Welcome to Excel 2013 statistical analysis video
number 48.
If you want to download this workbook Business 210
chapter 8 or the PDFs for chapter 8, click on the link
below the video.
Hey, we're in chapter 8.
We're going to talk about how to construct confidence intervals.
Now, here are four main topics that we'll cover.
But before we jump into chapter 8,
I want to jump over to the PDFs and compare what we
did in chapter 7 and chapter 8.
Now, the PD have start at page 1,
but I have two pages before that don't have a page number.
So here, let's remind ourselves of what we did in chapter 7.
First main point we did in chapter 7
is we knew what the population mean mu was,
or we had a good estimate of it.
So we, in our insurance cost example, so annual cost,
we knew that the population mean was $822.
We then added a margin of error and subtracted
a margin of error to get a lower and an upper limit.
We then made statements based on a 95% probability.
We said, the probability of getting an X bar
between $799 and $845 is 95%.
That allowed us to take a sample and compare our sample
to the sampling distribution of X bar.
So we get, if we went out and got a sample of $837,
well, that lies within this range here.
So we assume that the original claim is reasonable.
In essence, we say well, $837, that's just sampling error.
So we'll assume that the original value
seems reasonable.
However, if we got a value out here or down here,
for example, if we got an X bar equals to $850,
well then, the sampling error, the distance between these two
is so big, we then say, well, maybe this original claim
is not reasonable.
Now, very clear, this, and I have it written down here.
The statements we made in chapter 7
are we believe that 95% of the X bars
will lie between these two values.
Chapter 7 we knew, or had a good estimate of population mean.
Next page, but what do we do if we don't?
So we just flat out don't have a good way of estimating this.
Well then, we go out and we take a sample.
We add margin of error to either side
to get what's called a confidence interval.
And at 95%, we say we're 95% sure that the population, mu
lies in our interval.
Notice something here.
There's our X bar of $837.
But we're not directly comparing it
to our sampling distribution of X bar.
However, we are going to use the same techniques, Excel
techniques and probability techniques,
that we learned in chapter 7.
It's just, we can imagine that there
was a bell-shaped curve over here,
and we'll use all the same techniques.
But technically, we're not measuring it up
against this sampling distribution,
because we just don't know what mu is.
Now if we go back to this previous page
here, see our X bar, we were.
We were comparing right up here, because we knew mu.
That page right here.
All right, let's go to our next page here.
So in chapter 8, we don't know what the population mean mu is.
Now let's look at another example, printer cartridge
manufacturer example, that'll help us kind of think
of situations when we really can never calculate mu.
And thus, we have to learn how to create confidence intervals.
All right, when a printer cartridge manufacturer
wants an average number of pages per cartridge.
Now, think about this right here.
You're a manufacturer of printer cartridges,
and you're going to run an ad.
And you want to say something in the ad
like the average number of pages you can expect
is, or on the back of a box, right, that you delivered,
it's got to say what the average expected pages are.
Well, you can't know mu, right?
And the reason why is you can't calculate the population
mean, because they can't go out and print all the pages for all
the cartridges, right?
So that's where confidence intervals come in.
So we're still going to use point estimates.
So we'll go out and take a sample.
We'll get 2,409 pages.
That's the average from our sample,
or the mean from our sample of number of pages.
But we can't use this, or we can't compare it directly
to the sampling distribution of X bar.
So what do we do?
We're going to add a margin of error to our point estimate.
And we'll do this example a couple videos ahead.
We'll get a margin of error of 217 pages.
So we'll create a 95% confidence interval
that says we're 95% sure that the population mean
lies between 2,190 and 2,620.
So you might even see in the ad or on the back of the book
or the manual that comes with the printer cartridge,
it would say something like 2,400 pages with a margin
of error of 200 pages.
Now, the statements we can make in chapter 8.
Again, we are 95% sure that mu will
lie between the two limits.
Back in chapter 7, we were comparing X bar.
Now, 95% sure that mu will lie in our interval here, right?
Let's talk a little bit more about that 95%.
We'll go to the next page for 95% confidence intervals,
because we don't know the mu, technically,
would look like this.
Sometimes our interval would contain mu, right?
We don't know this.
Mu, here's the sampling distribution
of X bar up here, but down here, we
calculate based on our sample mean, our interval.
But notice this one theoretically
contains this one.
This one contains the mu.
So does this one.
But this one does not.
So this just flat out, we calculated our sample mean,
added on margin of error, and then we say,
we're 95% sure that our population mean lives
within this interval.
But it's the 5% risk that it doesn't
lie in this interval that's illustrated here.
So if you were to create 100 similar intervals,
95 would contain the population, mu and 5 would not.
Now, this assumes, that this analysis here assumes,
that the original population distribution
is normal or bell-shaped or near normal or bell shaped.
If it's not, then, as long as you
have a big enough N from the central limit theorem,
then about 95 intervals would have the population mean
and 5 would not.
All right, next slide.
Let's notice something for the printer cartridge example.
And we already mentioned this.
We cannot calculate population mean.
Why?
