Example finding critical t value

0:00- [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.

0:21We 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.

0:42Now 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.

1:02What 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.

1:27Sometimes 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.

1:47In 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.

2:07In 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.

2:26Now 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.

2:55Another 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.

3:21Let 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.

3:41So 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.