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  • 00:01

    Hello.Welcome to this lesson in mastering statistics.Here we're going to continue workingwith the idea of the confidence intervalbut will be applying it to estimatingthe population mean when we have large sample sizes.And we'll explain what all of these wordsmean here in a few minutes as we go along.Now just kind of remember back to the last couple of sections.We talked about the idea of what a confidence interval is.

  • 00:24

    And we did what I consider to be some baby problems reallyjust to kind of get you comfortable with whatthat concept really is.So by now you should know what a confidence interval is.It's simply a range of values that weare confident to a certain degreecontain our population mean.Now we might be looking at other parameters

  • 00:46

    of the population like standard deviation and soon later on down the road.But in general at first when talk about confidenceintervals, we have a population mean.And we want to find an interval that based on our samplingis going to contain that mean.And that's very useful when you'redoing surveys or statistical analysis of failures in a plantor something.You don't have time to look at everything.

  • 01:07

    So you just take a sample.And then from that, we're trying to estimate whatthe whole population is doing.So we kind of did that before.But we did some hand waving because in all of the problemsbefore we gave you the margin of error and so on.And so at this point you should know that the confidenceinterval is really the sample mean minus the margin of error

  • 01:31

    and also the sample mean plus the margin of error.Those two calculations, when you do minus and plus your marginof error, that gives you what we call the confidence interval.Of course the whole interval changes dependingon the level of confidence.If we're 90% confident 95% confidentthat's going to change how that interval looks.That all makes sense intuitively,but the problems we did before, well,kind of lacked some of the details.

  • 01:53

    Now we're going to get to teaching youhow to actually calculate that margin of error,how to calculate that confidence interval in sortof a real world kind of situation.So the first class of problems that you typically dois you're looking at estimating the population mean, whichis what we talked about a lot before.So you may have candy bars coming off an assembly line

  • 02:13

    and you're looking at the length of the candy bar.You know, the machine is making the candies.And they're never always going to be the same length.So we might measure the length of these candy bars coming off.And what we're trying to estimateis the average population mean of the candy bar lengthsin our factory, our whole factory which produces millionsof candy bars a month.Right.We can't look at all of them.

  • 02:33

    So what we want to do is we want to sample.Right.Now this is a little different.We talked about earlier it's a little different than when wedid our sampling distribution.The sampling distribution we have a sample sizeand we sampled the whole populationwith a certain sample size and wedrew some conclusions from that.Now that's useful.But what's a whole lot more usefulis if we don't sample the whole population,

  • 02:53

    if we just have the candy bars coming off the line,we have a sample size of let's say 100 candy bars.We look at 100 candy bars randomlyand then we calculate their average length.How will that relate to the whole population?How will that relate?How confident are we going to be that the samplesize, the calculation we get from that

  • 03:13

    for the length of those candy barsis going to relate to the mean length of the whole population?That's what samplings really useful for.And that's what this confidence intervals useful for.All right.Now there's two broad classes though.Two broad classes of estimating the populationmean with confidence intervals.The first class is when the sample size is greater than 30.

  • 03:34

    So if we looked at greater than 30 candy bars in our sample,let's say we looked at 45 candy bars or 60 candy bars, right,anything over 30.And that's what we're going to talk about here.And you'll see that it's very understandable.We'll do some problems and so on.But keep in mind that in a few lessonsI'm going to show you what do you haveto do when the sample sizes less than 30,

  • 03:55

    when it's less than 30.So let's go ahead and write our assumptions now.We're going to write some assumptions down herefor this particular lesson.This is what you call estimating population meanswith large number of samples or large sample sizes.That means sample size is greater than 30.All right.So what we're going to assume-- I apologize.This is going to be kind of a long lesson,

  • 04:16

    but we have to get through all of this stuffor you will not be able to do your problems.The first thing is the samples, whatever it is we're sampling,are random.In other words, if we were in a factory, for instance, maybewe have 15 machines making these candy bars.It would be kind of dumb to choose all of our samples

  • 04:37

    from one machine because that one machine might be,just the way it's designed, making candy bars thatare slightly longer than the other ones or slightly shorterthan the other ones.So if we're going to really sample the population of candybars properly in a factory, we wouldwant to take some samples from everywhere in the factory,from all the machines there.So we want to make sure all of our samples are totally random.

