INSTRUCTOR: Hello.Welcome to this lesson in Mastering Statistics.Here we're finally going to get our feetwet with real confidence interval type of problems.And I promised you before, and I'll reiterate it right now--these kinds of problems are not very hard now that we knowthe background material, we know the concept of a confidenceinterval, we know the concept of the critical value of z,and all the terms-- such as the margin of error,
INSTRUCTOR [continued]: and the point estimate, and things like that.So if you don't know that stuff, then things like a confidenceinterval seem very difficult. Let's work a problem here,and we'll see if they're quite very, very simple.So for the first problem, it says,"We ask 100 people how much a supreme pizza costs"in their area."The average answer is $25.99, with standard deviation
INSTRUCTOR [continued]: of $15.50.Construct a 99% confidence intervalthat contains the average price of a supreme pizza nationwide."All right, this is the classic type of problemthat you really are wanting to do when you'reusing a confidence interval.It's also very practical.Because clearly, if we want to figure outwhat the average price of a supreme pizza
INSTRUCTOR [continued]: is across the country, it would be impractical for usto go around every city and every pizza jointin the country and go figure out how much a supreme large pizzacosts, and then take the average of every placeacross the country.It's just not going to happen.So what we instead do is we take a small sample.In this case, we asked for 100 people,and we just asked those 100 people
INSTRUCTOR [continued]: what the price of their supreme pizza is.Now, we can average the data that we get,and we'll get a pretty good estimatefor the average price of a supreme pizza in the country.But clearly the data that we collect, the 100 peoplewe ask giving us that sample estimate for the priceof the supreme pizza, we expect itto be relatively close to the national average,
INSTRUCTOR [continued]: but we don't expect it to be exact.Of course we don't.So the idea of a confidence intervalis starting from that sample averagethat we have, from the 100 people, on plus or minus,on the top and on the bottom of that average that we getfrom our data, what is an appropriate range to give us,in this case, a 99% confidence that the national average is
INSTRUCTOR [continued]: going to lie in that band?Because you're never going to know the real answer.The best you can do is bound the problem.And here, we're using a very high-- 99%--confidence interval.All right, so in order to use the techniques that we outlinedin the last sections, we have to verify that a couple of thingsare true.First thing is, we do not know the sigma,the standard deviation, of the population.
INSTRUCTOR [continued]: We don't know that.In fact, the problem just says we ask 100 people.It gives us the average answer of those 100 people, $25.99.It gives us a standard deviation of $15.50.But you have to realize that that standard deviationin the problem, $15.50, that's the standard deviationof the 100 people that we ask, all right?That, you can always calculate.
INSTRUCTOR [continued]: If you get 100 numbers, you can alwayscalculate the mean of those numbersand the standard deviation of those numbers.We've done that many, many times in the past.So the standard deviation they'retalking about in this problem is the standard deviationof our sample.And that will be used in our calculation, as well.But we don't know the standard deviationof the whole population.We already said it's very, very unlikelythat you would ever know that.
INSTRUCTOR [continued]: The second thing is the sample size is greater than 30.In this case, it's 100.So that means we have, quote unquote, "large sample size,"and we can use the technique that we learned aboutin the last section to proceed with the confidence interval.And of course, we want to find the confidence intervalfor the mean.And so in this whole class of problems,we're always talking about finding confidence intervalfor the mean.
INSTRUCTOR [continued]: Later on, we'll talk about finding different typesof confidence intervals.But for now, we're just going to be talkingabout the mean of some number-- in this case,the price of that pizza.So in this case, for this particular problem,the sample mean, we always denote it as x bar, right?The sample mean is x bar.
INSTRUCTOR [continued]: And that was given to us at $25.99, right?Now, this is what we're going to call the point estimate.And that's just a fancy word that basically says,hey, we talk to 100 people.We get their average price of a pizza.We average over those 100 people, and we get $25.99.
