.

00:00:34.296 --> 00:00:36.318
Here we have the sugar content in grams

00:00:36.318 --> 00:00:37.617
for each of those drinks.

00:00:37.617 --> 00:00:39.210
And here we have the
caffeine in milligrams

00:00:39.210 --> 00:00:41.324
for each of those drinks.

00:00:41.324 --> 00:00:42.764
And then we are asked,

00:00:42.764 --> 00:00:46.096
the individuals in this data set are,

00:00:46.096 --> 00:00:48.128
and then we have three choices.

00:00:48.128 --> 00:00:50.178
Ben's Beans customers.

00:00:50.178 --> 00:00:51.943
Ben's Beans drinks.

00:00:51.943 --> 00:00:54.276
Or the caffeine contents.

00:00:54.276 --> 00:00:55.264
Now, we have to be careful.

00:00:55.264 --> 00:00:57.990
When someone says the
individuals in a data set,

00:00:57.990 --> 00:01:00.700
they don't necessarily mean
that they have to be people.

00:01:00.700 --> 00:01:02.265
They could be things.

00:01:02.265 --> 00:01:04.612
And the individuals in this data set,

00:01:04.612 --> 00:01:06.754
each of these rows, they're referring

00:01:06.754 --> 00:01:11.153
to a certain type of drink
at Ben's Beans coffee shops.

00:01:11.153 --> 00:01:15.327
So the different types of
drinks that Ben's Beans offers,

00:01:15.327 --> 00:01:18.439
those are the individuals
in this data set.

00:01:18.439 --> 00:01:21.328
So they're Ben's Beans drinks.

00:01:21.328 --> 00:01:25.189
Next, they ask us the data set contains,

00:01:25.189 --> 00:01:27.427
and they say how many variables

00:01:27.427 --> 00:01:30.730
and how many of those
variable are categorical.

00:01:30.730 --> 00:01:34.077
So if we look up here,
let's look at the variables.

00:01:34.077 --> 00:01:35.292
So this first column

00:01:35.292 --> 00:01:37.793
is essentially giving
us the type of drink.

00:01:37.793 --> 00:01:38.729
This wouldn't be a variable,

00:01:38.729 --> 00:01:40.854
this would be more of an identifier.

00:01:40.854 --> 00:01:44.823
But all of these other columns
are representing variables.

00:01:44.823 --> 00:01:46.661
So, for example, type is a variable.

00:01:46.661 --> 00:01:48.593
It can either be hot or cold.

00:01:48.593 --> 00:01:52.864
And because it can only take
on one of kind of a number

00:01:52.864 --> 00:01:55.719
of bucket, it's either
going to be hot or cold.

00:01:55.719 --> 00:01:58.337
It's going to fit in
one category or another.

00:01:58.337 --> 00:01:59.912
And you don't just have two categories,

00:01:59.912 --> 00:02:01.746
you could have more than two categories.

00:02:01.746 --> 00:02:03.943
But it isn't just some
type of variable number

00:02:03.943 --> 00:02:06.119
that could take on a
bunch of different values.

00:02:06.119 --> 00:02:09.460
So this right over here
is a categorical variable.

00:02:09.460 --> 00:02:12.346
Calories is not a categorical variable.

00:02:12.346 --> 00:02:14.993
You could have something
with 4.1 calories.

00:02:14.993 --> 00:02:17.431
You could have something with 178.

00:02:17.431 --> 00:02:19.492
Things aren't fitting into nice buckets.

00:02:19.492 --> 00:02:22.651
Same thing for sugars
and for the caffeine.

00:02:22.651 --> 00:02:25.004
These are quantitative variables

00:02:25.004 --> 00:02:27.536
that don't just fit into a category.

00:02:27.536 --> 00:02:29.885
And so here I would say
that we have four variables,

00:02:29.885 --> 00:02:32.106
one, two, three, four.

00:02:32.106 --> 00:02:34.150
One of which is categorical.

00:02:34.150 --> 00:02:37.150
So that would be choice A over here.


Introduction to vertex form of a quadratic

WEBVTT
Kind: captions
Language: en

00:00:00.233 --> 00:00:01.165
- [Instructor] It might not be obvious

00:00:01.165 --> 00:00:02.784
when you look at these three equations

00:00:02.784 --> 00:00:04.850
but they're the exact same equation.

