all right in this video we are actually kind of taking a step back from hypothesis test for just a quick second and we are going to actually revisit confidence intervals specifically for variants so we save our variant stuff until the very end it makes things a little bit funky in the timeline but makes the most sense to do confidence intervals for variance and hypothesis testing for variance at the same time so that's what's going on here so you'll see on your formula sheet you have your confidence interval for variance equation and you also have your degrees of freedom and so you might notice that there's some funky letters in here that we haven't seen before um so this is Kai squared it's a squiggly Little X thing this is Kai squar and this is Kai like KY kind of thing um so we're going to jump in to what this is so the Ki Square distribution is something like this so it kind of looks normal but it's definitely skewed um but you're using the same theory that we've been using for confidence intervals and hypothesis testing for as long as we've been going um so you have your one minus Alpha in the center with Alpha over two on each Edge and yeah so variance is um covered by the kai squar distribution um your estimator is going to be S squar just like when we were talking about um confidence interval for mean our estimator was xbar from our sample pop from our sample and then our parameter that we are trying to estimate here is Sigma squar so this is similar to when we talked about mean and we had xar and mu just to kind of relate that for you all right so we're going to jump into an example so just like with your standard normal and your T distribution you have a z and T table we also have a Ki squ table um that helps us read our probabilities um so take note that the kai Square table reads from this direction and then you'll see when we get into our equation that these are kind of backwards from what we've been doing so in the past we've had the smaller number here and the larger number here but now these are swapped um and you'll see that when we do our example this is because we are now dividing by them so we want to divide by the larger number so that we get the smaller number on this side and divide by the smaller number here to get the larger number here so don't get freaked out by that um just follow the equation and you'll be okay all right the following are the weights and decagrams of 10 packages of grass seed distributed by a certain company finding 95% confidence interval for the variance of the weights of all such packages of grass seed distributed by this company assuming a normal population so I'm going to copy down our equation so that we know what we're working towards we have n -1 * s^ 2 over k^ 2 Alpha / 2 less than Sigma 2 less than nus1 s^2 over k^ 2 1 - Alpha over 2 so that's what we're working towards so our n is going to be our number of samples so n is equal to 1 2 3 4 5 6 7 8 9 10 it's also given to us right here and S squar that's our sample standard or stample sample variance and it's not given to us we're given just a whole bunch of random numbers so we have to actually calculate our sample variance and so s s is going to be equal to n * the sum of x i 2 minus the sum of x i^ 2ar / n * n minus one we have this equation long time ago um if you need it I will give it to you but that's that and so when you do that you would have n is equal to 10 then the sum of X i^ squ you would take each X so like 46.4 Square it and then add it to the next one so then you would have 46.1 squar plus 45.8 SAR and so on and you ultimately end up getting 21 27 3.12 and then in this case you're going to take the sum of all x's and then Square it so you're going to do 46.4 plus 46.1 plus 45.8 dot dot dot dot dot and then square that number so that gives us 4 61.2 squared divided by 10 * 9 and that is 0.286 and then we need our Ki squar value so we need Ki squar of so our Alpha 95% means that 1 minus Alpha is equal to 0.95 which means Alpha is equal to 0.05 so k^ 2 of alpha over 2 is going to be k^ 2 of 0.02 25 and our degrees of freedom is equal to n minus one so that's N9 so now I need to find k^ squ of 0.025 with n degrees of freedom so I'm going to go to my table k^ squ of .025 with 9 degrees of freedom 19.23 and then I have k^ 2 1us Alpha over 2 and this is k^ 2 of 0.975 9 so 9759 is [Music] 2.7 now I have everything for my equation so I'm going to have n minus1 is 9 * 0.28 6 ided by Alpha over 2 so that's going to be this 19.23 less than Sigma 2 less than 9 * 0.286 ID 2.7 and again be careful of what you're plugging in so we solved for s s so we don't need to square this value when we move it down here it's already s squ and vice versa if this is the square root version you got to make sure that you square it down here and so then we end up getting our final answer to be 0.135 less than Sigma squar less than 0.953 and then we would do our conclusion we would say we are 95% confident that the true population variance is between 0.135 and 0.953 and I would want to add in my units here decagrams and decagrams all right that is it for confidence interval for variants