Hello.Welcome to this lesson in mastering statistics.Here we're going to talk about the concept of the confidenceinterval.It dovetails in with what we have talked aboutbefore when we were talking about the central limittheorem.You'll see how that dovetailing, how that connection happenshere a little bit.But first, I want to explain what a confidence interval is.We'll have to do some definitions
so that you understand what we're talking about.And then over the next several lessons,we'll develop the ideas of the confidence interval,and you'll start to see why it's so important.What would have said since the very beginning of the masteringstatistic sequence is that the purpose of statistics,in most cases, is really that youwould like to learn something about a large population,but you don't have the means to go actually talk
to everybody in the population.If you could do that, them you'd know everything.But you don't usually have time or money to do that.So what you usually do is you go take a sample, which we weretalking about somewhat before.They usually want to go take a sample.And from that sample, you would liketo see if that sample is somehow representativeof the population.
For instance, if I go take a sample 100 peopleand calculate their IQs, if I go give them a test,then I would like to see, OK, is this representativeof the population?If it's not, then what range of values of the IQwould work for the population given that I've taken a sampleand I've gotten the IQ scores that I've gotten there.So a lot of times, what you're trying to do,
is you're taking a small sample, and you're studying,and you're trying to blow that up and figure outsome information and study some valid resultsfor the population that you're studying.So, in this case, what we're trying to do.Just to simplify it and make it a little more concrete,let's talk about the mean, for instance.Because the mean is a very, very common thing that you study.
For instance, if we're going to talk about IQs,maybe we have a city or a state or a nation,and we would like to know what the average IQof that population is.Now, clearly we cannot give everybody an IQ test,in the whole population of the country.But what we could do is we could gotake a sample of 100 people, or 200 people.We could give them an IQ test.
Now, from that, we would get scores.And we could calculate the samplemean-- calculate the sample mean of the people thattook the test.And that sample mean probably is prettyclose to the population mean-- the IQof the whole population-- probably.But clearly, it depends on the number of samples.I mean, if I go take five people, randomly,
off the street and give them an IQ test and get an average,am I going to claim that that's representativeof the whole country?No, it's only five people.What if I do something absurd, just give one person an IQ testand say, well, his score is the average IQ of the United Statesof America.Clearly, that's just not going to work.But if I give us test to maybe 3,000 people,or a test to 10,000 people, then I
might feel a lot more confident about my results.Clearly, somewhere in the middle,there exists a number of samples thatwill give me a good answer-- what I consider to be good.But I'm throwing out lots of weird termsthat I haven't really defined.How do we know if it's good enough?How do we know that what we get from our sample is goingto be pretty close to what we're studying, our population mean?
So whether or not we're studying IQs, or whether or not we'restudying the average height of people,or how they're answering on a test, or whatever,the idea is we want to study a population.But instead of doing everything with every personin that population, we're going to take a sample,we're going to learn about that sample,and we're going to try to figure outsome information about the population from the sample.
And in order to do that, we're goingto talk about things like a level of confidenceand confidence interval.All right, first, I already said this.We want to use a sample set and use it to estimatethe population mean.So we give our IQ test.We want to get an average IQ that comes outof that test of 100 people.And we want to use it to estimate the population mean.
So a term you're going to run into on any test or any bookis called the point estimate.Is just a term.I could just tell you what it is,but I think it's better is to go ahead and write down.A point estimate is a single number estimate
of a population-- that's what popis-- parameter-- single number estimate of a populationparameter.What I mean by that is, if I tell you, hey,I want to study the IQ-- I want to figure outwhat the mean IQ is everybody in the country, all right?Now, what I'll do is I'll go grab 1,000 students.I'll call that my sample size, or my sample.
And I go give those 1,000 people a test.And, of course, everyone's going to score differently.But what I do is I average them, and I come upwith a sample mean, all right?What I'm going to say, then, is the point estimatefor the average IQ of everybody in the country,I'm going to just say that it's equal to the samplemean, all right?Now, we know that that's not right.I mean, I give the test to-- let's say it's 100 people.I give the test to 100 people, let's say
the average IQ is 92, all right, or 103, or whatever.Now, I can sit there and use it as a pointestimate for my population mean, for the mean IQ of everybodyin the country, and I know that's a pretty darngood starting point, especially to havea large number of samples.But I know that's not going to be exactly correct.I mean, what are the odds that I givethese small number of people a testand the average result I get is exactly
equal to the average result of that IQ of everybodyin the country?The idea that happening is really low.But the best point estimate-- point estimate--for the population mean is equal to the sample mean.It's the only place you know how to start.I mean, you're taking the sample data.So I could say the best point estimate for the population
mean-- and that's Mu, by the way,the population mean is always denotedMu-- is the sample mean X bar.So you know how we've been doing so far.We've always said, well, we take a sample of 50 people,we average the results, we get somethingcalled X bar, all right?
