V17 Sampling Distribution Theory

Welcome to Part 12 of our video series on sampling distribution theory. In this video, we're going to cover the F distribution. We're going to cover the F probability table or, otherwise known as, the table of critical values of the F distribution. We are going to cover how to use this table and do some practice examples with it as well. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, so, this is what the f distribution looks like, or a picture of it. It looks very similar to the Chi Square distribution which we covered in the previous video. It is a continuous distribution as well. Okay, it is skewed to the right also because it has a long right tail. Okay, therefore, it is not symmetric… is not a symmetric distribution like the T distribution is or the Z distribution. Okay, it ranges from 0 to positive Infinity along the x-axis just like the Chi Square distribution does. But, where it differs from the Chi Square distribution is that it has two parameters: nu one and nu two. So, it has… the F distribution has two different degrees of freedom. Okay, each one is the sample size minus one.

Okay, however, the order in which you specify the two different degrees of freedom matters, and we will see that in a… in an example problem. Okay, the convention is to write the critical value of the F distribution as F sub alpha, just like you see right there. So, F sub alpha sits along the x-axis. Alpha, again, is the area or probability to the right of f sub alpha. Okay, so, alpha here represents all the area under the curve to the right of the critical value, F sub alpha. That is the convention. Okay, so, this is one of the F probability tables in the back of your book, and it differs somewhat from the table of critical values for the Chi Square distribution. And, the reason for that is because we now have two different values for nu, or two parameters, nu one and nu two. Nu one values are specified across the top of the table. Nu two values are specified down the left hand side. Okay, and, because these column and row headers are now taken up by values for nu one and nu two, that limits the alpha values that can be specified in these tables. Okay, so, the… for this particular table, show the alpha value is .05. And that you know because the subscript there is shown as 0.05.

Okay, let me draw your attention to the visual legend. Okay, hard to see but F sub alpha is your critical value that sits along the x-axis. The area to the right of it under the curve is alpha. Okay, so, alpha refers again to the area to the right of the critical value. Okay, now, in the back of your book, there are actually two pages associated with alpha of .05. Okay, the first page has nu values- nu one values- that range from… from 1 up to 9, and then, that's page one, and then the second page picks up from there with a nu one value starting at 10 all the way up to Infinity. Okay, the values for nu two remain the same across both pages. Okay, values for f, of course, sit in the middle of the table and are read from the middle of the table. Okay, so, let's try an example. What is the F value for nu 1 equal 4 and New 2 equal 6 degrees of freedom with an area of 0.05 to the right? Okay, so, that means alpha equal .005. Okay, the convention is alpha is area to the right. So, we just simply need to read this off of the table. Okay, we need to make sure we're in the alpha equal .05 table, and we know that because this header up here has a subscript of 0.05. Okay, so, we're in the right table. We just have to find nu 1 of four, right there, nu 2 of six, right there, and read off the value that sits at the intersection of those two points. And that value is 4.53.

Alright, in the back of your book, there are also two pages of critical values for alpha equal .01. Okay, so, you can see right there- F sub .01. Okay, so, there are critical values for alpha of .01 as well, and there's two separate pages. Okay, first page goes from nu 1 equal 1 up to nu 1 equal 9. The second page picks up from there- goes from nu 1 equal 10 out to infinity.

Okay, okay, so, in the back of your book, you know, as far as your book is concerned, critical values are only given for two different alpha values: 0.05 and .01. And that's just a matter of space in the back of the book. Other alpha values could certainly be specified, but there's limited space in the back of the book. Okay, and…and that's because, you know, our column and row headings are now taken up by two different values of nu, or two parameters, and so that's why you know the alpha values are limited in this case. Okay, so, in reality, generally we allow software to calculate probabilities associated with the F distribution. However, it still is good to be able to use the F table and have an understanding of it.

Okay, so, in terms of being able to directly read critical values of f out of these tables, you can only directly read critical values for f sub .05 and F sub .01- the two different alpha values that are available. However, there's a bit of a silver lining. You can also calculate critical values for two additional values… critical values of f. Those having an alpha of 0.95 and 0.99. And what you'll notice here is that these two additional alpha values are complimentary, if you will, to the ones that are in the table, right? .95 and 0.05 add to one. .99 and 0.01 also add to one. Okay, so, um… um… theorem… by theorem 87 in your book, that is how we can… we can use that to calculate these two additional critical values for f. And eight… 87 reads as follows: the f of 1 - alpha nu one nu 2, okay, is equal to the reciprocal of f of alpha. Let's take a note that this subscript, 1 minus alpha, and alpha are complimentary, and that they add to one, right? And that's why I was saying this- and this add to one, right? Just as .99 and 0.01 add to one.

Okay, so, it's equal to the reciprocal, but the order of the nu’s is now changed. Instead of nu one, nu two, it's in the denominator: nu 2, nu 1. Okay, so, let's try an example that will help to clarify this. Okay, say we want to determine the value of f having 11 and 7 degrees of freedom and an area of .99 to the right. Okay, so, what we're trying to find out is f of N- .99, 11, 7. That's what we're trying to determine. Okay? Okay, so, that's going to be equal to the reciprocal of the f value having a complementary alpha subscript, right? Go back to our theorem here: if 1 - alpha is .99, then alpha in the denominator is going to be .01, and those values we can read directly from the table. But, you do have to watch in that, here, nu one is 11, nu two is 7. Those get reversed in the denominator, so you say 7 comma 11. And, remember, on the first slide, or… or an early slide, we said the order of specification of the nu… the nu values matters. Okay? Okay, so, when we then finish this out, we need to find the f critical value alpha .01, 7, 11. Okay, easy enough, we can read that directly out of the table. This is our alpha equal .01 table. We know that because it's specified right there. Okay, nu one is seven. Nu one is across the top. Nu 2 is 11, down the left hand side. We're going to read off the f value… val that's at the intersection of those two. Okay, 4.89. So, in the denominator is 4.89. Okay, and when we do the math on that we get .204. Okay, so, that's the critical value from the F distribution having an area of 0.99 to the right and 11 comma 7 degrees of freedom. Just fill this in 1 over .01, 7, 11 over 4.89.

Okay, we wish to thank the National Science Foundation for supporting our work under Grant 233582. Thank you for watching.