  • 04:57

    However, you need to do that in your particular problem.If you're doing a survey of people,you want to make sure you call peoplefrom different parts of the country.Don't just call people in the urban centers or people thatlive in a certain neighborhood.You want to get as broad cross section as possibleotherwise this doesn't really become very accurate.Now in this particular case, the sample size,which we're calling it is greater than 30.

  • 05:19

    And we call that quote, I put quote unquote large.And you may ask yourself, how do they come up with 30?How do they know 30 is large?Well, it basically ties into what we did beforewith sampling distributions.And if you remember when we did sampling distributionsthat a sample size of 30 there was a nice break

  • 05:41

    point whenever we could draw some conclusionswhen the sample size is greater than 30.Now sampling distributions is a little bitdifferent than confidence intervals here.But the two things are related through statistics, you know.We're learning these things in statistics.There's a connection between those two thingsthat's kind of behind the scenes herethat's been proven mathematically.And it just turns out that when you have over 30,

  • 06:02

    a sample size of over 30, that the procedure that we'reabout to outline becomes reliable enoughfor us to use consistently.I mean, could use 29?Well, probably not too much of a problem.Could use 28?Probably not too much of a problem.But the accepted consensus is that when the sample size isgreater than 30, what we're about to outline hereis accurate enough for most situations.

  • 06:25

    So it's not really a law of natureso much as it is a study has been done, the mathematics havebeen proved, that this is a good place to gofor this particular procedure.When we're dealing with sample sizes less than 30,we'll have a different procedure that we'll deal with later.All right.It will be totally different.The third thing, and there's the only three assumptions here

  • 06:45

    in this lesson, the populations, I'llcall it population standard deviationwhich we call sigma is unknown.And I'm going to put unknown here like this, abbreviated.Unknown, what does that mean?Well, because in some statistics booksthey break it down even further and they ask you,

  • 07:07

    hey do you know the population standard deviation?Because if you do, you could do another techniqueto calculate confidence interval.Here's the deal though.Most of those methods are kind of absurd to use anyway.Because if you think about it, how do you calculate,or how would you ever know the population standard deviation?It's very, very, very difficult for you ever to know that.

  • 07:27

    I mean, when you think about what the standard deviation is,the standard deviation is telling youon average what is the spread of your data about the mean,right.So if I'm looking at candy bar lengths in a factoryand I'm making a million candy bars a day or monthor whatever it is, then in order for meto know the standard deviation, Iwould already have to know the lengths of all the candy bars

  • 07:48

    in order to calculate that quantity.So it's extremely unlikely for youever to know what the population.It's really unlikely for you to know anythingabout the population with certainty.But the standard deviation is one of those thingsthat you calculate by knowing the values of all the datapoints.So in some books you might see doyou know the standard deviation, do you notknow the standard deviation of the population.

  • 08:11

    And so there's some things you cando if you do know the standard deviation,but the chances of that ever happening in real lifeare basically almost never going to happen.So in this particular set of circumstances,we're going to assume that we don't know that because that'sreal life and that's what we're trying to do here.All right, so these are our assumptions.The samples are chosen randomly.

  • 08:32

    The sample size is greater than 30.We call that large number of samples.And we do not know what the population standard deviationis.OK, when these things are true, thenwhat we're going to end up doing is outlining a procedureto calculate the confidence interval to estimate the mean.That's what we're doing.All right.Now, I'm going to throw something

  • 08:52

    at you that I haven't talked about before,but I need you to kind of absorbed itand take it in and realize that I haven't covered itbefore with you because we haven'thad a need to use it before.So I'm going to introduce some things here real quickly.And then as we require more details,I'll show you those details as we go.So when these things are true, then

  • 09:13

    we can use something called, I have never mentioned thisto you before, it's called a t-distribution.And sometimes you see this thing,this t-distribution might also be called a studentt-distribution, to find, remember capital E, which

  • 09:37

    is a margin of error.And remember this whole confidence interval businessit really boils down just to findingwhat the margin of error is.Because don't forget we're going to sample the candy bars,we're going to go pick 65 of them off an assembly line.We're going to measure their lengths.We're going to calculate an average length.That's going to be the average of that sample.

