- [Instructor] We are asked what is the critical value,
t star or t asterisk, for constructing
a 98% confidence interval for a mean
from a sample size of n is equal to 15 observations?
So just as a reminder of what's going on here,
you have some population.
There's a parameter here,
let's say it's the population mean.
We do not know what this is, so we take a sample.
Here we're going to take a sample of 15,
so n is equal to 15, and from that sample
we can calculate a sample mean.
But we also want to construct a 98% confidence interval
about that sample mean.
So we're going to go take that sample mean
and we're going to go plus or minus some margin of error.
Now in other videos we have talked about
that we want to use the t distribution here
because we don't want to underestimate the margin of error,
so it's going to be t star times
the sample standard deviation divided by
the square root of our sample size, which in this case
is going to be 15, so the square root of n.
What they're asking us is
what is the appropriate critical value?
What is the t star that we should use in this situation?
We're about to look at, I guess we call it a t table
instead of a z table, but the key thing to realize
is there's one extra variable to take into consideration
when we're looking up the appropriate critical value
on a t table, and that's this notion of degree of freedom.
Sometimes it's abbreviated df.
I'm not going in depth on degrees of freedom.
It's actually a pretty deep concept,
but it's this idea that you actually have a different
t distribution depending on the different sample sizes,
depending on the degrees of freedom,
and your degree of freedom is going to be
your sample size minus one.
In this situation, our degree of freedom is going to be
15 minus one, so in this situation our degree of freedom
is going to be equal to 14.
This isn't the first time that we have seen this.
We talked a little bit about degrees of freedom
when we first talked about sample standard deviations
and how to have an unbiased estimate
for the population standard deviation.
In future videos we'll go into more advanced conversations
about degrees of freedom, but for the purposes
of this example, you need to know that
when you're looking at the t distribution
for a given degree of freedom, your degree of freedom
is based on the sample size and it's going to be
your sample size minus one when we're thinking about
a confidence interval for your mean.
Now let's look at the t table.
We want a 98% confidence interval
and we want a degree of freedom of 14.
Let's get our t table out, and I actually
copied and pasted this bottom part and moved it up
so you could see the whole thing here.
What's useful about this t table
is they actually give our confidence levels
right over here, so if you want a confidence level of 98%,
you're going to look at this column,
you're going to look at this column right over here.
Another way of thinking about a confidence level of 98%,
if you have a confidence level of 98%,
that means you're leaving 1% unfilled in
at either end of the tail, so if you're looking
at your t distribution, everything up to and including
that top 1%, you would look for a tail probability
of 0.01, which is, you can't see right over there.
Let me do it in a slightly brighter color,
which would be that tail probability to the right.
Either way, we're in this column right over here.
We have a confidence level of 98%.
Remember, our degrees of freedom,
our degree of freedom here, we have 14 degrees of freedom,
so we'll look at this row right over here.
So there you have it.
This is our critical t value, 2.624.
So let's just go back here.
2.264 is this choice right over here, and we're done.