- [Charlie] In the next part of the course,
we're gonna talk about these two cases known
as variance known or variance unknown.
So this short screencast is gonna kind of give you
an idea of what that really means.
So up until now, we've assumed that we know sigma,
which is the population standard deviation.
That's different from sample standard deviation,
which is lowercase s.
Generally this will never be known with absolute certainty.
However, many processes have a long history,
and sigma is essentially known.
So let's say there's been a factory producing a widget
for 10 years, and based upon that long history of data
you really know a standard deviation of the population.
Also as n gets big, the sample standard deviation s
approaches the population standard deviation sigma.
So the previous analysis can be used with large samples.
Even if the population standard deviation is unknown,
if you have a really big sample
then you can assume that the variance is known.
I wanted to kinda demonstrate this with an example.
This is a simulation in Excel.
I have some true population parameters.
I'm just assuming that these are
the true population parameters mu, which is 20,
and the variance, which is two.
And I've got up to I think 100 different samples
in Excel I have just randomly chosen
through a random number generator.
I'm calculating items that follow this distribution.
And what I'm doing for each line in this table
is I'm taking the average of up to
and including that sample.
So this third row would be there's three samples.
The fourth row would be if there's four samples.
I'm also calculating the variance, the sample variance.
Now again, the true parameters are 20 and two
for the average of standard deviation of the population.
And you see that when we have very few samples up to
at least five to 10, the sample average
and the sample variance in this particular simulation
are not matching the true parameters.
On the left here, this is the sample average,
which goes to the left axis.
It approaches 20 when we get up to very large sample size.
And the variance also kinda goes back and forth
until we're up to a certain sample size.
I can go ahead and recalculate,
and that updates all of the formulas on this sheet.
You can also press F9 if you want.
And you see that depending upon the simulation,
here's kinda an interesting one.
The sample average and sample variance,
it takes a long time to actually approach
the true population parameters, if at all.
So in this one, the average here
is approaching a little over 20,
and the variance is approaching a little less than two.
So bottom line, if you have few samples down here,
then you really don't know the sample variance.
Average tends to approach the actual average
a lot faster than variance.
So with small sample sizes as I just demonstrated,
sigma is not known, so we use
the sample standard deviation s
along with a new distribution that I'm gonna talk about
in a subsequent screencast called the t-distribution.
And this is known as the variance unknown case.
And as I noted above, if variance is unknown
but n is large, then we can use the normal distribution,
and this would be the variance known case.
I know this is confusing,
so I have a little flow chart here.
We start. Is the true population variance known?
That's what sigma squared is.
That's not the sample variance.
That's this population variance.
If it's known, then we can use the normal distribution
as we've done previously.
And in this case we can calculate a z value,
x bar minus mu over sigma divided by root n,
or you can use Excel norm dot dist function.
If the variance is not known,
then we can ask ourselves is n sufficiently large?
And some people will say n has to be greater
than 75 to 100, but your textbook recommends
if n is greater than or equal to about 40.
If that's the case, then we're gonna also assume
that this is the variance known case.
Again, in this simulation, when the sample size gets bigger
than about 40, our sample variance
is getting close to the population variance.
However, if the sample size is not greater than 40,
then you're gonna have to use something known
as a t-distribution.
This is known as the variance unknown case
where we calculate a t statistic, this formula up here,
and again I'll be talking about this
in a subsequent screencast.
But when we have the variance unknown case,
the analysis is a little bit different
than the variance known case.
This screencast just gives you an introduction
to the variance known and the variance unknown cases
that we'll be talking about in the future.
Thanks for watching.