V15 Sampling Distribution Theory

Welcome to Part 10 of our video series on sampling distribution theory. In this video, we're going to cover the T probability table, also known as the table of critical values for the T distribution. We're going to review how to use this table and do some practice… practice examples using it. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, so, this is the table of critical values of the T distribution, or more simply, the T probability table. Okay, I want to point out a few key things about it. First, down the left hand side of this table, you have your degrees of freedom, okay?

Okay, they… they're specified by the Greek letter nu. Okay, across the top of this table, you have values for alpha. Okay, alpha is the area to the right of T along the x axis. Okay, so, let me draw your attention to the visual legend which, when you're using any of these tables, you want to take a look at the visual legend first. As you can see from the visual legend, there is T that sits along the x axis. Okay, and alpha is the area under T that's shaded to the right of T. Okay, so, in the middle of the table are actual values of T. These are also known as the critical values of T. Okay, and they are denoted as T sub alpha, as you can see here along the x axis.

The critical value is denoted by T sub alpha, okay, because it has an area, or probability, of alpha to the right of it. Okay, so, let's try an example. What is the value of t with alpha equal to area to the right and 8 degrees of freedom? Okay, so we're going to find our value of alpha across the top: 2. We're going to go down along the left side and find our degrees of freedom, which is 8, and read the value off that's at the intersection of those two points. In this case, T-alpha: 0.889. Okay, and this value of T sits along the x-axis, okay?

Okay. There are actually two pages in the back of your book of critical values of the T distribution, okay. This is the second page, and what it contains are additional values of alpha. Okay, across the top, the degrees of freedom are the same as in the prior table. The second page of critical values for T values… of T are in the middle… have additional values of alpha across the top. Okay, so let's do some practice examples. Let's find the value of t such that 15% of the area is to the right for a sample size of 15. Okay, so, remember with t, our degrees of freedom are n minus 1. Okay, so, in this problem, the degrees of freedom are going to be 15 minus 1, or 14. Okay, so, we are going to find nu of 14, which is right there, okay, and we want to find a value of T. Remember, values of T are in the center of the table. Okay, but we want to find this value of t such that 15%... percent of the area is to the right. Okay, so, we know, again, let's look at our visual legend, alpha refers to the area to the right because that's what's shaded. So, in this case, alpha equals .15. So, we are going to read off the value that's at the intersection of those two points and that t value, and, if you want to write it like this, 1.76.

Okay, all right, let's try a different example. Find the value of t with N equal 4 that has 95% of the area to the left of it. Okay, so, again, in this case, degrees of freedom is going to be n minus 1, or 4 – 1, or 3. Okay, so, let's mark that right there: degrees of freedom is three, or nu is three. Now we want a value of T. Remember values of T are shown in the center, that has 95% of the area to the left of it. Okay, so, what the problem is telling us is that this value of T has 95% of the area to the left of it. Okay, that means this value of T has 0.05 to the right of it, right, because the total area under the curve must add to one. Okay, so, our alpha, in this case, is .05. Alright, right there, okay? So, we're going to read off the value at the intersection of those two. In this case, T is, if you want to write it as T .05, equal 2.353.

Okay, let's try another example. Let's say you're asked to find the value of t for 17 degrees of freedom… that it has an area of 0.001 to the left. Okay, so, again, let's draw a picture. That's very helpful. Okay, so, this is a value of T having an area of 0.01 to the left, okay? So, that's a very small area. Okay, so, that value of T is way out in the left hand tail, okay, and, in fact, we know it's negative, correct? Because it sits to the left of the mean of zero. Okay, so, if it's got an area of 0.001 to the left, okay, that means it has an area of .999 to the right.

Alright, so, if we go into our table, however, we see that the alpha only goes as high as 40 to the right. Okay, so, what do we do in this case? Okay, so, in this case, we have to make use of sym… the symmetry of the T distribution. Okay, if… because of symmetry, we know that there is a corresponding value of T, okay, that is the mirror image, if you will, of the value of T that we're interested in finding. That's a mirror image in terms of magnitude, okay, such that it has an area of 01 to the right of it. Okay, now, whereas the value we're looking for is negative, this one's going to be positive. But, they're going to be equal in… in… in… in magnitude, okay? Easy enough, we do have alpha of .001. So, we go into the table for an alpha of 1 and go down to 17 degrees of freedom, and we see that the value of T at the intersection of those two values is 2.567. Okay, so, this value of t with 0.01 to the right is 2.567.

Okay, as we said, due to symmetry, then the value that we're looking for is actually equal to -2.567, okay? Meaning if 2.567 has an area of .01 to the right of it, okay, then, due to symmetry -2.567 also has an area of .01 to the left of it.

Okay, let's try one additional practice example. Okay, this one asks us what is the area under the T distribution curve between a t of 2.715 and T of 4.029 for 7 degrees of freedom. Okay, so, this type of problem is one that really benefits from drawing a picture, so we're going to do that. Okay, this is our T curve. Going to draw in our two critical values for t.

Okay, and it's asking us, or telling us, there are seven degrees of freedom. Okay, so, we're going to just underline that right now. Okay, so, it's asking us for the area under the curve between these two values of t. Okay, so, this is the area that it's asking us for, and I'm going to highlight this in green here. Here this is what we are trying to solve for, okay? If you recall, if we take all of the area under t to the right of 2.715, okay, and we subtract off all of the area under t to the right of 4.029, okay, what we are left with is the area in between. Okay, and that's what we're being asked to solve for.

Alright, so, let's do that. Okay, so, for 7 degrees of freedom, the area to the right of 2.715, which is right there, is 0.015. Okay, so, I'm going to put that in blue actually. I am going… going to erase the green so all the area under t to the right of 2.715 is 0.0015. That's this alpha value right there, okay? For 7 degrees of freedom, all of the area to the right of 4.029 is an alpha of .0025. Okay, so, that is just going to superimpose that… that area is .0025.

Okay, so, doing the math, the area in between those two values of T is equal to 0.0015, which is all the area to the right of 2.715 under T, subtract off the area to the right of 4.029, which is .0025. Okay, when we figure out the math on that, our final answer becomes 0.125. Okay, that is again equal to the area that I'm going to color now in green. That is the area in between, under T, in between 2.715 and 4.029.

We thank the National Science Foundation under Grant 233582 for supporting our work. Thank you for watching.