  • 09:58

    The sample x bar that's going to be the average lengthof those candy bars.Now to find the confidence intervalat a certain level of confidence for the populationof the lengths of those candy bars,we need to know the margin of error.And then we take the mean and we subtract the margin of error.And we add the margin of error.That creates our confidence interval.OK.So what we're saying is in order to find the margin of error,the thing that you need to solve the problem,

  • 10:20

    you use something that we've never used before.It's called a t-distribution.And we're going to get to that t-distribution in justa second.I didn't want to really introduce it to youearly because it's kind of pointless to talkabout it unless you know what it's used for.This t-distribution is something that's been shown and provenin statistics to actually be able to calculatethese confidence intervals in a very reliable way.However, let me go ahead and do a number 2 here.

  • 10:43

    However, number two, but it's a bigbut here, when n, which is the sample size,is greater than 30, which is what we're doing here,we can use a normal distribution to approximate what

  • 11:06

    we call this thing called the t-distribution.So I guess what I'm trying to sayis that when we choose sample sizes greater than 30for the purpose of calculating a confidence interval,really what you're doing is you're using what we callthe student t-distribution or the t-distributionto find the margin of error.That is going to give you the most accurate answer.However, when the sample size is greater than 30,

  • 11:28

    this is where this comes in.When the sample size is greater than 30,we can use a normal distribution to actually approximatethe t-distribution.So the bottom line is and what you're alwaysgoing to be doing on all of your problems and all of your examsis when the sample size is greater than 30,what you're really going to end updoing is using a normal distribution to approximate.We already know how to deal with normal distributions.

  • 11:48

    So there's not a lot of new informationyou need to know to be able to solve these things.But I wanted to break it down and show youthat really what it is is a t-distribution that governs it.Which you don't know yet and we'lltalk about it in a minute, but youcould use a normal distribution as a high approximation.And just keep in the back of your mindas we go forward and several lessons from now,we're going to talk about how to find this confidenceinterval when the sample size is less than 30.

  • 12:10

    What if we can only pull 6 candy bars off of the assembly line?Well, in that case you're no longerallowed to use the normal distributionas an approximation.And then you have to use the actual t-distributionto get there.And so I'm going to save what that t-distribution lookslike for when we needed.So for now for these problems, whenwe have sample size greater than 30,we're not really going to use a t-distribution.

  • 12:31

    We're going to use a normal distributionas a high approximation which is not cheating.That's what all of your statistics textbooksare going to tell you to do because that'swhat's typically accepted.So let's outline what we call a procedure.And I hate writing things down like a cookbook formulationbut for this, it's really useful.So procedure.

  • 12:51

    I can cut through a lot of this stuffand just tell you what to do, but then it would justbe all in words in the air and I want to have somethingto hold onto here.So step one, the first thing you needto do when you're calculating a confidence interval isyou need to figure out do you want a 90% confidence interval?Do you want a 75% confidence interval?How much confidence do you really

  • 13:11

    need to make sure that your populationmean is within that bound?That's going to govern everything.That's going to govern all the calculations.So you need to figure that out.So I'm going to say step one, settleon what we call the level of confidence,level of confidence.And we call that, that variable, if you want to call it that c.

  • 13:33

    If you remember that was a level of confidence.And that's going to be, for instance 95%, 90%, et cetera,whatever it is.Now if it's 95%, c is 0.95.You move the decimal when you do the calculations.If we're doing 90% confidence interval,then the variable c is 0.90.But anyway, that's step number one.

  • 13:54

    Let me switch colors to try to break up this wall of texthere.Step number two is-- let me just write it downand then I'll draw a picture.We need to find a critical value of z.Remember the z-chart, the standard z-score.We need to find the critical value of z which we're goingto call critical value of z.

  • 14:22

    Critical value of z.Let me draw you a picture to show you how that really works.When we say find a critical value of z what basicallyis happening here-- Let me draw a normal distribution for youreal quick right under here.So here's a normal distribution.Let me go ahead and put this in red.That'll help you visualize things a little bit.This is very simple concept, but here is the right hand side,

  • 14:43

    here is the left hand side of a normal distribution.Actually doesn't look too bad.All right, what we're basically saying is the following.When you're doing this level of confidence business,if I draw here and here, I'm goingto call this value z sub c, critical value of z.

