V16 Sampling Distribution Theory

Welcome to part 11 of our video series on sampling distribution theory. In this video, we are going to cover the Chi Squared distribution, the Chi squared probability table, or also known as the table of critical values of the Chi Square distribution, how to use this table, and some practice examples with the table. I'm Renee Clark from the Swanson School of Engineering at the University of Pittsburgh.

Okay, on this slide is a picture of what the Chi Square distribution looks like. Okay, it is a continuous distribution, and you remember us in an earlier video talking about continuous data or continuous random variables. It is skewed to the right because it has a tail to the right. Okay, so, it is not a symmetric distribution. T and z are symmetric distributions, right, that both took a bell-shape. Okay, chi square is not symmetric. Okay, it ranges from 0 to Infinity along the x-axis. Okay, its parameter is nu, Greek letter nu, which nu refers to the degrees of freedom.

Okay, and, like the T distribution, nu is equal to n minus one, or sample size minus one. Okay, the convention is to write the critical value of the Chi Square distribution as follows. Chi Square sub alpha, okay, which sits along the x-axis. Okay, alpha is the area or probability to the right of the critical value Chi Squared Sub Alpha. Okay, so, alpha refers to this area or probability that is to the right of the critical value Chi Square sub… sub alpha.

Okay, so, let's talk a little bit more about the Chi Square distribution. It should be somewhat intuitive to you that exactly 95% of the area under the Chi square distribution, or under the Chi square curve, lies between these two values of chi square that sit along the x-axis. Okay, let's… let's explore why. Okay, so, remember these alpha values. These are alpha values. They refer to area to the right. Okay, so, in gold here, we see the critical value of Chi Square sub n -75. Okay, what that means is that particular value of Chi Square along the…along the x-axis is situated such that .975 of the area is to the right of that particular Chi Square value.

Okay, let's take the critical value of Chi Square sub .025 in green, okay. That is a critical value of Chi Square such that there is an area of 0.025 to the right of it, and I'm sort of shading that in in green here… here. That's a point along the x-axis such that there's an area of .025 to the right of it. Okay, that's why it sits further to the right on the x-axis than this critical value, because it has less area to the right of it. Okay, and as we've talked about before, if you want to get the area between two points, such as this point and this point, you basically take all the area to the right of that one, subtract off all the area to the right of that one, and what you're left with is the area in between. Okay, so, mathematically, the way to get that is area… all the area to the right of the first one, .975, which is the subscript, subtract off all the area to the right of that one, okay, and you're left with .095. Okay, that's why we say exactly 95% of the area under the curve lies between those two particular critical values. Okay, so, why this is significant is that it… these two critical values bound, if you will, 95% of the area under the Chi Square curve. So, putting that another way, that means that there's a very high probability of acurr… currence of a Chi Square value between those two particular critical values because 95% of the area under the curve lies between those two values.

Okay, high probability of occurrence of a Chi Square value between those two points. Okay, so, you might ask, well, but, why those two particular alpha values. Aren't there other alpha values that, when subtracted, give us a difference of 0.95? Absolutely. So, for example, if you were to subtract .976 and .026, you would also get 0.95. Okay, so, why are we focusing on those two particular values for alpha or those two critical values? So, the reason is because those two alpha values are shown in the table right there, and right there, and this is the table we're going to talk about… this next on the next slide. This is the table of Chi Square critical values and alpha values, which is set up just like the table for t.

Okay, so, those two particular values of alpha are in the table and, as we saw with t, only certain values of alpha are in the table or else the table would be infinitely wide, right? For every particular value of Alpha that could be, right, because alpha is just a probability. It's an area, so it's continuous. Okay, so, that's why we're focusing on those two particular values of alpha because they're actually in the table and we can easily read critical values.

Okay, alright, so, let's try some practice problems with the Chi Square table. This table, again, is set up very similarly to the T table, okay? Going to direct your attention to the visual legend. Okay, critical value of Chi, or Chi Square sub alpha, sits along the x-axis. Alpha is the area to the right under the curve- to the right of that critical value under the curve, okay? So, again, paying attention to your visual legend with any of these tables is very important. It tells you how to interpret…interpret the values you're getting out of the table. Okay, so, let's try a practice problem.

What is the value of Chi Square? Values of Chi Square are shown in the middle of the table. Degrees of freedom, or nu, down the left. Values of alpha, or areas, across the top. Okay, what is the value of Chi Square? So, we want to read one of these values having 8 degrees of freedom. So, nu is 8. I'm going to underline that right there. Such that there is an area of .9975 to the right. Okay, alphas. What is the convention? Alpha is area to the right. So, very simply, we find that alpha value right there. Okay, and we are going to then simply read off the critical value of Chi Square that's at the intersection of those two points, okay? So, Chi Square .975 for 8 degrees of freedom is 2.18. Sits right there at that intersection.

Okay, let's try a second practice problem. Okay, before I do that, just like the T table in the back of the book, the Chi Square table spans two different pages. On the second page, the degrees of freedom are the same. But, the second page provides additional values of alpha. Okay, so, practice problem. What is the value of Chi Square for a sample size of 14 having an area of .90 to the left? Okay, couple of things. In order to get nu, or degrees of freedom, sample size minus one. Okay, so, that's going to be equal to 14 – 1, or 13. Okay, so, we're dealing with 13 degrees of freedom. So, I'm just going to underline that right there. Okay, so, we're trying to find a value of Chi Square sits in the middle of the table having an area of .90 to the left. Okay, so, let's draw in Chi Square here. Okay, so, that's a lot of area to the left. Probably sits somewhere around in here.

Okay, so, we're looking for that 0.90 to the left. Now, alpha, if you remember from our visual legend, alpha refers to, of course, area to the right. Okay, so, if this particular Chi Square value has 0.90 to the left, by subtraction it must have 0.10 to the right. Okay, so, we are looking at a Chi Square… we looking for the Chi Square value of…with an alpha of 0.1. Okay, so, great. We are going to find our alpha of 0.1 across the top and read the value off that's at the intersection of those two points. So, that particular critical value of Chi square that sits along the x-axis is 1.9812. Okay, it's a larger value, okay, because it… it's further… it's somewhat far over to the right having only 0.1 area to the right of it.

We wish to thank the National Science Foundation under Grant 2335802 for supporting our work. Thank you for watching.