- [Instructor] It is election season,
and there is a runoff between candidate A
versus candidate B.
And we are pollsters.
And we're interested in figuring out, well,
what's the likelihood that candidate A wins this election?
Well, ideally, we would go to the entire population
of likely voters right over here,
let's say there's 100,000 likely voters,
and we would ask every one of the them, who do you support?
And from that,
we would be able to get the population proportion,
which would be, this is the proportion that
support,
support
candidate A.
But it might not be realistic.
In fact, it definitely will not be realistic
to ask, well, all 100,000 people.
So instead, we do the thing
that we tend to do in statistics is,
is that we sample this population,
and we calculate a statistic from that sample
in order to estimate this parameter.
So let's say we take a sample right over here.
So this sample size,
let's say n equals 100.
And we calculate the sample proportion
that support candidate A.
So out of the 100, let's say that 54
say that they're going to support candidate A.
So the sample proportion here is 0.54.
And just to appreciate
that we're not always going to get 0.54,
there could've been a situation
where we sampled a different 100,
and we would've maybe gotten a different sample proportion.
Maybe in that one, we got 0.58.
And we already have the tools in statistics
to think about this,
the distribution of the possible
sample proportions we could get.
We've talked about it
when we thought about sampling distributions.
So you could have the sampling distribution of
the sample proportions,
of the sample
proportions,
proportions.
And it's going, this distribution's going to be specific
to what our sample size is,
for
n is equal to 100.
And so we can describe
the possible sample proportions we could get
and their likelihoods with this sampling distribution.
So let me do that.
So it would look something like this.
Because our sample size
is so much smaller than the population,
it's way less than 10%,
we can assume that each person we're asking,
that it's approximately independent.
Also, if we make the assumption that the true proportion
isn't too close to zero or not too close to one,
then we can say that, well, look,
this sampling distribution is roughly going to be normal.
So we'll have a normal, this kind of bell curve shape.
And we know a lot about the sampling distribution
of the sample proportions.
We know already, for example,
and if this is foreign to you, I encourage you to
watch the videos on this on Khan Academy,
that the mean of this sampling distribution
is going to be the actual population proportion.
And we also know what the standard deviation
of this is going to be.
So,
let me, maybe that's one standard deviation.
This is two standard deviations.
That's three standard deviations above the mean.
That's one standard deviation, two standard deviations,
three standard deviations below the mean.
So this distance, let me do this in a different color,
this standard deviation right over here,
which we denote as the standard deviation of
the sample proportions, for this sampling distribution,
this is, we've already seen the formula there.
It's the square root of p
times one minus p,
where p is, once again, our population proportion
divided by our sample size.
That's why it's specific for n equals 100 here.
And so in this first scenario,
let's just focus on this one right over here,
when we took a sample size of n equals 100
and we got the sample proportion of 0.54,
we could've gotten all sorts of outcomes here.
Maybe 0.54 is right over here.
And the reason why I had this uncertainty is
we actually don't know what the real
population parameter is,
what the real population proportion is.
But let me ask you maybe a slightly easier question.
What is,
what
is
the probability,
probability
that
our sample proportion of 0.54
is within,
is within two times two standard deviations
of p?
Pause the video, and think about that.
Well, that's just saying, look, if I'm gonna take a sample
and calculate the sample proportion right over here,
what's the probability that I'm within
two standard deviations of the mean?
Well, that's essentially going to be
this area right over here.
And we know, from studying normal curves,
that approximately 95% of the area
is within two standard deviations.
So this is approximately 95%.
95% of the time that I take a sample size of 100
and I calculate this sample proportion,
95% of the time,
I'm going to be within two standard deviations.
But if you take this statement,
you can actually construct another statement
that starts to feel a little bit more,
I guess we could say inferential.
We could say there,
there is
a 95%
probability
that
the population proportion p is
within,
within two standard deviations,
two standard deviations of p-hat,
which is equal to 0.54.
Pause this video.