INSTRUCTOR [continued]: That's a pretty good starting pointto estimate the national mean for the pizza prices.That's why we call it a point estimate.We're going to build an interval from that, a confidenceinterval, but this is the seed.This is the starting point.The second thing we know is that n is equal to 100.That's how many people we asked to begin with.And what we wish to find out-- well,
INSTRUCTOR [continued]: actually, one more thing we know.We know the standard deviation of the data that we collectedfrom those 100 people is $15.50.Notice that we're denoting this s.And this is kind of what we come back toin some of the earlier classes in Mastering Statistics.Your terminology and the way you write things down on your paperis very important.We represent the standard deviationof a small sample of people with an s.
INSTRUCTOR [continued]: We represent the standard deviationof an entire population-- which would be the national pizzanumbers, for instance-- that would be the sigma.This is what we don't know.Because we don't know all the data in the country.But we do know this, because this is justthe standard deviation about the mean.So in other words, the main pizza price of these 100 people
INSTRUCTOR [continued]: is $25.99.The standard deviation about the mean of that 100 people is$15.50, so pretty widespread.What it's telling you is of the 100 people that you ask,some of them have pizza prices well above this,and some of them have pizza prices well below that.Pretty broad standard deviation.And that makes sense.
INSTRUCTOR [continued]: You have some expensive pizza placesand you have some cheap pizza places.All right?Now, what we really wish to do is we really want to find--and I'll switch colors here.We really want to find the margin of error, whichwas called E. We really want to find that,
INSTRUCTOR [continued]: because once we figure what this is,the confidence interval is going to be--we'll have the sample mean that we have,and then we will have x bar plus this guy,and then x bar minus the margin of error.This is what constructs our confidence interval.So how do we find this margin of error?Really, the whole problem boils down
INSTRUCTOR [continued]: to figuring out what that is.And in this case, the margin of erroris equal to this critical value of Z multipliedby the standard deviation, s, dividedby the square root of the sample size.All right.Now, when you think about it, we alreadyknow what the standard deviation of the data we collected is.It's given to us in the problem.We know what the sample size is.
INSTRUCTOR [continued]: That's given to us in the problem.This critical value of Z is almost never given to youin the problem.But we know that we're after, in this case--I didn't write it down, but the problemsaid-- a 99% level of confidence,or a 99% confidence interval.And if you remember, we talked about howto take that confidence level and come back
INSTRUCTOR [continued]: to find that critical value of Z. We went through allthe details of that, and we built a table, whichI've reproduced on the board.These are the very common levels of confidencethat you will come across.In this case, a 99% confidence level,which means c, level of confidence 0.99, 2.575is the critical value of z.
INSTRUCTOR [continued]: You can use this chart for all of your problems.If you have a level of confidence that's listed here,you can always use these without having to derive itover and over and over again.If you're asked for an oddball level of confidence,like 93, or 93%, then you would haveto use the method we outlined to find that.So here, let's use 2.575.
INSTRUCTOR [continued]: So what we're going to have for the critical value of Z, 2.575.s, which is the standard deviation, 15.50.And the square root of the number of samples is 100.So the square root of 100.And so basically, the bottom line is whenever you take15.50, divide it by the square of 100--
INSTRUCTOR [continued]: which is 10 on the bottom here-- and multiple by this guy here,the margin of error that you actually get is 3.99.What you're really always seeking in these problemsis to figure out what that margin of error is.So then you can say for a 99% confidence interval-- that's
INSTRUCTOR [continued]: what's CI means, confidence interval-- whatyou do is you take your point estimate, which is your samplemean, 25.99-- so 25.99-- and you add to it 3.99.When you add these guys together, you get $29.98.And then you do the same thing again, 25.99 minus 3.99,
INSTRUCTOR [continued]: and then what you get there is $22 even.So what you have figured out is that the meanof the population-- you know what I'll do,is I'll switch colors, just to kind of drill it home.The population mean-- which notice,now, we're not calling x bar.We're calling it mu here.This is the mean of the population,which means it's the average price of a pizza nationwide.
