Example finding critical t value

We are asked what is the critical value t star (t asterix) for constructing a 98% confidence interval for the mean from a sample size of n, which 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. 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 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 was going to be 15, so square root of n.

But what they're asking us is: well, what is the appropriate critical value? What is the t star that we should use in this situation? And so we're about to look at a, 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 degrees of freedom.

Sometimes it's abbreviated df, and I'm not going to go 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 1.

So in this situation, our degree of freedom is going to be 15 minus 1. Thus, our degree of freedom is going to be equal to 14. This is 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 1 when we're thinking about a confidence interval for your mean.

So now, let's look at the t table. We want a 98% confidence interval, and we want a degree of freedom of 14. So let's get our t table out. I actually copy and pasted this bottom part and moved it up so you can 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% is: 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 you can't see it right over there. Let me do it in a slightly brighter color, which would be that tail probability to the right.

But either way, we're in this column right over here. We have a confidence level of 98%, and remember, our degrees of freedom; our degree of freedom here is 14. So we'll look at this row right over here, and so there you have it.

This is our critical t value: 2.624. So let's just go back over here, and so there you have it, we have 2.624 as our answer.