Appreciate that these two are equivalent statements.
If there's a 95% chance that our sample proportion
is within two standard deviations of the true proportion,
well, that's equivalent to saying that there's a 95% chance
that our true proportion is within two standard deviations
of our sample proportion.
And this is really, really interesting
because if we were able to figure out what this value is,
well, then we would be able to create
what you could call a confidence interval.
Now, you immediately might be seeing a problem here.
In order to calculate this,
our standard deviation of this distribution,
we have to know our population parameter.
So pause this video,
and think about what we would do instead.
If we don't know what p is here,
if we don't know our population proportion,
do we have something that we could use as an estimate
for our population proportion?
Well, yes, we calculated p-hat already.
We calculated our sample proportion.
And so a new statistic that we could define
is the standard error,
the standard error of our sample proportions.
And we can define that as being equal to,
since we don't know the population proportion,
we're going to use the sample proportion,
p-hat times one minus p-hat,
all of that over n.
In this case, of course, n is 100.
We do know that.
And it actually turns out,
I'm not going to prove it in this video,
that this actually is an unbiased estimator
for this right over here.
So this is going to be equal to 0.54
times one minus 0.54,
so it's 0.46,
all of that over 100.
So we have the square root of .54
times .46
divided by 100,
close my parentheses, Enter.
So if I round to the nearest hundredth, it's going to be,
actually, even if I round to the nearest thousandth,
it's going to be approximately 5/100.
So this is going to be,
this is approximately 0.05.
So another way to say all of these things is, instead,
we don't know exactly this,
but now we have an estimate for it.
So we could now say with
95%
confidence, and that will often be known
as our confidence level right over here,
with 95% confidence between,
between,
and so we'd want to go two standard errors
below our sample proportion
that we just happened to calculate.
So that would be 0.54 minus two times 5/100.
So that would be 0.54 minus 10/100,
which would be 0.44.
And we'd also want to go two standard errors
above the sample proportion.
So that would be that plus 10/100.
And 0.64
of voters,
of
voters
support,
support
A.
And so this interval that we have right over here,
from 0.44
to 0.64,
this will be known as our confidence interval,
confidence
interval.
And this will change,
not just in the starting point and the end point,
but it will change the actual length
of our confidence interval,
will change depending on
what sample proportion we happened to pick
for that sample of 100.
A related idea to the confidence interval
is this notion of margin of error,
margin
of error.
And for this particular case,
for this particular sample,
our margin of error,
because we care about 95% confidence,
so that would be two standard errors.
So our margin of error here is two times our standard error,
would just be 0.1 or 0.10.
And so we're going one margin of error
above our sample proportion right over here
and one margin of error
below our sample proportion right over here
to define our confidence interval.
And as I mentioned, this margin of error is not going
to be fixed every time we take a sample.
Depending on what our sample proportion is,
it's going to affect our margin of error
because that is calculated, essentially,
with the standard error.
Another interpretation of this is
that the method that we used
to get this interval right over here,
the method that we used to get this confidence,
to get this confidence interval,
when we use it over and over,
it will produce intervals,
and the intervals won't always be the same.
It's gonna be dependent on our sample proportion,
but it will produce intervals
which include the true proportion,
which we might not know and often don't know.
It'll include the true proportion 95% of the time.
I'll cover that intuition more in future videos.
We'll see how the interval changes,
how the margin of error changes.
But when you do this calculation
over and over and over again,
95% of the time,
your true proportion is going to be contained
in whatever interval you happen to calculate that time.
Now, another interesting question is,
is, well, what if you wanted to
tighten up the intervals on average?
How would you do that?
Well, if you wanted to lower your margin of error,
the best way to lower the margin of error
is if you increase this denominator right over here.
And increasing that denominator means
increasing the sample size.
And so one thing that you will often see
when people are talking about election coverage is,
well, we need to sample more people
in order to get a lower margin of error.
But I'll leave you there,
and I'll see you in future videos.