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## Confidence intervals for means

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# Example finding critical t value

AP Stats: UNC‑4 (EU), UNC‑4.Q (LO), UNC‑4.Q.1 (EK)

## Video transcript

- [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.

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