  • 15:04

    And this one, of course, will be negative z subc because this is a mirror image.When we have the normal distributionand we have standard z-scores, negative and positive, whateverthis value is, this is the negative of the same valuehere.When we look at the shaded region between these two

  • 15:25

    values of z, the shaded region here the area is c.The area of c.So what I'm basically trying to sayis if you settle on a level of confidenceat 95% that means c, the level of confidence, is 0.95.And that means that the area between these two values of z

  • 15:49

    is 0.95.So that's why you have to figure outwhat your level of confidence is first.So what it basically boils down to isyou want to find your level of confidence.And whatever you decide upon is goingto be the region of the shaded areahere on the board between this value and this value.So you want to figure out what values of zcorrespond to giving you an area in here.

  • 16:09

    Of course if you pick a higher level of confidence,this gets farther and farther out.And the level of z, negative z and positive z,are going to be farther away.If I have a very low level of confidence like 70%,then c would be 0.7.And then this would move in.And the level of z and negative zwill be a smaller number to contain that smallerarea there.

  • 16:30

    All right.So what you're doing is you're locking this down.That defines the area on the normal distributionto give you what we call this critical values of z.So basically the values of z correspondto your level of confidence through the normal curvelike this.And that's a property.This is just sort of how the level of confidence is derived.So we need to figure out what these values are.

  • 16:52

    OK, and then step three.Step three, the margin of error for the interval.Remember that's all we need, the margin of error.Once we know that, we can solve the whole thing.The margin of error, e, is this critical value of z

  • 17:12

    that we just found multiplied by s over square root of n.And I'm going to label everything herebecause this is really important stuff.OK.Z sub c is the critical value.N is the sample size, which you already know.

  • 17:33

    And s this is the sample standard deviation.Sample standard deviation.So let's think about this for just a second, this equation.Remember, all we need is the margin of error.Once we know the margin of error,we can find the confidence interval.Because Is here's how it's going to go down.

  • 17:53

    Let's say we're doing the candy bar deal where we'retrying to estimate the population mean of the candybars in this factory.So we go in and we say we want a 90% percent level of confidenceand calculate the confidence intervalfor the mean coming off this assembly line.So what we do is we go find-- say we pull 65 candy barsoff the line.We measure their lengths.And we average them together into a sample mean.

  • 18:16

    So we know that that's a pretty good, probablya pretty good estimate of what the factory is doing.But we'd like to know how accurate that isand what the confidence interval is about whatthe whole factory is doing.All right.So then what we do is we say, first thinglet's go and figure out what level of confidencewe want this confidence interval to be.And we already said it's 90%.So c is 0.9.Right.

  • 18:37

    Then we come down to the normal distribution.And we say, all right, we want a level of confluence of 0.9.So the area here is 0.90.And then we look at our z-chart table,and I'll show you how to do that in a minute.But basically you want to figure out what these numbers are,z sub c, that are going to correspondto locking in area of 0.9 in here

  • 18:57

    because that's our level of confidence.Once we have this number that comes from the z-chart table,then we stick z sub c in here.We stick the sample size, it was 65in our little verbal example, so square root of 65.S is the standard deviation.But it's not the standard deviationof the whole population of the factory.Remember, when we use s, it's the standard deviation

  • 19:18

    of the sample.So we take our 65 candy bars that we've sampledand we calculate the standard deviation about the meanof that little bitty sample.So we're always going to know whats is because we've sampled it.And we can quickly calculate the standard deviationof a small sample.So we have numbers everywhere.We calculate a margin of error.Once we have a margin of error, it's very simple

  • 19:39

    because then you take the sample meansubtract the margin of error add the margin of error.Bam.That's a confidence interval that we say is 90% certainwill contain the population mean of the length of the candy barsin our factory.Now I know that's a verbal example.I know that's a little bit of hand wavingbecause you've got to do a real problem to get comfortablewith it.But that is the point.