00:00:04.850 --> 00:00:06.925
They've just been
algebraically manipulated.

00:00:06.925 --> 00:00:08.633
They are in different forms.

00:00:08.633 --> 00:00:09.858
This is the equation

00:00:09.858 --> 00:00:13.400
and sometimes called standard
form for a quadratic.

00:00:13.400 --> 00:00:16.000
This is the quadratic in factored form.

00:00:16.000 --> 00:00:18.567
Notice this has been
factored right over here.

00:00:18.567 --> 00:00:20.192
And this last form

00:00:20.192 --> 00:00:22.150
is what we're going to
focus on in this video.

00:00:22.150 --> 00:00:25.025
This is sometimes known as vertex form

00:00:25.025 --> 00:00:25.958
and we're not gonna focus

00:00:25.958 --> 00:00:28.334
on how do you get from
one of these other forms

00:00:28.334 --> 00:00:29.683
to a vertex form in this video,

00:00:29.683 --> 00:00:32.250
we'll do that in future videos,
but what we're going to do

00:00:32.250 --> 00:00:36.000
is appreciate why this
is called vertex form.

00:00:38.425 --> 00:00:43.025
Now to start, let's just remind
ourselves what a vertex is.

00:00:43.025 --> 00:00:45.667
As you might remember from other videos,

00:00:45.667 --> 00:00:49.758
if we have a quadratic,
if we're graphing y

00:00:49.758 --> 00:00:54.533
is equal to some quadratic
expression in terms of x,

00:00:54.533 --> 00:00:57.649
the graph of that will be a parabola,

00:00:57.649 --> 00:00:59.467
and it might be an upward opening parabola

00:00:59.467 --> 00:01:02.300
or a downward opening parabola.

00:01:02.300 --> 00:01:03.683
This one in particular

00:01:03.683 --> 00:01:06.417
is going to be an upward opening parabola,

00:01:06.417 --> 00:01:10.141
and so it might look something like this.

00:01:10.141 --> 00:01:12.058
It might look something

00:01:14.733 --> 00:01:16.900
like this right over here.

00:01:18.250 --> 00:01:21.550
And for an upward opening
parabola like this,

00:01:21.550 --> 00:01:25.025
the vertex is this point right over here.

00:01:25.025 --> 00:01:27.000
You could view it as this minimum point.

00:01:27.000 --> 00:01:30.200
You have your x-coordinate of
the vertex right over there

00:01:30.200 --> 00:01:31.750
and you have your y-coordinate

00:01:31.750 --> 00:01:33.842
of the vertex right over here.

00:01:33.842 --> 00:01:36.717
Now the reason why this
is called vertex form

00:01:36.717 --> 00:01:40.292
is it's fairly straightforward
to pick out the coordinates

00:01:40.292 --> 00:01:42.817
of this vertex from this form.

00:01:42.817 --> 00:01:44.383
How do we do that?

00:01:44.383 --> 00:01:46.400
Well, to do that, we just have
to appreciate the structure

00:01:46.400 --> 00:01:47.783
that's in this expression.

00:01:47.783 --> 00:01:48.933
Let me just rewrite it again.

00:01:48.933 --> 00:01:53.100
We have y is equal to three
times x plus two squared

00:01:53.958 --> 00:01:54.791
minus 27.

00:01:56.008 --> 00:01:58.150
The important thing to realize

00:01:58.150 --> 00:02:01.149
is that this part of the expression

00:02:01.149 --> 00:02:03.308
is never going to be negative.

00:02:03.308 --> 00:02:05.808
No matter what you have
here, if you square it,

00:02:05.808 --> 00:02:08.667
you're never going to
get a negative value.

00:02:08.667 --> 00:02:10.267
And so this is never going to be negative

00:02:10.267 --> 00:02:13.125
and we're multiplying it by
a positive right over here.

00:02:13.125 --> 00:02:14.758
This whole thing right over here

00:02:14.758 --> 00:02:18.508
is going to be greater
than or equal to zero.

00:02:19.850 --> 00:02:20.741
So another way to think about it,

00:02:20.741 --> 00:02:24.458
it's only going to be
additive to negative 27.