INSTRUCTOR [continued]: It falls between a lower bound of $22and an upper bound of $29.98.So the way you do it is you read from the center out.The average price of a pizza is greater than $22and the average price of a pizza is less than $29.98.This is how you would write it with greater-than, less-than.
INSTRUCTOR [continued]: Or you can just write the interval as 22, comma, $29.98.So you put parentheses with a comma in the middle.It's the same thing.So this is the 99% confidence interval--so, 99% confidence interval.What we're saying is that we are 99% certainthat the mean of the population is going
INSTRUCTOR [continued]: to fall between these numbers.The average price of a supreme pizzanationwide is going to fall between these numbers.We're not 100% sure of it, but we're pretty darn sure of it.Now, let me kind of take just a momentto point something out for you.Notice that the entire calculationrests on what this margin of error really comes out to be.
INSTRUCTOR [continued]: And in the calculation of this margin of error,there's only three things that really apply.One of them is Z sub c, which is basically dependent on whatlevel of confidence you have.So when we have a 99% confidence level, this guy's locked down.And then we have a sample standard deviation,which is given to you by the data
INSTRUCTOR [continued]: that you collected from the survey or whatever.In this case, it was given to you by the problem.So this is pretty much fixed by the problem.And then we have the number of samples.If we lock this down and we lock this down,then think about this.If you increase the number of samples,at the same level of confidence--this number stays the same.If you increase this number, then it's
INSTRUCTOR [continued]: going to drive this whole calculation down.It's going to make the margin of errorgo down if you make the number of samples go up.So what that means is if everything else isheld constant, if I increase the number of samples,I drive down my margin of error.And since I'm adding the margin of errorand subtracting the margin of error,it's going to make this interval get smaller and smaller if Imake a large number of samples.
INSTRUCTOR [continued]: Now, that makes total sense to you.If you're doing a survey-- think about it for a second.If you're doing a survey and you want to figure out and estimatewhat the national prize of a supreme pizza is,you obviously want to sample as many people as possible.Now you have your answer.Why do I want to do that?Why is it better to do that?You kind of know that it's better to ask more people.But mathematically, what this is telling
INSTRUCTOR [continued]: you is if I ask more and more and more people,I'm going to be a whole lot more confidentthat the mean of that population isgoing to fit in a tighter band.In other words, right now I don'tknow if the price of that pizza nationallyis between 22 and 29.But if I ask a million people, if I make n 1 million,
INSTRUCTOR [continued]: I'm going to be a whole lot more sure, and that band,that average pizza price nationwide, is goingto come down because I'm going to have a whole lot more datato base this calculation on.And this is how that calculation plays out.If you ask a whole lot more people, the margin of errorgoes down, because I have more data,because I'm basically more confident in whatI'm calculating here.And that's going to drive this confidence interval down.
INSTRUCTOR [continued]: It's still the same thing.It's still telling you that you're99% certain that the average price of a pizzanationwide is in the band.It's just that when you drive up the number of samples,the confidence interval, the actual length of the intervalgoes down, down, down.Because of course, the more people you ask,the more confident you are, and the smaller wiggle room youhave in the difference between the samples that you collect
INSTRUCTOR [continued]: and reality that you have.All right?So that's really good problem.Let's do another one here.This one says, "78 students surveyedsaid that they study an average of 15 hours per weekwith a standard deviation of 2.3 hours.What is the margin of error for a 90% confidence interval
INSTRUCTOR [continued]: for the student population?"So again, you're given that the sample size is78, which is greater than 30.That means that we have "large samplesize," so we're allowed to use the methods that we'velearned here.And we're given and that of those 78 people,an average of 15 hours per week studying is what they have.So we have the sample mean of 15 hours.
INSTRUCTOR [continued]: We're also given the standard deviation.This standard deviation is not from the populationof students in the whole school or whatever.This is the standard deviation of the 78 people that we ask.Obviously some people are going to study more than 15 hoursa week.Some people are going to study less than 15 hours a week.But the standard deviation gives youa measure of how spread apart that datais about the mean, which is 15 hours, in this case.