  • 19:59

    Now let's focus in on this a little bit and kind of justgive a quick example of just this little part of it,finding the critical value of z.So let's say we want a 95% confidence interval.I'm going to call that CI, 95% confidence interval.So what it means is what we're really after since it's

  • 20:20

    z sub c, we say z sub c is z 0.95this is the critical value correspondingto the level of confidence that we have.We want to figure out what that z value is.So what it means is if I draw a normal distribution, which I've

  • 20:41

    done here, what it means is I really want to figure out--I know it's going to be symmetrical whatever it is.So this I'm going to call z sub c.And over here, this I'm going to call negative z sub c.And what I'm basically saying is the shaded area between z sub cand negative z sub c the area is 0.95

  • 21:03

    because I've locked the level of confidence at 95%.So 0.95 that is going to be the area here.I need to look into my chart and figure out what z-valuewill give me an area of 0.95.So you have to go backwards.You know the area.Usually when you're using a z-chart tableyou're trying to figure out the area because youwant the probability.

  • 21:23

    You're going backwards now.You know what the area is and youwant to figure out what value z does that.All right, so the way you need to approachthis is realize-- the way you need to work on thisis to realize that if the area here is 0.95, if the area hereis 0.95, then the area right here,just this shaded area over in this tail, this area,

  • 21:47

    is going to be what?It's going to be, well if this is 0.95,if I know the entire area under the curve is 1,then this is 0.05 divided by 2 because 0.91 minus 0.95 right.And this area is exactly the same thing.This area is also 0.95 divided by 2.

  • 22:08

    So this area right here is really 0.025and this area here is really 0.025.This is just very simple math because weknow what the area is here.We know the area under the entire curve is just one.So the area under here must be 0.025.The area under here must be 0.025.

  • 22:29

    So if we know, I'll just shade this now, but this area upto the value of z sub c here is 0.25, then,and that's the area to the left of the z value,then all I have to do is look this upin the z-chart on the back of your statistics book.So you just go and search around the body of the tablethe main chart.You look up 0.025.

  • 22:50

    And what you're going to figure outis that when you do that you're going to get a value of zis negative 1.96.Because the way busy chart is set up is this is zero.Everything here is negative z's and everything hereis positive Z's, right.So if I have a very small area like here in this extreme tail

  • 23:13

    here, what this value is going to give me is negative 1.96.And then that means that this value is positive 1.96.So what this is negative 1.96, this is positive 1.96.Now when we go and use this value here

  • 23:34

    in our equation to calculate the margin of error,we just use the positive value.The critical value z sub c is just the positive one.So typically, what you're-- like in this example we foundthe negative one first because the chart,the z-chart is set up to give you the area to the left.So we put this area in, we get a z valuethat comes out negative.When we stick it in our formula we just use the positive value.

  • 23:56

    That's the way it's all set up.If you actually accidentally put the negative value in here,you're just going to get a negative margin of error.And the margin of error is plus or minus about your mean.So it doesn't really matter as longas you know what the absolute value of your margin of errorreally is.So this is how you would in practicality take a confidence

  • 24:17

    level and figure out what the critical values of z are.So now you have all the tools because if you do a samplingyou're going to know what the sample standard deviation is.You're going to know what the sample size is.And now I've shown you how to figure outthe critical value of z sub c.But think about it for a second.This whole process of figuring outwhat the critical values of z is,

  • 24:38

    I showed you how to do it so that you would understandwhat the process really is and what we're doing.But this process, you don't reallyhave to do it for every problem.The reason is because typically whenyou're doing statistical problems youonly really want to use certain levels of confidence.You end up using those over and over again.For instance you might use a 95% confidence

  • 24:59

    level, or 90% confidence level, or maybean 85% confidence level, or maybe a 99% cost confidencelevel.But you're generally not going to use oddball confidencelevels.Like you're not going to choose 79.5% confidence level.You're just not going to usually do that.When you think about surveys and thingsyou've experienced in real life with statistics,

  • 25:20

    it's always like 99% confidence or 95% confident, or 90%confident, something like that.This whole process of finding the values of zis really independent of any problem you have.This is just a problem of if I know my confidence level that'sthe area what's the value of z to give me that.If I change this to 90% confidence level,

  • 25:41

    the area would change.I would look up a different value of z for this tail.If I were looking at an 85% confidence level,I would find a different area here and a different value of zhere.And since you end up using the same ones over and overand over again, then really you don'tneed to do this whole process every time.So what you can do is define your level of confidence,

  • 26:03

    which is what we're calling c, and just make a little table.And this is typically provided in most statistics books.And this would be the critical value z that you would need.So if you are looking for an 80% confidence level,that would be a level confident of 0.8,and the critical value z is 1.28.If you're looking for 85%, or 0.85, it would be 1.44.