00:02:24.458 --> 00:02:28.117
So your minimum point for
this curve right over here,

00:02:28.117 --> 00:02:30.275
for your parabola, is going to happen

00:02:30.275 --> 00:02:32.425
when this expression is equal to zero,

00:02:32.425 --> 00:02:35.267
when you're not adding
anything to negative 27.

00:02:35.267 --> 00:02:37.583
And so, when will this equal zero?

00:02:37.583 --> 00:02:39.742
Well, it's going to be equal to zero

00:02:39.742 --> 00:02:42.492
when x plus two is going
to be equal to zero.

00:02:42.492 --> 00:02:43.325
So you could just say,

00:02:43.325 --> 00:02:45.800
if you wanna find the
x-coordinate of the vertex,

00:02:45.800 --> 00:02:49.583
well, for what x value
does x plus two equal zero?

00:02:49.583 --> 00:02:51.542
And of course we can
subtract two from both sides

00:02:51.542 --> 00:02:54.692
and you get x is equal to
negative two, so we know

00:02:54.692 --> 00:02:57.200
that this x-coordinate right
over here is negative two.

00:02:57.200 --> 00:02:59.508
And then what's the
y-coordinate of the vertex?

00:02:59.508 --> 00:03:01.933
You could just say, "Hey,
what is the minimum y

00:03:01.933 --> 00:03:03.875
"that this curve takes on?"

00:03:03.875 --> 00:03:06.417
Well, when x is equal to negative two,

00:03:06.417 --> 00:03:10.375
this whole thing is zero and
y is equal to negative 27.

00:03:10.375 --> 00:03:12.933
Y is equal to negative 27,

00:03:12.933 --> 00:03:15.875
so this right over here is negative 27.

00:03:15.875 --> 00:03:17.600
And so the coordinates of the vertex here

00:03:17.600 --> 00:03:20.417
are negative two comma negative 27.

00:03:20.417 --> 00:03:22.183
And you are able to pick that out

00:03:22.183 --> 00:03:26.109
just by looking at the
quadratic in vertex form.

00:03:26.109 --> 00:03:28.633
Now let's get a few more
examples under our belt

00:03:28.633 --> 00:03:30.842
so that we can really get
good at picking out the vertex

00:03:30.842 --> 00:03:34.350
when a quadratic is
written in vertex form.

00:03:34.350 --> 00:03:36.534
So let's say let's pick a scenario

00:03:36.534 --> 00:03:39.100
where we have a downward opening parabola,

00:03:39.100 --> 00:03:42.925
where y is equal to, let's
just say negative two

00:03:42.925 --> 00:03:44.675
times x plus five,

00:03:47.150 --> 00:03:49.417
actually, let me make it x minus five.

00:03:49.417 --> 00:03:51.167
X minus five squared,

00:03:53.150 --> 00:03:55.400
and then let's say plus 10.

00:03:57.766 --> 00:04:00.108
Well here, this is gonna
be downward opening

00:04:00.108 --> 00:04:02.675
and let's appreciate why that is.

00:04:02.675 --> 00:04:06.842
So here, this part is still
always going to be non-negative

00:04:07.833 --> 00:04:09.934
but it's being multiplied
by a negative two,

00:04:09.934 --> 00:04:12.833
so it's actually always
gonna be non-positive.

00:04:12.833 --> 00:04:14.742
So this whole thing right over here

00:04:14.742 --> 00:04:18.909
is going to be less than or
equal to zero for all x's,

00:04:19.817 --> 00:04:22.592
so it could only take away from the 10.

00:04:22.592 --> 00:04:24.275
So, where do we hit a maximum point?

00:04:24.275 --> 00:04:25.517
Well, we hit a maximum point

00:04:25.517 --> 00:04:27.275
when x minus five is equal to zero,

00:04:27.275 --> 00:04:29.783
when we're not taking
anything away from the 10.

00:04:29.783 --> 00:04:33.151
And so, x minus five is equal to zero.

00:04:33.151 --> 00:04:34.400
Well, that of course is going to happen

00:04:34.400 --> 00:04:35.925
when x is equal to five,

00:04:35.925 --> 00:04:38.875
and that indeed is the
x-coordinate for the vertex.

00:04:38.875 --> 00:04:41.249
And what's the y-coordinate
for the vertex?

