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## Confidence intervals for the difference between two proportions

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# Confidence intervals for the difference between two proportions

AP Stats: UNC‑4 (EU), UNC‑4.I (LO), UNC‑4.I.1 (EK)

## Video transcript

- [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.

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