everyone. We just learned how to create a confidence interval for a population mean, mu, when we know our population standard deviation, sigma. But often, we won't actually know what sigma is. So how can we create a confidence interval then? Well, whenever sigma is unknown, we can actually use our sample standard deviation s in order to estimate sigma. Now in this case, our sampling distribution of xbar, which now relies on s instead of sigma, follows what's called a t distribution. Now here, we're gonna walk through what a t distribution is and how to find its associated critical values so that we can ultimately construct a confidence interval, whether we know sigma or not. So let's go ahead and get started here. Now taking a look at our familiar normal distribution, we know that it has this symmetric bell shaped curve with a mean of 0 and a standard deviation of 1. Now taking a look at our t distribution over here, we can see that it has a pretty similar shape to our normal distribution, but it's much shorter and wider. So it still has a mean of 0, but it has a standard deviation that will always be greater than 1 because it actually depends on the sample size. Now our t distribution also has an additional parameter to consider called degrees of freedom, which we're ultimately going to use to find our critical t value. Now it's not hugely important that you understand exactly what degrees of freedom are. But it is important that you know that our t distribution has n, our sample size, minus 1 degrees of freedom. So as our sample size increase, our degrees of freedom increase as well. And here on our curve, we can see that as our degrees of freedom increase from 1 to 30 to 100, our curve starts to look more and more like the curve of our normal distribution. So as our degrees of freedom increase, t gets more and more normal. Now the most important thing with our t distribution here is that we're able to find our critical t value, which we can ultimately do by looking it up in a table or by using our calculator the same way we did for our critical z value. Let's get a bit more familiar with this process by working through an example here. Here, we're asked to find and compare the critical values for a 95% confidence interval with an alpha over 2 of 0.025, where sigma is known and where sigma is unknown using a sample size of 31. Now starting with part a here, we want to find our critical value to build a confidence interval with a known value of sigma. Now whenever we know that population standard deviation, we need to use our critical z value, z alpha over 2. Now remember, to find z alpha over 2, we just need to know what alpha over 2 is, which we're given in our problem as 0.025. So coming up here to our z table, remember that we're using that alpha over 2, which is in the body of my table here, in order to ultimately find z. So here with this 0.025, I can see that going out to the left and up to the top that this corresponds to a critical z value of negative 1.96. Now remember that this is a negative value because looking at our curve here, we know that this is the area to the left of whatever our boundary is. So this ultimately gives us negative z alpha over 2. Now that means that our positive z alpha over 2 is then positive 1.96. Now moving on to part b here, we want to find our critical value to build our confidence interval when we don't know what sigma is, when sigma is unknown. Now whenever we don't know that population standard deviation, we instead need to know our we instead need to use our critical t value, t alpha over 2, or you may also see this written as t c. Now in order to find our critical t value, we still need to know what alpha over 2 is. It's the same thing, 0.025, given to us in our problem. But we also need to know one more thing, our degrees of freedom. Remember that for our t distribution, we have n minus 1 degrees of freedom. So here with a sample size of 31, 31 minuteus 1 gives me 30 degrees of freedom. So coming up here to my t table, I now need to use alpha over 2 and my degrees of freedom in order to ultimately find t. Now because our t table was specifically designed for use with confidence intervals, we can see these alpha over 2 values across the top here and our degrees of freedom down the side. So we no longer have to worry about this being the area to the left because it's ultimately just the area in either tail, alpha over 2, given to us in this table. So here, looking across the top for my alpha over 2 value of that 0.025, I know that my value will then be in this column. And then with 30 degrees of freedom here, we have this row. Now we can see that those intersect right here, giving us a critical t value of t of 2.042. Now depending on what t table you're working with, some of them might explicitly give you that confidence level. But remember, it's always a good idea to know what alpha over 2 is because you might not be given that. That. Now comparing our critical t value to our critical z value, we can see that our t value is greater for the same level of confidence. And this will always be the case because our t distribution is wider and ultimately has a greater margin of error. Now, one final thing that I want to mention here is that we can only use either of these 2 critical values, z alpha over 2 or t alpha over 2 to construct a confidence interval if our sample is random and either x is normal or our sample size n is greater than 30. So remember to double check for those things before you proceed in creating a confidence interval using these critical values. Now that we know how to find our critical values, whether we know sigma or not, let's continue getting a bit more practice. I'll see you in the next one.