INSTRUCTOR [continued]: So what's the margin of error for a 90% confidence interval?Well, we go back to our calculation.The margin of error E, which is Z subc times the standard deviation of the samples that we have,divided by the square root of n.Now, in this case, it's a 90% confidence interval.So this number, Z sub c, is completely bound to the fact
INSTRUCTOR [continued]: that this is 90%.It doesn't have anything to do with anything else.So a 90% confidence interval is going to give us 1.645.So what we get is 1.645.The standard deviation of the samples-- which in this case,is 78 students-- the standard deviation is 2.3 hours.So that just goes there.And the number of samples is 78.
INSTRUCTOR [continued]: So we just put a square root of 78 in there.And then what we end up getting iswe take this divided by the square root of 78, times this.We get 0.428 hours.This is a margin of error.This is what was asked in this particular problem.So if the problem says construct a confidence interval,then you have to do the whole interval.
INSTRUCTOR [continued]: If the problem says, tell me what the margin of erroris, then you just go through and find what that margin of erroris.And you know what that implicitly means.It just means if you take the point estimate, whichis the sample mean, you take the margin of errorless than the mean and higher than the mean--so you add it and you subtract it-- that gives youthe confidence interval.So in this particular case, you couldstop here and just circle this and say,
INSTRUCTOR [continued]: this is the margin of error.Because that's what was asked.But because we're getting practice,let's go ahead and calculate and find the entire confidenceinterval at 90%.So in this case, the sample mean thatcame out of this survey of students is 15 hours a week.And so for the confidence interval,what we're going to do is we'll say,sample mean plus the margin of error
INSTRUCTOR [continued]: is 15 plus 0.428-- which, of course, is 15.428.And of course, we could have x bar minus the margin of error,which is 15 minus 0.428.And when we do that subtraction, we would get 14.572.
INSTRUCTOR [continued]: And so-- I'll go ahead and draw this guy in another color here.The population mean-- notice we changed symbolsand we're using mu-- is going to be greater than thelower bound here, 14.572, and lessthan the upper bound, 15.428.And you could write it like this,or you can just write it with parentheses.
INSTRUCTOR [continued]: 14.572, comma, 15.428.This is a confidence interval at 90%.And what it means is we're 90% confident that if we somehowknew all of the data of all the students in the school,or whatever my population is, that I would be 90% confident
INSTRUCTOR [continued]: that if I could take a survey of every student,then the population mean of how many hours a week they studyis going to fall between 14.572 and 15.428, right?Because what we're using is we'reusing our sample mean of 15 hours as a point estimate.And then we go plus and minus the margin of errorthat we calculate.In this case, the margin of error
INSTRUCTOR [continued]: is dependent upon the level of confidencewe have, the standard deviation of our data,and how many samples we take.So it's kind of the same process over and over and over again.We're going to do some more problemsand get a little more practice.But these two problems really embody the whole spiritof what a confidence interval is.You want to learn about the population,so you take a small sample.But you want to have a sample more than 30
INSTRUCTOR [continued]: in order to use this.And this process is very simple.Look at how simple the math is here.We're just doing very simple multiplication and division,got a square root in there, and then some additionand some subtraction.And from that, and a relatively small number of peoplebut still greater than 30, you can reallyestimate a whole lot of very useful pieces of informationfrom that.And you can then specify how confident
INSTRUCTOR [continued]: you are that the population mean of whatever itis you're studying is going to fall between a band like this.So make sure you understand these problems.Work them yourself.Make sure there's no confusing bits here.And then we will move on to the next section,continue working some additional problems.
Series Name: Mastering Statistics, Vol 3
Publisher: Math Tutor DVD
Publication Year: 2014
Segment Num.: 1
Brian Gibson introduces and explains two confidence interval word problems. He demonstrates that the confidence formula can be used to determine any one of the missing variables.
Looks like you do not have access to this content.
Brian Gibson introduces and explains two confidence interval word problems. He demonstrates that the confidence formula can be used to determine any one of the missing variables.