So then, what we do is we use that point estimateor that sample mean, and we just say, hey, as a starting point,the population mean is probably pretty close to that.Clearly we know it's not exact, but it'sa great starting point.Now, what we're going to do is as follows,is we're going to figure out what is the error associatedwith that, and what is our level of confidenceabout what we've done?All right?
So we know that it's not 100% accurate.We know there's an error associated with this pointestimate.And we're going to quantify that here in a minute.Now, recall, I've talked about central limit theorem a lotin the last several lessons.You might be looking at this concept and central limitconcept and getting a little bit confused.But just remember that, in the case of the central limittheorem, what that was is we had a population and we sample it.
We have a sample size.I'll say the sample size is 50.But if you remember, I was very clear.In the central limit theorem, you took a sample of 50 people,took the mean of that-- sample mean.Take another sample of 50 people and calculate that sample mean,and do another sample, another sample.In every case, you're grabbing 50 people,if that's your sample size.You're getting all the sample means.But for the central limit theorem,
you have to sample the whole population.So you keep grabbing samples until there'snobody left to grab.And then where we talked about the resultsof what the central limit theorem really means.But in this case, we're not doing that.I'm not saying that we're going to samplethe whole population here.What I'm saying is there's a populationof 300 million people in this country,and I want to study and figure out
what the average IQ of everybody in the country is.So what I'll do is I'll go grab one sample, only one,but a sample size-- let's say the sample size is 100,or a sample size is 200 people.So I grab one sample, which is 200 people.And I give them a test, I average the results,and I get an average IQ.I'm going to say that that is a pointestimate for the average IQ, or the mean IQ,
of everybody in the country.I know it's not perfect, but I'm goingto say that that's my starting point.Now, we've said this a few times,but we know that this point estimate,the sample mean saying it's the population mean,we know that it's not perfect.So we know that there's a margin of error associated
with using the sample mean, this one group of measurements,as the population mean for whatever we're trying to study.We know there's an error associated with it.And we call that the margin of error.And we denote that capital E in your calculation.So you'll be calculating the margin of error,and you just call it capital E. And it is the largest distances
from the point estimate that willcontain the population mean Mu.
What I'm basically trying to say hereis, look, what we're trying to figure out,in the case of average IQ, we wantto calculate what the average IQ is of everyone in the country.But there's no way to calculate that with 100% certainty,because we'd have to give a test to everybody.So what we do is we give a test to let's say 200 people,and then we get their average result.And we say that's a point estimate.
So my tip of my finger is the samplemean that we get by giving those IQ testsand averaging the results right here.But we know that this answer is not exactlyequal to the population mean IQ.We know that.We know that it's a little bit fuzzy.And we know that the real mean, of the real populationfor the IQ, could be a little bit lower,or it could be a little bit higher, right?
So it's going to vary on either side of this point estimate.But somewhere around this point estimatethat we've actually measured, somewhere in that cloud--hopefully not too far away-- is the real population mean.That's unmeasurable, because you can't survey everybody.So what we say is there's going to bea margin of error associated with this point estimate.
So this point estimate's the tip of my finger.The estimate E, or the error, I should say,the margin of error E is the largest distancefrom the point estimate that willcontain the population mean.So what I'm saying is the largest distancemeaning lower or greater than this point estimate.Somewhere in that interval actuallyexists our population mean that we're trying to chase down.
So that margin of error is the distanceon either side of the point estimate, the one-way distance,either side of that point estimatethat forms an interval-- we'll talk about that a minute--that's going to contain what we're trying to hunt for.So I'll underline that.And the next thing we want to talk aboutis the interval estimate-- interval estimate.
And that is the range of values.This pen's going dead, so I'll switch here.A range of values that contains the population parameter.
So I've already hinted about this, but we're saying,hey, we take an IQ test to 200 people,we get an average result. That's our point estimate.Then we have an error that's below that estimateand above that point estimate.So below and above, you form an interval.You form a range of values of the lowerbound and an upper bound.