  • 26:27

    If you're looking for 0.90, which is 90%, then 1.645.And then if you're looking for 95%, it would be 1.96.And then 98, it would be 2.33.And 0.99, which be 99% confidence interval, 2.575.

  • 26:53

    So I went through this whole processup here to tell you how to do it so that you understoodthe way the machinery is working in the math behind it.I don't like just throwing a chart at youand say, hey, use this in your calculations.You have no idea where it comes from.All of these numbers are calculatedusing this technique.In fact, if you look at 95% confidence interval or 0.95,

  • 27:14

    you get 1.96 which is what we found here.If you change the level of confidence, all you're doingis changing the amount of area under this curveand you're finding a new value of z to correspond with that.These values of z basically you justend up using in your problems.Because 9 times out of 10, you'regoing to use 80%, 85%, 90%, 95%, 98%, 99% confidence interval.

  • 27:35

    You're not generally going to be using confidenceintervals different than that.If you are asked to find and use a 97% confidence interval,then you would use a technique that we outlined.But if not, then you would just use this right here.There's one very, very important thingthat I want to tell you because it might lead to confusionfor no really good reason.It's not a complicated thing, but sometimes different books

  • 27:58

    will show you things different ways.Some books, a lot of books, will denote this critical valuez of z with a little 0.95 because thisis the level of confidence that we are interested in.And so when you see z sub c, like for instancein this problem we got 1.96 based on what we have here,but in this case, all right, some books instead

  • 28:21

    of having it written is z sub 0.95,you will also see this written as z sub 0.025.Now don't freak out about that.It's the same value of z.It's still 1.96.That's still what goes in the calculation.But sometimes in some books you'llsee it written with the smaller number

  • 28:42

    here because when you think about it,there's two ways to denote this position on the graph here.One is to put the subscript with the area between.That's what we've done here.The other one is to denote the little bitty tailarea that's off to the left over here, which is 0.025.It's the same area that's over here, there.So this is, you know, it's not really one better

  • 29:03

    than the other.It's just in some books you will seeit written as with the level of confidence thereand some you'll have it written with the extra area heredivided by 2.Remember 0.025 came because the area here plus the areahere was 0.05 in this case, 1 minus 0.95.We divide by 2 to get this tail area.

  • 29:24

    All right.So you might also see that as writtenas z alpha divided by 2.And remember I think we've done this before.If not, you can just kind of just to know that alphais 1 minus c.So basically what's going on hereis the level of confidence c is the shaded area here.1 minus c is all of the area outside of this guy

  • 29:47

    here, right.And when we divide this by 2, thatgives us just the tail area over here,which is in this case 0.025.So just to solidify that for you a little bit,let's change it from 95% confidence interval.Let's say we're doing 90% confidence interval.Right.So that means the level of confidence is 0.90.

  • 30:11

    And that means that alpha is 1 minus c, 1 minus 0.90,which means 0.10.If we want to draw a picture, which a picture is alwaysworth a thousand words in situations like this,basically we have a normal distribution.Draw this down like this.Draw this down like this.What's basically happening is we still are locking down

  • 30:34

    a central area here.This central area here.This area is 0.90.That is all true.Right, so one way to write this guy here,z sub c and z sub c like this, one way is to say z sub 0.90,

  • 30:57

    right.Because that denotes what value of zis required to lock down an area 0.9 between there.But another way that you would see it writtenis to take the other area outside of this guy, whichis we already know what it is, 1 minus c gives us 0.1.So the area just to one side is going to be 0.1 divided by 2

