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## Confidence intervals for proportions

# Example constructing and interpreting a confidence interval for p

AP.STATS:

UNC‑4 (EU)

, UNC‑4.D (LO)

, UNC‑4.D.1 (EK)

, UNC‑4.D.2 (EK)

## Video transcript

- [Instructor] We're told
that Della has over 500 songs on her mobile phone, and she wants to estimate what proportion of the songs
are by a female artist. She takes a simple random sample, that's what SRS stands for,
of 50 songs on her phone and finds that 20 of the songs sampled are by a female artist. Based on this sample, which of the following is a 99% confidence interval for the proportion of songs on her phone that
are by a female artist? So like always pause this video and see if you can figure it out on your own. Della has a library of
500 songs right over here. And she's trying to
figure out the proportion that are sung by a female artist. She doesn't have the time
to go through all 500 songs to figure out the true
population proportion, p. So instead she takes a sample of 50 songs, n is equal to 50, and from that she calculates
a sample proportion, which we could denote with p hat. And she finds that 20 out of the 50 are sung by a female, 20 out of the 50 which
is the same thing as 0.4. And then she wants to construct
a 99% confidence interval. So before we even go about constructing the confidence interval, you wanna check to make
sure that we're making some valid assumptions or
using a valid technique. So before we actually calculate
the confidence interval, let's just make sure that
our sampling distribution is not distorted in some way, and so that we can with confidence make a confidence interval. So the first condition is to
make sure that your sample is truly random. And they tell us that it's
a simple random sample, so we'll take their word for it. The next condition is to assume that your sampling distribution
of the sample proportions is approximately normal. And there you wanna be
confident or you wanna see that in your sample you
have at least 10 successes and at least 10 failures. Well here we have 20 successes which means well 50 minus 20, we have 30 failures. So both of those are more than 10, and so meets that condition. And then the last condition is, sometimes called the independence test or the independence rule or the 10% rule. If you were doing this
sample with replacement, so if she were to look at one song, test whether it's a female
or not and then put it back in her pile and then look at another song, then each of those observations
would truly be independent. But we don't know that. In fact we'll assume that she
didn't do it with replacement. And so if you don't do
it with replacement, you can assume rough independence for each observation of a song
if this is no more than 10% of the population. And so it looks like it is
exactly 10% of the population, so Della just squeezes through on our independence test right over there. So that out of the way
let's just think about what the confidence
interval's going to be. Well it's going to be her sample proportion plus or minus, there's going to be some critical value, and this critical value
is going to be dictated by our confidence level we wanna have, and then that critical value times the standard deviation of
the sampling distribution of the sample proportions
which we don't know. And so instead of having that, we use the standard error
of the sample proportion. And in this case it would be p hat times one minus p hat all of that over n our sample size, all of that over 50. So what's this going to be? We're gonna get p hat, our
sample proportion here, is 0.4 plus or minus, I'll save the z star
here, our critical value for a little bit. We're gonna use a z-table for that. And so we're gonna have
0.4 right over there, one minus 0.4 is times 0.6 all of that over 50. So we can already look at some choices that look interesting here. This choice and this choice
both look interesting, and the main thing we
have to reason through is which one has a correct critical value. Do we wanna go 1.96 standard
errors above and below our sample proportion? Or do we wanna go 2.576 standard errors above and below our sample proportion? And the key is the 99% confidence level. Now if we have a 99% confidence level, one way to think about it is, so let me just do my best shot at drawing a normal distribution here. And so if you want a 99% confidence level, that means you wanna contain the 99%, the middle 99%, under the curve
right over here, that area. And so if this is 99%, then this right over
here is going to be 0.5%, and this right over here is 0.5%. We want the z value that's
going to leave 0.5% above it. And so that's actually going to be 99.5% is what we wanna look up on the table. And that's because many z-tables, including the one that
you might see on something like an AP Stats exam, they will have the area
up to and including, up to and including, a certain value. And so they're not going to
leave this free right over here. So let's just look up
99.5% on our z-table. All right, so let me move
this down so you can see it. All right that's our z-table. Let's see we're at 99.
okay it's gonna be right in this area right over here. And so that is 2.5, looks like 2.57, or 2.5, 2.58 around that. And so this right over here is about 2.57, it's between 2.57 and 2.58, which gives us enough information
to answer this question. It's definitely not going to
be this one right over here. We have 2.576, which is
indeed between 2.57 and 2.58. So let's remind ourselves, we've been able to construct
our confidence interval right over here. But what does that actually mean? That means that if we were
to repeatedly take samples of size 50 and repeatedly use this technique to construct
confidence intervals, that roughly 99% of those intervals constructed this way are going to contain our
true population parameter.

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