So I took this little snippet of text here from the Wikipedia page on ellipses. So, and they say, more generally, for an arbitrary oriented ellipse, centered at v, is defined by the solutions (x- v) transposed A (x- v) = 1. So V is the center of the ellipse. And then the first Igon Value of the matrix A, is the length of the major axis and the second Igon Value is the length of the minor axis. And if we say the points that satisfy less than or equal to one, those are all the points on the interior of the ellipse. And I think at this point in the class you would notice this equation is the same equation as the. Mahalova's distance between X and V, where a is a weight matrix, and also it would look a lot like the S statistic. So take for example if I were to take the hypothesis H not, K Beta = m versus Ha: K beta Not equal to M. Which we've discussed previously. Remember our variance of K beta hat is equal to K x transpose x inverse k transpose * sigma squared. And then what we saw was that, so then (k*beta hat- m) transpose * k( x transpose x) Inverse x transpose, ahh, not x transpose, k transpose, k transpose * sigma squared * (k beta hat- m). Inverse here, that follows a chi squared distribution. With the number of rows of K degrees of freedom. And then if we divide that by its degrees of freedom, which is, let's just say that the rank of K is let's say V, so divide by its degrees of freedom, then divide it similarly by, ( n- p ) s squared over sigma squared, which is also a kai squared, and then divide it by its degrees of freedom which is ( n-p Then we have the ratio of two Ki squared this part is the Ki squared with three degrees of freedom, this part is an independent Ki squared with n- p degrees of freedom. And then let's organize terms and again we have already done this but it's good for review I think, if we organize terms what we see Is that that sigma squared cancels with that sigma squared, that n- p cancels with that n- p. And then we get that K beta hat- M transpose, K x transpose x inverse K transpose inverse K beta hat minus M, divided by v s squared. It follows an f distribution with the v numerator degrees of freedom and n minus p denominator degrees of freedom. So we would Reject this hypothesis, K Beta equal to m, if this is greater than or equal an F critical value of 1 minus alpha v numerator degrees of freedom and n minus p denominator degrees of freedom. And so we would accept We'd accept the null hypothesis, we'd accept this hypothesis if it was, let's say if it was less than or equal to. Sorry, it probably should have been greater than for the other direction. Okay, so I could divide this by F 1-alpha. It's already Divided by vs squared and divided by F, 1 minus alpha, v, n minus p. And then we get less than or equal to 1. Okay, now we're just back up to this equation again, and so, what we see is that The set of values, Of m for which we fail to reject a null hypothesis, Forms, An ellipsoid. They form in the interior of an ellipsoid. Okay, and so that is given a term that's called a confidence Ellipsoid. Okay, and so a confidence ellipsoid is just a confidence interval, the generalization of a confidence interval to more than one dimension, okay. And we see that the form is pretty easy. Here it's our k beta hat is observed. Our m is the variable that's varying. And then this entire matrix right here is the A matrix from before, from the Wikipedia definition. So if you had say software that would for a given taken K beta hat and a, and plot a hyper ellipse, say in two or three dimensions, then you could create hyper ellipses for the settings. And we'll show in a coding example next how to do that in three dimensions, just because it's kind of neat. But there's a couple of times that you might want to do this, a good example is when and say linear regression where you want a joint confidence interval for the intercept and the slope. But also notice if you have two X not values, let's say x not 1 and x not 2 transpose, right. If I take those times Beta hat then I get say y not, 1 and Y knot 2. So if I want to predict my confidence surface at 2 points, okay then I might want to do that a joint confidence surface for that, okay. And then just to get our confidence ellipsoid and this is just a special case of this formula. And of course if you wanted to generalize a confidence, a prediction ellipsoid I think you could all do this at this point given this knowledge. So in the next lecture why don't we go through and create a confidence ellipsoid In this specific case. And I think it'll be pretty neat because we can use RGL and show how the confidence ellipsoid looks in three dimensions.