00:04:41.249 --> 00:04:43.508
Well, if x is equal to five
and this thing is zero,

00:04:43.508 --> 00:04:45.783
you're not gonna be taking
anything away from the 10

00:04:45.783 --> 00:04:48.208
and so y is going to be equal to 10.

00:04:48.208 --> 00:04:52.483
And so the vertex here is x equals five,

00:04:52.483 --> 00:04:53.721
and I'm just gonna eyeball it,

00:04:53.721 --> 00:04:56.492
maybe it's right over here, x equals five.

00:04:56.492 --> 00:04:58.125
And y is equal to 10.

00:04:58.125 --> 00:05:00.491
If this is negative 27,
this would be positive 27,

00:05:00.491 --> 00:05:01.792
10 would be something like this.

00:05:01.792 --> 00:05:05.325
I'm not using the same
scales for the x and y-axis,

00:05:05.325 --> 00:05:06.683
but there you have it.

00:05:06.683 --> 00:05:09.183
So it's five comma 10

00:05:09.183 --> 00:05:13.508
and our curve is gonna
look something like this.

00:05:13.508 --> 00:05:17.333
I don't know exactly where
it intersects the x-axis

00:05:17.333 --> 00:05:21.275
but it's going to be a
downward opening parabola.

00:05:21.275 --> 00:05:23.109
Let's do one more example

00:05:23.109 --> 00:05:24.842
just so that we get really fluent

00:05:24.842 --> 00:05:28.634
at identifying the
vertex from vertex form.

00:05:28.634 --> 00:05:31.158
So let's say, I'm just gonna make this up,

00:05:31.158 --> 00:05:33.908
we have y is equal to negative pi

00:05:35.575 --> 00:05:37.908
times x minus 2.8 squared

00:05:42.242 --> 00:05:43.075
plus 7.1.

00:05:46.700 --> 00:05:49.950
What is the vertex of the parabola here?

00:05:49.950 --> 00:05:52.017
Well, the x-coordinate is
going to be the x value

00:05:52.017 --> 00:05:55.592
that makes this equal
to zero, which is 2.8.

00:05:55.592 --> 00:05:56.925
And then if this is equal to zero,

00:05:56.925 --> 00:05:58.867
then this whole thing is
going to be equal to zero

00:05:58.867 --> 00:06:01.342
and y is going to be 7.1.

00:06:01.342 --> 00:06:03.029
So now, you hopefully appreciate

00:06:03.029 --> 00:06:05.450
why this is called vertex form.

00:06:05.450 --> 00:06:08.808
It's quite straightforward
to pick out the vertex

00:06:08.808 --> 00:06:12.142
when you have something
written in this way.


Democratic ideals in the preamble of the US Constitution

WEBVTT
Kind: captions
Language: en

00:00:00.597 --> 00:00:01.872
- [Instructor] This over here is a picture

00:00:01.872 --> 00:00:03.499
of the Constitutional Convention,

00:00:03.499 --> 00:00:06.130
which we mentioned happened in 1787.

00:00:06.130 --> 00:00:08.562
The original intent of the
Constitutional Convention

00:00:08.562 --> 00:00:11.755
was to revise the
Articles of Confederation,

00:00:11.755 --> 00:00:15.093
but folks like Alexander
Hamilton and James Madison

00:00:15.093 --> 00:00:17.987
really wanted to replace the
Articles of Confederation.

00:00:17.987 --> 00:00:21.697
You can see it's being presided
over by George Washington.

00:00:21.697 --> 00:00:23.737
And it starts, the preamble says:

00:00:23.737 --> 00:00:25.998
"We the People of the United States,

00:00:25.998 --> 00:00:28.509
"in Order to form a more perfect Union,

00:00:28.509 --> 00:00:32.059
"establish Justice, insure
domestic Tranquility,

00:00:32.059 --> 00:00:33.989
"provide for the common defense,

00:00:33.989 --> 00:00:36.051
"promote the general Welfare,

00:00:36.051 --> 00:00:38.031
"and secure the Blessings of Liberty

00:00:38.031 --> 00:00:41.757
"to ourselves and our Posterity,
do ordain and establish

00:00:41.757 --> 00:00:46.037
"this Constitution for the
United States of America."

