- [Instructor] Let's review calculating
confidence intervals for proportions.
So let's say I have a population,
and I care about some proportion.
Let's say I care about the proportion
of folks that are left-handed.
I don't know what that is,
and so I take a sample of size n,
and then from that sample,
I can calculate a sample proportion.
That's why I put that little hat on top of it.
It's a sample proportion that's estimating
our true proportion.
Now I wanna construct a confidence interval,
but before I go down the path,
I need to actually set up my conditions for inference,
make sure that I meet them.
And we've done this many times.
So the first condition for inference
is the random condition.
I need to feel good that this is truly a random sample
from the population.
The second one is often known as the normal condition,
and that's the condition that hey,
in order to feel like the sampling distribution
for the sample proportions is roughly normal,
n times our sample proportion
should be greater than or equal to 10
and n times one minus our sample proportion
should be greater than or equal to 10.
We've seen that multiple times before.
And then the third one is the independence condition,
and there's two ways to meet this.
Either the individual observations in our sample
should be done with replacement
or if it's not done with replacement,
we can feel pretty confident about this
if our sample size is no more than 10%
of the size of the entire population.
But let's say that we meet these conditions for inference,
what do we do?
Well we come up, we set up a confidence level,
confidence level,
for our confidence interval that we're about to construct.
And let's say we said it was a 95% confidence level.
That would mean that 95% of the time
that we went through this exercise,
the confidence interval that we get would
actually overlap with the true population proportion.
And 95% is actually a fairly typical one.
But from that confidence level,
you can calculate a critical value.
And the way that you do that,
you just look up in a z-table,
and once again all of this is review.
You would say hey how many standard deviations
above and below the mean of a normal distribution
would you need to go in order to
get say 95%, that confidence level of the distribution.
And now we're ready to calculate the confidence interval,
confidence interval.
It is going to be equal to
our sample proportion
plus or minus our critical value,
our critical value,
times the standard deviation
of the sampling distribution
of the sample proportion.
Now there is a way to calculate this exactly
if we knew what p is.
If we knew what p is,
this would be the square root of p times one minus p over n.
But if we knew what p is,
then we wouldn't even have to do this business
of constructing confidence intervals.
So instead, we estimate this.
We say look, an estimate of the standard deviation
of the sampling distribution,
often known as the standard error,
an estimate of this is going to be
the square root of,
instead of the true population parameter,
we could use the sample proportion.
So p hat times one minus p hat,
all of that over n.
Now the whole reason why I did this,
this is covered in much more detail and much slower
in other videos,
is to see the parallels between this and a situation
when we're constructing a two-sample confidence interval
or z-interval for a difference between proportions.
What am I talking about?
Well let's say that you have two different populations.
So this is the first population,
and it has some true proportion of the folks
that let's say are left-handed,
and then there's another population.
So let's call that p two.
You know maybe this is
freshmen in your high school or college
and maybe this is sophomores,
so two different populations,
and you wanna see if there's a difference between
the proportion that are left-handed, say.
And so what you could do,
just like we've done here,
is for each of these populations,
you'll take a sample here, we'll call that n one,
and then from that sample you calculate
a sample proportion, let's call that p one.
And then from this second population,
we do the same thing.
This is n two.
Notice n one and n two do not have to be
the same sample size.
That's a common misconception when doing these things.
These could be different sample sizes.
And then from that sample,
you calculate the sample proportion.
Now after you do that,
you would wanna check your conditions for inference.
And it turns out that the conditions for inference
would be exactly the same.
Do both of these samples meet the random condition?
Do both of these samples meet the normal condition?
And do both of these samples
meet the independence condition?
And if both samples meet these conditions for inference,
then we would have to calculate our critical value.
And you would do it the exact same way.
I'll just write it down again.
So first, you need to check all of these.
Then you would take your confidence level,
confidence level,
and from that get a critical z.
And then you're ready to say what your confidence
interval's going to be.
So your confidence,
confidence interval,
interval for p one
minus p two, so it's the confidence interval
for the difference
between these true population proportions.
That is going to be equal to the difference
between your sample proportions,
so p hat one minus p hat two,
p hat two, plus or minus,
plus or minus your critical value right over here
times the standard deviation
of the sampling distribution of the difference
between the sample proportions.
So it would be p hat one minus
p hat two.
And so we already know how to calculate this,
how to calculate this, how to calculate this.
How do we calculate that?
Well I will just give you the formula first,
but then we just have to appreciate
that this just comes outta the properties
of standard deviations and variances
that we have studied in the past.
So the standard deviation
of the sampling distribution
of the difference between the sample proportions,
it is a mouthful,
this is going to be approximately equal to,
approximately equal to the square root
of p hat one times one minus p hat one
over n one, over n one,
plus, plus, p hat two
times one minus p hat two
over n two.
And then you put that there.
You have constructed your confidence interval.
And once again how would you interpret that?
Well let's say your confidence level is 90%,
and from that you're able to construct
this confidence interval.
That would mean that 90% of the time
that you go through this exercise,
your confidence interval would overlap
with the true difference
between these population parameters,
the true difference between these population proportions.
Now where did this thing come from?
Well you might notice some similarities here.
This part over here
is an estimate or it's approximately equal to
the variance of the sampling distribution
of the sample proportion for our first population.
And then this right over here
once again is approximately going to be equal
to the variance of the sampling distribution
for the sample proportions for this population,
for p two.
How did I know that?
Well look if this is approximately the standard deviation,
you square that, you approximately get the variance.
And so the big takeaway is,
is that the variance for the sampling distribution
of the difference is just the sum
of the variances of each of those sampling distributions.
That's a lotta big mouthful and I know
it can get confusing, but hopefully that makes sense.
And that's where this formula comes from,
and so it's really not that much more to remember.
In the next few videos we're gonna do many more examples,
both looking at these conditions
and calculating confidence intervals and critical values.