And we're saying, based on the theory of statistics,that somewhere in this interval exists our population meanthat we're hunting for.In other words, somewhere in this intervalexist the mean IQ of everybody in the country.That's called an interval estimate.And then, the last thing we talkedabout-- I'll put this on the other board over here--
is a level of confidence.All right, level of confidence, and thisis just how certain we are that the interval estimate contains
the mean.Now, in this case, we're talking about the mean.But we'll do confidence intervals for other parameterslater on.So finally, we have the last thing, the confidence interval,putting everything together-- confidence interval.All right, finally putting it together.
And I'll change colors to make this, hopefully,a little bit clearer.The confidence interval is the interval estimatethat goes with a specific level of confidence-- LOC, all right?
So it's level of conference.So basically everything's on the board.I'm going to explain a little bit furtherand we're going to wrap it up.Basically, the confidence intervalis just a range of values that we'resaying the population mean lives inside.So for instance, if I'm trying to findthe average IQ of everybody in the country,I will construct a confidence intervalthat will go from here to here.
And I'm going to say that I am pretty darnsure that the mean will land inside of this confidenceinterval, somewhere.All right?How confident in mind that the meanwill be inside of this confidence interval?Well, that's what the level of confidence is.So I may be 95% sure that my confidence interval containsthe mean of the population.Or I may only be 80% sure that this confidence interval--
that the 80% confidence interval containsthe mean of my population.Or, if I've done my homework really, really welland sampled tons and tons of people,I may be 99% confident that my confidence intervalhere, that I will construct, containsthe mean of the population.So whatever it is you're studying,whether it's average length of pencils coming off an assembly
line, average IQ, height of people, whatever,if you have a very large population of stuffthat you can't really study every single personin the population, or every single entity, frequentlywhat you do is you sample the population.And you construct a confidence intervaland say, well, I'm going to go and grab that,calculate something.And I'm going to say, I'm 97% confidentthat this range of values is going
to contain the population mean.And the rest of the lessons here aregoing to be showing you how to calculate the confidenceinterval and how to be sure that the population mean fallswithin it.I'll give you a hint.What we're going to use is the point estimate,which will be the sample mean in the center of the confidenceinterval.And the margin of error on both sidesis going to basically end up constructing the interval.
Now, in picture form-- pictures are alwaysbetter than words-- what we have is a sample mean.So if we're doing an IQ test, we give an IQ tests to 200 people,we calculate the mean.That's the point estimate here.So I'll put a little arrow here.This is the point estimate for the population
mean, which is denoted Mu.So we're saying, hey, we're prettysure the population means pretty close to that,but obviously it's not exactly right.There's a margin of error associated with it.And that margin of error extends in this direction to X barplus E. E is that margin of error, right?And it extends this direction to X bar minus E.
This picture contains everything that we've reallytalked about here.It's basically, we have a point estimatein the center of the interval.We have a margin of error.So you add it to the point estimate.You add it to the sample mean, basically, you got.And then you get the upper end of your confidence interval.And then you subtract it here.So you have the sample mean, minus that same margin
of error, and you get a lower bound.The lower bound to the upper bound gives you an interval.We're saying that the population mean, Mu,falls somewhere inside of here.And how sure are we of that?Well, this could be a 95% percent confidence interval,or it could be a 99% confidence interval,
or it can be an 82% confidence interval, or whatever.So you have an interval of values, we're saying,that our population parameter fits inside.And we have a level of confidence associated with it.So a lot of times, when you see pollson television, what they're doingis they're sampling a fairly small amount of people,and their extrapolating and saying, well, wethink the election's going to go to this candidate.
And here's the margin of error.Notice that they usually say thatin the polling, the margin of error.What we're going to be learning about how to calculatethat margin of error and how to be surethe confidence interval is right.So let's just stop here.This Section was full of a lot of definitions.And I hate doing that.I hate just hit you with a lot of definitions.But if I skip all this stuff, and I justjump into confidence interval, you're going to be confused.
Here, you shouldn't be confusing at all.You should know roughly what a point estimate is, roughlywhat the sample mean is, what the big-picture goal is here.And in the next section, I'll teachhow to calculate these things.And you'll find it is very, very simple once you understandthe concept of the terminology whatwe're doing here with the confidenceinterval in statistics.
Jason Gibson introduces the concept of the confidence interval, incorporating definitions for point estimate, confidence, and margin of error.
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Jason Gibson introduces the concept of the confidence interval, incorporating definitions for point estimate, confidence, and margin of error.