  • 31:20

    is 0.05.So an equivalent way of denoting the position on this curveis to say z 0.05.And sometimes you see that written as z sub alpha over 2because alpha is the area outside here.So you divide by 2 just to get one half.There is no right answer to this.In both cases 0.9 right here, z sub 0.9,

  • 31:43

    with the level of confidence as 0.9 it's still 1.645.So this is 1.645.Well this value is 1.645.There's no difference in the values.It's just how you write down the actual position of z.The subscript is the only thing that's different.So in some of your books you might see it

  • 32:04

    as written z alpha over 2.You need to kind of realize all they're saying isthey're looking at the tail area that's1 minus the level of confidence and dividing it by 2 to giveyou one half of the tail area.That's locking down this positionof z exactly the same way that z sub 0.9 is locking down the guyhere.It's just a way of writing down the position

  • 32:25

    and denoting and keeping track of what level of confidenceyou have.So in your problems in this course,we're going to be writing as z subc with the level of confidence being here.That's just what I'll kind of use in my notation,at least for now.But in some of your books you might see a different subscripthere.But just remember that it's goingto give you the same value.So whatever you get for this value of z

  • 32:46

    is what goes in here.So the calculation for the margin of error doesn't change.The number you pull off the chart doesn't change.It's just how it's written down and referencingwhat area of the curve some books do different than others.All right, so we're going to close it down in this sectionand be done with it.It was a long section.I apologize for that a little bit.

  • 33:06

    But we have to get through some of this informationand some of these concepts for you to really understand it.The bottom line is, just a super quick recapand then we'll do some problems in the next section.If we're doing confidence intervals,we have two broad classes, when the sample sizeis greater than 30, which is what we're doing here,and when the sample size is less than 30.If the sample size is greater than 30, what you're really

  • 33:29

    are doing is the student t-distribution,which we haven't learned yet.But since the sample size is greater than 30,we can use the normal distribution to approximate it.And we're kind of getting into this a little bit here.So the procedure is we find a level of confidence.We use that to find the critical value of z.Once we have that, we calculate the margin of error.And then we do plus and minus the margin of error

  • 33:49

    like we've done before to figure out what the confidenceinterval is.And then we just did some examples and sortof showing you exactly how you find this z sub c here.And it all boils down to the area between these two valuesis going to be equal to whatever the confidence level is.So you can tabulate the very common confidencelevels and then, of course, get these associated

  • 34:12

    critical values of z.And that's going to be what you're goingto use in your calculations.So 9 times out of 10, especially all the ones in this class,where we're using 90%, or 95%, or 85% reallyyou're just going to go grab the value of zand stick it in your calculation straight from here.I didn't want to just give that to youbecause you wouldn't have any idea where it came from.Now you know where it kind of comes from.

  • 34:32

    And then of course if on a crazy situationyour teacher gives you a problem whereyou're looking at a 92% confidence interval,it's not in this table, 92%.So the way you would do it is to go look at the area.Basically you find this tail area, look it up in your chart,get the critical value of z to use in your calculations.So that's why I wanted to kind of go over it as well.

  • 34:53

    So make sure you understand this.We just kind of outlined the procedure.I expect you to be a little bit confused right now.I hope you're not, but I expect that a lot of people will be.Let's go work some problems, and you'llsee the procedure for finding a confidence interval is notthat hard.Let's go ahead and do that now.Lock it down, give you some skills and some practice,and then we'll just kind of go from there.And hopefully you're understanding

  • 35:14

    will improve with every problem that we work.

Video Info

Series Name: Mastering Statistics, Vol 3

Episode: 11

Publisher: Math Tutor DVD

Publication Year: 2014

Video Type:Tutorial

Methods: Estimation methods, Confidence intervals, Population mean

Keywords: mathematical computing; mathematical formulas; reference materials; representation of reality

Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:

Keywords:

Abstract

Brian Gibson explains how to calculate confidence intervals for estimating the population mean for large samples. He refers to the z-value tables commonly found in any statistics book, but he also demonstrates how to find the values using the appropriate formula.

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Estimating Population Means (Large Samples): Part 1

Brian Gibson explains how to calculate confidence intervals for estimating the population mean for large samples. He refers to the z-value tables commonly found in any statistics book, but he also demonstrates how to find the values using the appropriate formula.

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