Welcome to Excel 2013 statistical analysis video
number 47.
Hey, if you want to download this workbook for Chapter 7,
and this is the second file, click on the link
below the video.
Hey, we got to talk about our last example for Chapter 7
and talk about the central limit theorem.
Now we've already done a bunch of examples
using the central limit theorem, but let's review and do
one last example for our Chapter 7 before we move
on to Chapter 8.
Hey, central limit theorem, in selecting a random sample
of size n from a population, the sampling distribution
of x bar or sample means can be approximated
by the normal distribution as sample size becomes larger.
Hey, if the distribution is symmetrical in the first place,
then and greater than 10 will work.
If it's thick tailed or skewed, we can use n greater than 30.
If it's heavily skewed, we can use n greater than 50.
So as long as our n is big enough, we just take a sample
and compare it to the standard normal curve.
So use of the central limit theorem,
we can reason about the sampling distribution
of x bar with absolutely no information about the shape
of the original distribution.
This means we could take one sample
and compare it to the standard normal curve using norm dot
s.dist, or the normal curve norm dot dist.
Now most of the time we're using normal curve.
But each time we're checking to see if our sample
result is reasonable or not.
If the sample mean seems reasonable,
then the original process or claim seems reasonable.
If the sample mean does not seem reasonable,
then the original process or claim is not reasonable.
So we want to come over and see our last example.
And just to get a visual of what that reasonable means,
here's a distribution.
If we go ahead and get a sample mean in here,
it seems reasonable.
Way out here, it doesn't seem reasonable.
And we'll explain more about that later in this video.
Now here's our example, and it's similar to the one
we did earlier with some slightly changed numbers.
Hey, this example is the same example
we'll start with next chapter.
So an insurance company claims that the average cost
per year of the policy is $822.
The standard deviation for the population is known to be $202.
What is the probability of getting the sample mean
plus or minus about $23?
If we take a random sample and get
a sample mean of $850 with n equals to $300,
is the original claim of $822, does it seem reasonable?
Now here in this example, we have the sample data.
So right down here, sample x bar.
Remember, it's a point estimator of the population parameter.
We're going to go ahead and take our sample.
It had to be a random sample.
We have our data over here, Control Shift Down Arrow,
Shift Enter to put that in the cell and jump up.
So we actually took our sample here of $850.
Now in this problem the margin of error is listed up here.
But let's go ahead and calculate that.
We need to calculate our standard error.
As we saw in our last video, standard error, here's
the formula here.
Most of the time you're not using the correction factor.
And we want to check.
But first, let's calculate our n equals--
and I'm going to use count, because we're counting numbers.
Control Shift Down Arrow, Shift Enter.
So that sample had a size of 300.
Now we can check to say, hey, is our sample size compared
to our population size, is it less than or equal to 5%?
You bet.
So we don't need to use that correction factor.
Now let's calculate our standard error.
We'll need that when we're calculating our probability,
because remember, this is the sampling
distribution of our x bars.
So we need to dramatically reduce
the sigma using our standard error formula equals sigma.
That's given in this chapter.
Next chapter we'll see what to do when it's not given.
And we'll take the square root of our sample size.
So there we go, $11.66.
I'm going to come up here and yes, we
had our margin of error.
But oftentimes, we're calculating on a margin error
based on a number of standard deviations.
So in this case, we want to have two standard deviations
on either side of our population means.
So I'm simply going to multiply that times 2.
Right down the middle of this curve
is the actual population mean mu,
which of course, is equal to our expected x bar.
So that-- broop-- right down the middle,
we're going to subtract some and add some.
Let's calculate our low x bar.
There's the population mean for our expected x bar.
And we're going to subtract our margin of error x
bar on the upper side.
We're going to take our population mean or expected x
bar and add two times our standard error, and Enter.
Now we should be able to already guess
what the probability is, because remember, z is
number of standard deviations.
And so many of the calculations we've done so far in this class
are all about how many standard deviations.
And two standard deviations is about 95%.
But let's go ahead and do our formula.
We do norm dot dist.
Hey, it says x.
But we're talking x bar here.
And we're perfectly allowed to put x bar in there.
The bigger, comma, our mean, not standard deviation
of the population.
It's our standard error.
Comma 1 for cumulative.
That'll go from the low end, Broop!
All the way up to this bigger x bar, minus norm dot dist.
It says x, but we know these are x bars.
That's the lower.
The second one that gets subtracted
is always the probability of the lower.
Comma mean, population or expected x
bar standard deviation.
That is the standard error.
Comma 1, close parentheses, Control Enter.
It says $0.95, but we know that's number formatting.
And I'm going to go up to Home, dropdown for General
or Control Shift Tilde or Grave Accent.
So the probability that we could go out and get
an x bar between this lower and upper is 0.95.
Visually, that's what this means here.
Broop!
There's the lower.
There's the upper.
All of this probability in here, this
is where most of the values lie.
So if we go out and get an x bar in here, we can say,
hey, you know, it seems reasonable we'd get one here.
Now there's a 2.5% risk on either side
that we could get a really small x bar.
But it's so unlikely that we could get an x bar out here
that when we go and hit 850 and it's not in this interval--
it's actually way out here--
then we say you know, it's so unlikely
that we could get that 850 in an actual sample, that we're
going to say the original claim of 822 doesn't seem reasonable.
Now we could say this a few ways.
We are 95% sure that our x bar will
lie between about 799 and 845.
We also can say to explicitly recognize
that there's some risk, we run a 5% risk
that are x bar will lie outside the interval 799 to 845.
Down here, our x bar of 850 is outside our interval.
And because the chances of this are so small, like 5%,
we say that it does not seem reasonable
that we could take a sample and have gotten 850.
Our sampling error is not acceptable.
Let's look at a picture down here.
These are all the x bars.
This is the sampling distribution of x bar.
There's our mean.
There's are standard error, the lower, the upper.
Now our sample, we went out and got it--
and even though there's a chance it's out here--
we are going to say that it seems too unlikely to get this,
so the original claim is not reasonable.
But if we had gone out and gotten an x bar of 837,
it's within this range, we would then say hey,
the original claim seems reasonable.
Now in Chapter 9 we'll use these techniques here
in hypothesis testing, and we'll be even more careful
with the language.
We'll say things like, even though 2.5% of sample means
would occur out in this range right here,
because getting a sample of 850 is so unlikely,
we say that from our sample evidence,
it seems unlikely that our original claim seems
reasonable.
Because samples almost always have sampling error,
the sampling error with x bar equals
850 is not acceptable, whereas the x bar with 837,
that sampling error is acceptable.
Now in this chapter we have created point estimates
and compare them to the sampling distribution of x bar.
And we make statements like we are
95% sure that the x bar lies between 799 and 845.
Now next chapter, when we don't know the population mean
or standard deviation for the population,
we're not going to be able to make statements like this.
But we'll go out and take a sample.
And instead of comparing x bar directly to a known population
mean, we'll make what's called a confidence interval.
And we'll have similar statements,
but they'll be called confidence interval.
All right.
I can't wait for Chapter 8.
See you next video.