00:00:46.037 --> 00:00:49.241
Once again, pause this video,
and think about whether

00:00:49.241 --> 00:00:53.626
you see ideas of popular
sovereignty, limited government,

00:00:53.626 --> 00:00:56.006
social contract, natural rights,

00:00:56.006 --> 00:00:57.548
going on even in this preamble,

00:00:57.548 --> 00:01:00.194
or even from the fact
that they took the trouble

00:01:00.194 --> 00:01:02.761
to create this Constitution.

00:01:02.761 --> 00:01:04.490
Let's start at the beginning.

00:01:04.490 --> 00:01:06.675
It starts with "We the People."

00:01:06.675 --> 00:01:07.892
We, the people, are the ones

00:01:07.892 --> 00:01:10.285
that are creating this Constitution.

00:01:10.285 --> 00:01:12.643
And not only does it start
with "We the People,"

00:01:12.643 --> 00:01:14.508
but "We the People" is
intentionally written

00:01:14.508 --> 00:01:17.152
in this very, very large
writing right over here.

00:01:17.152 --> 00:01:19.638
This is a picture of the Constitution.

00:01:19.638 --> 00:01:21.516
It's really all about "We the People."

00:01:21.516 --> 00:01:23.331
The people are sovereign.

00:01:23.331 --> 00:01:27.999
This idea of popular sovereignty
comes out loud and clear

00:01:27.999 --> 00:01:31.121
in not just the Declaration
of Independence,

00:01:31.121 --> 00:01:33.802
but also the US Constitution.

00:01:33.802 --> 00:01:35.609
The fact that "We the People"

00:01:35.609 --> 00:01:38.265
are setting up this government,

00:01:38.265 --> 00:01:40.187
this is all about social contract.

00:01:40.187 --> 00:01:42.444
They are forming a government.

00:01:42.444 --> 00:01:45.525
They're forming a social
contract with a government

00:01:45.525 --> 00:01:48.333
that is going to protect,

00:01:48.333 --> 00:01:50.638
that is going to establish justice,

00:01:50.638 --> 00:01:52.593
provide for the common defense,

00:01:52.593 --> 00:01:54.615
promote the general welfare.

00:01:54.615 --> 00:01:55.703
Let me make this clear.

00:01:55.703 --> 00:01:57.722
That is this, this is all social contract.

00:01:57.722 --> 00:01:59.527
This is what we expect this government

00:01:59.527 --> 00:02:01.204
that we're creating to do.

00:02:01.204 --> 00:02:02.812
"Promote the general welfare,

00:02:02.812 --> 00:02:06.061
"and secure the Blessings of Liberty

00:02:06.061 --> 00:02:08.886
"to ourselves and our Posterity."

00:02:08.886 --> 00:02:11.809
Now what about things
like limited government?

00:02:11.809 --> 00:02:15.976
Well, just the very fact
that we have a Constitution

00:02:16.862 --> 00:02:20.186
is a sign of limited government,

00:02:20.186 --> 00:02:22.490
that it isn't just a pure democracy,

00:02:22.490 --> 00:02:26.619
that whoever is governing
is going to be constrained.

00:02:26.619 --> 00:02:28.296
There, the rights of the government

00:02:28.296 --> 00:02:31.817
are going to be described
by this constitution.

00:02:31.817 --> 00:02:34.531
We also talk about the
Blessings of Liberty,

00:02:34.531 --> 00:02:37.587
so this is another
reference to natural rights.

00:02:37.587 --> 00:02:38.881
The Declaration of Independence is

00:02:38.881 --> 00:02:40.860
a little bit more clear
about natural rights,

00:02:40.860 --> 00:02:42.198
or a little bit more explicit,

00:02:42.198 --> 00:02:45.288
but the Blessings of
Liberty does talk about,

00:02:45.288 --> 00:02:49.642
or that's maybe in reference
to, natural rights.

00:02:49.642 --> 00:02:51.660
I will leave you there.

00:02:51.660 --> 00:02:53.821
As we study US Government,

00:02:53.821 --> 00:02:55.470
both the Declaration of Independence

00:02:55.470 --> 00:02:57.633
and even more so the US Constitution

00:02:57.633 --> 00:03:00.755
are going to be things
that we keep going back to

00:03:00.755 --> 00:03:04.128
to understand how we are trying to form

00:03:04.128 --> 00:03:07.266
a more perfect Union, and what is in line

00:03:07.266 --> 00:03:11.259
with the vision of our Founding
Fathers, and what isn't?