Because we can't possibly test every cartridge
made or count every page for every cartridge in every house,
not possible.
In this case, we do sampling and create a confidence interval.
Other examples, we do not know how many meals married
couples eat out each week.
There is no way to calculate that, no database you
can go to get all that data.
We do not know what the mean amount of profit
is per auto sold in the US.
So three examples here, including
in our cartridge example, where confidence intervals is
a great way to estimate the population, mu.
Here's our formal definition of confidence
interval, an estimate of the population parameter
that provides an interval believed to contain
the value of the parameter.
Right, our formula, in the general form, will be this,
point estimate plus or minus a margin of error.
So for X bar, we'll have calculate
our X bar, plus or minus.
And we'll learn how to calculate the margin of error
in just a moment.
Now, we've been talking about 90%,
95%, 99% confidence interval.
So the confidence level, or confidence coefficient,
this textbook says confidence coefficient,
I tend to say confidence level, that's probability
that your population parameter is in the interval so, 90, 95,
99.
Now, that other little bit of probability 10, 5, 1
is called level of significance, or alpha.
And that's the risk, or the probability,
that the interval does not contain the population
parameter.
Our next slide will show our official formula
that we'll use, number 6 on the page, number 8,
I think in the counter down at the bottom the PDF.
Here's our confidence interval for mu.
Now, we're going to have two of these formulas.
And deciding which formula to use depends on sigma.
Sigma is the population standard deviation.
If you know it or you have a good estimate, which is not
usually the case, then you use the Z distribution
in our formula.
Now, sigma known, sometimes you have
a lot of population data amassed,
so you can calculate or have a good estimate.
Or in quality control calculations,
sometimes it's safe to assume that you know sigma.
So for some cases where we know sigma or it's a good estimate,
we can use Z. In a couple of videos ahead,
we'll see how to use the new distribution
called T, which is the more common usage, because you
oftentimes don't know mu or sigma.
But confidence interval when for calculating the interval for mu
when sigma known.
All right here's our formula.
Here's our variables.
Now, X bar plus or minus, that whole thing
is the margin of error.
Now, this little part right here we've seen before.
That's standard error, right?
Sigma divided by the square root of N.
That's the standard deviation of the sampling distribution of X
bar times, and what's Z?
Z is number of standard deviations.
Now, we'll talk about this alpha divided by 2 in just a moment.
Z we've dealt with a lot so far in this class.
So it's Z times the standard deviation for the sampling
distribution of X bar.
Now what's this alpha divided by 2?
Let's go back a bunch of slides, I think all the way
to-- go to one, no sorry, two.
Now here's our picture.
This is our sampling distribution of X bar.
And again, we're not comparing when
we're doing confidence interval directly to this.
But all the same math works.
So watch this.
If we're talking about 95%, the alpha, or the risk, is 5%.
Well, half of it has to be there.
And half of it has to be here.
So alpha divided by 2 is that little bit right at top.
So when we say Z of alpha divided by 2,
we mean Z on the upper end.
Notice, because the distribution is symmetric,
if you have Z on the upper end, you just
slap a minus sign in front of it for the Z on the lower end.
All right, I don't remember which slide we were on.
Oh, there it is.
All right, so that's Z on the upper end.
There is all the definition.
Notice they define confidence level as 1 minus alpha.
That's because alpha is the risk mu not in the interval,
so usually, you think about your risk up front.
And then you take 1 minus that to get your confidence level.
That means how sure you are that the population parameter's
within your interval.
Now, in our next video, we'll calculate all of these
and see Excel.
We're going to have method 1, 2, and 3.
NORM.INV function, we've already used a bunch.
We're actually going to be able to calculate
the upper limit and the lower limit
directly using the NORM.IN.
Now, in the textbook, they don't show you this.
They show you this method, 2 and 3.
But I'm going to show this, just because if we've already
been using these, and we can use these to calculate
an upper and lower limit.
Method 2, calculate the Z upper.
We saw this one earlier also, NORM.S.INV.
That gives you that Z. And then you already
have the data for calculating this,
so you multiply those together.
And that's the margin of error.
So method 1, 2.
And method 3, here's a brand new function
called CONFIDENCE.NORM.
A couple videos ahead, we'll learn about the T distribution.
And it will be called CONFIDENCE.T.
But CONFIDENCE.NORM means Z distribution.
Now notice, here we're going to have to calculate,
for the NORM.INV, we're going to have to calculate
the standard error.
For the NORM.S.INV, all we need is the probability
from negative infinity up to alpha divided by 2.
But notice this, we don't have to make many calculations.
We have alpha.
Since sigma is known, we're given that,
and we have this the sample size,
you just plug these in and CONFIDENCE.NORM,
because it's got sigma and N will
calculate the standard error.
And alpha, obviously, can divide by 2 and do all that.
So this function's real nice, because you can quickly
get what's the margin of error.
And that's directly the amount added and subtracted
from your sample mean.
All right, so in our next video, we'll
finally get down to some calculating.
We'll use all the data from this page here.
All right, see you next video.