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- [Instructor] We are asked
what is the critical value,

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t star or t asterisk, for constructing

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a 98% confidence interval for a mean

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from a sample size of n is
equal to 15 observations?

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So just as a reminder
of what's going on here,

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you have some population.

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There's a parameter here,

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let's say it's the population mean.

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We do not know what this
is, so we take a sample.

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Here we're going to take a sample of 15,

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so n is equal to 15, and from that sample

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we can calculate a sample mean.

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But we also want to construct
a 98% confidence interval

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about that sample mean.

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So we're going to go take that sample mean

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and we're going to go plus or
minus some margin of error.

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Now in other videos we have talked about

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that we want to use
the t distribution here

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because we don't want to
underestimate the margin of error,

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so it's going to be t star times

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the sample standard deviation divided by

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the square root of our sample
size, which in this case

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is going to be 15, so
the square root of n.

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What they're asking us is

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what is the appropriate critical value?

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What is the t star that we
should use in this situation?

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We're about to look at, I
guess we call it a t table

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instead of a z table, but
the key thing to realize

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is there's one extra variable
to take into consideration

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when we're looking up the
appropriate critical value

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on a t table, and that's this
notion of degree of freedom.

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Sometimes it's abbreviated df.

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I'm not going in depth
on degrees of freedom.

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It's actually a pretty deep concept,

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but it's this idea that you
actually have a different

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t distribution depending on
the different sample sizes,

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depending on the degrees of freedom,

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and your degree of freedom is going to be

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your sample size minus one.

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In this situation, our degree
of freedom is going to be

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15 minus one, so in this
situation our degree of freedom

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is going to be equal to 14.

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This isn't the first time
that we have seen this.

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We talked a little bit
about degrees of freedom

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when we first talked about
sample standard deviations

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and how to have an unbiased estimate

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for the population standard deviation.

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In future videos we'll go into
more advanced conversations

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about degrees of freedom,
but for the purposes

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of this example, you need to know that

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when you're looking at the t distribution

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for a given degree of freedom,
your degree of freedom

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is based on the sample
size and it's going to be

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your sample size minus one
when we're thinking about

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a confidence interval for your mean.

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Now let's look at the t table.

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We want a 98% confidence interval

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and we want a degree of freedom of 14.

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Let's get our t table out, and I actually

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copied and pasted this
bottom part and moved it up

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so you could see the whole thing here.

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What's useful about this t table

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is they actually give
our confidence levels

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right over here, so if you
want a confidence level of 98%,

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you're going to look at this column,

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you're going to look at
this column right over here.

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Another way of thinking about
a confidence level of 98%,

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if you have a confidence level of 98%,

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that means you're leaving 1% unfilled in

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at either end of the
tail, so if you're looking

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at your t distribution,
everything up to and including

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that top 1%, you would
look for a tail probability

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of 0.01, which is, you
can't see right over there.

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Let me do it in a slightly brighter color,

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which would be that tail
probability to the right.

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Either way, we're in this
column right over here.

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We have a confidence level of 98%.

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Remember, our degrees of freedom,

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our degree of freedom here,
we have 14 degrees of freedom,

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so we'll look at this row right over here.

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So there you have it.

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This is our critical t value, 2.624.

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So let's just go back here.

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2.264 is this choice right
over here, and we're done.