- 00:02
Hello, welcome to this lesson in Mastering Statistics.We're going to continue talking about the confidence interval.We'll work a couple of simpler problems.We're not going to get into the full gloryhere on how to calculate these confidence intervals just yet.I want to solidify some more fundamental concepts with themfirst.These problems will be fairly simple in retrospect,but they'll be important for you to build your knowledge

- 00:24
and understanding.Here we have the first problem, a survey of 200 malesshows that they read on average of 15.7 hours per week.If the margin of error is 2.2 hours at a 95% confidencelevel, construct the confidence interval.So there's two key things that youneed to understand before you can solve this problem.

- 00:45
The first thing is, we discussed previously,that when we take a survey like that, or a sample like that,and get the sample mean, that the pointestimate for the population mean is goingto be equal to the sample mean.And here we're sampling 200 males,showing how often they read, or how much they read,15.7 hours a week.We're trying to find a confidence interval.

- 01:06
That confidence interval is goingto tell us how many hours per weekthe population will read at.And it's going to be about a lower boundon an upper bound that will contain the populationmeans, what we're trying to do.So let's write things down.We know from the problem that the sample meanis equal to 15.7 hours.That is, what we have surveyed, 200 males, right?

- 01:29
And we also know the margin of error, E,we said we would denote that with E is 2.2 hours.So this is hours, and hours.And we need to calculate a confidence interval.Very, very simple to do, really, once you understandand remember that the mean plus the margin of error

- 01:52
is 15.7 plus 2.2, that will give you 17.9.And the mean minus the margin of error is 15.7 minus 2.2,and you get 13.5.Now, if you remember from previous discussions,we said that whatever we get for the sample mean,we're going to take that as a point estimate

- 02:14
for our population mean, 15.7.So in other words, we survey these 200 people.We're going to assume, since the survey resultedin an answer of 15.7 hours, we'regoing to assume that the center of our confidence intervalis at that point.We're going to assume as a point estimatethat the population mean of the whole population is 15.7.

- 02:35
We know that's not right, that's why we'regiven the margin of error.And so this confidence interval isgoing to extend up from the point estimate to 17.9.And it's also going to extend below that guy, to 13.5.So do you see how the point estimatelies in the center of this thing that we're calling

- 02:56
the confidence interval?So the way you write it down is, youwould say 95% confidence interval isthe following-- the way would write it down,

- 03:17
is you would say that the populationmean is less than 17.9 and greater than 13.5.The way you read this-- for those of youwho haven't done a math class in a while-- whenyou have two inequalities like that,and you have the thing of interestin the middle-- this is the population mean.This is everybody in the population.The population is probably going to be

- 03:38
males, in whatever country or city you're talking about.And the parameter you're looking at is how many hours per weekthat they read.So we're saying the population meanis going to fall between these limits here.So you read it from the inside out.You go here, and you read this way,the mean is greater than 13.5, the mean is less than 17.9.

- 04:01
So you can write it that way.Another way you could write it, in terms of interval notation,is just put the lower number, 13.5 comma,and then the upper number.I like representing it like this,because I like seeing the actual variable there,but this is perfectly fine, too.You're listing a confidence interval.So this whole answer is really it.

- 04:21
Of course, you could choose to display it either way here.Now, why is it a 95% confidence interval?Well, we haven't really gotten into the details of that yet.In this problem, we were told that the margin of errorwas 2.2 hours at 95% confidence.So we're bypassing some details.In future problems, I'll give youthe tools to be able to calculate all that yourself.

- 04:44
Here you've been given the margin of error,and you need to construct the interval.So for your purposes, you just subtract the margin of error,and add the margin of error.Just to make it absolutely explicit,the distance from the point estimate to the lower guyis the margin of error.The distance from the center to the upper part

- 05:05
is also the margin of error.So you go from the point estimatedown one margin of error, and up one margin of error.That's how you construct the confidence interval.So similar type of problem, we'll just do it for practice.A survey of 600 people finds that they sleep an averageof 10.5 hours per night.If a margin of error at 98% confidence interval

- 05:27
is 1.3 hours, construct the confidence interval.So we're given how many people we sampled,we're told what the sample mean is-- 10.5 hours in this case.We're given the level of confidence, 98%,and were given the margin of error, which is 1.3 hours.So really, it's the same sort of thing.The sample mean, after we do the surveying, is 10.5 hours.

- 05:51
The margin of error is 1.3 hours.And in this particular problem, itdoesn't really matter the confidence levelor the number of people we surveyed,that's just extra information.In future problems, you'll use that informationto calculate a confidence interval.But here you've given the margin of error,so all you really have to do is say, OK, well, 10.5 plus 1.3

- 06:13
is 11.8.And 10.5 minus 1.3 is 9.2.You go plus 1 margin of error and minus a margin of error.And to write that down, you can say weare 98% confidence interval, that the population

- 06:34
mean is going to be greater than 9.2 and less than 11.8.Or, if you want to write it more as an interval,open in parentheses and say 9.2, comma, 11.8.Either one is acceptable.So to interpret the results, what you're sayingis you constructed an interval that ranges from 9.2 to 11.8

- 06:57
in this case.And this is the number of hours per nightthat you get sleep, right?And we're saying that the population average is goingto fall between these numbers.And we're 98% certain that if we could actuallytalk to everybody in the country,and get an answer from everybody,

- 07:17
and average it together, that the average number of hoursper week that we get from that is going to fallbetween these boundaries here.So you see, it's really impossible to do that.If we could just get answers from everybody,then we wouldn't have to ever use statistics too much.So what we do then, is we sample of a portion of people,we get an answer.And then we calculate this confidence interval and say,well, we don't know everything, but we're pretty darn sure.

- 07:40
How sure?In this case we're 98% sure that the population isgoing to fall in this interval.That's what the confidence interval is,and that's why it's important.It's very, very useful, especiallywhen you're looking at manufacturing.You might say you're 98% confident that your defectrate is going to fall within a certain range there--

- 08:01
of however many defects per week, or whatever,but you can't study every single cell phone coming off the line,so what you do is, you sample maybe 500 of them,and get a good number based on that.Now these two problems have been pretty simplified,because I've been given the sample mean,I've been given the margin of error.Now, in future problems, you're goingto learn how to calculate this margin of error yourself.And that margin of error is going

- 08:22
to be dependent on the level of confidence that you have there.And it's also going to be dependent upon your samplesize.So we haven't done any of that in these problems,I'm just kind of crawling before we walk.Once you get that margin of error, though, all you dois, you add it, and you subtract it to get the confidenceinterval that you have.So follow me onto the next section,where we'll build these skills.

- 08:42
You'll continue learning about confidence intervals,and how to calculate them.And we'll do it one step at a time,and I think you'll see this topic is one of the most usefulin all of statistics.

### Video Info

**Series Name:** Mastering Statistics, Vol 3

**Episode:** 10

**Publisher:** Math Tutor DVD

**Publication Year:** 2014

**Video Type:**Tutorial

**Methods:** Confidence intervals, Population mean

**Keywords:** mathematical computing; mathematical formulas

### Segment Info

**Segment Num.:** 1

**Persons Discussed:**

**Events Discussed:**

**Keywords:**

## Abstract

Given the point estimate and margin of error, Jason Gibson demonstrates how to compute and express a confidence interval.