- [Instructor] In a previous screencast,
I talked about how we can put together confidence intervals
on the mean when the variance is known.
In this case, we're going to talk about
how we can make these confidence intervals
when the variance is unknown.
In particular, we're going to look at this example.
We collect a sample of the weight in grams
of 5 Chipotle burritos.
These are all chicken burritos.
We do not know the true population mean
nor the population variance.
It's a very small sample.
Sample size is only 5,
and so we can't assume that the sample variance
is the same as the population variance.
And so, in this case, this is the variance unknown case.
We can collect sample statistics.
We get our sample standard deviation of 36.7,
our average x bar of 534.8.
And we want to put together a 95% confidence interval
on the true mean of the population,
so all Chipotle chicken burritos.
As I emphasize in one of the previous screencasts,
whenever we have t sub alpha, n minus 1,
this corresponds to the t-value that has alpha probability
to the right of it, not to the left.
And you have to be really careful
when you're determining these t-values.
In fact, some Excel functions,
I believe the T.INV function,
will give you the cumulative up to that point.
But if we're looking at these t-values in tables,
for example, in your book,
these correspond to t-values having alpha
to the right of it.
Also, because of the symmetry of the t-distribution,
we know that t alpha, nu is equal to -t 1 minus alpha nu.
For the variance known case,
we put together the following two-sided confidence interval.
So the probability that mu, the true population mean,
is between two values is 1 minus alpha.
For example, if alpha is 0.05,
that's a 95% confidence interval.
These confidence intervals don't have to be 95%.
We could do 98% confidence intervals.
We could do 90% confidence intervals, 99 and so on.
So you can tune or choose what alpha is.
For the variance unknown case, we make a couple changes.
Instead of sigma, the true population standard deviation,
we use s, which is the sample standard deviation.
We don't use the standard normal distribution,
instead we use the t-distribution
because we have more uncertainty.
The t-distribution has n minus 1 degrees of freedom,
and so we use those t alpha divided 2, n minus 1 parameters
in our confidence interval,
and we can calculate our confidence interval
as shown down here.
So let's work through our example.
What is a 95% confidence interval
on the true population mean?
We can use our equation here.
If you look this up, t alpha divided by 2
with n minus 1 degrees of freedom,
that's gonna be t 0.025,4.
We can go into a table in the back of our book.
We have 4 degrees of freedom,
that's the fourth line here.
Alpha is 0.025.
Even though our alpha for our two-sided confidence interval
is 0.05, we have to divide that by 2.
So in this example,
we're gonna be splitting 0.05 into two tails.
So we're gonna be using the 0.025 where it intersects 4,
and that's right here.
So our t-value, this is also known as percentage points,
is 2.776.
So if we substituted 2.776 for both of these values,
we used our s and our X bar calculated above with our n,
our population size of 5,
we would end up with our confidence interval,
our 95% confidence interval here, is 489.2 to 580.4.
That means, based upon our sample
of these 5 chicken burritos, the mass,
we can be 95% certain that the true population mean
of chicken burritos is between 489.2 and 580.4,
and we would run a 5% risk
that the true main is not in that interval.
By the way, we can also use the T.INV.2T,
that means two tails function in Excel.
I would caution you on using the T.INV function in Excel
because it is cumulative from the left.
Hopefully, this screencast gave you a better idea
of how we can calculate confidence intervals
on the mean when the variance is unknown.