all right in this video we are talking about hypothesis testing for variant so we've talked about confidence intervals for variance we've talked about all the other hypothesis tests now we're bringing those kind of two ideas together to talk about hypothesis testing for variance so you have your equations here you have your null hypothesis is equal to a value you can do all three types of tests um you have your comparan values for your critical value versus your test statistic value and then you have your test statistic formula here all right um again it's going to be the same hypothesis test method but instead of the previous tables you've been using you're going to be using variance your Ki squar table and everything else is going to stay the same so then we have our Kai Square table and we're going to jump into an example a manufacturer of car batteries claims that the life of the company's batteries is approximately normally distributed with a standard deviation equal to 0.9 years if a random sample of 10 of these batteries has a standard deviation of 1.2 years do you think that Sigma greater than 0.9 year do you think that Sigma could be greater than 0.9 years you use a 0.05 level of significance all right so some things to kind of be careful of here number one is this is a claim this does not mean that Sigma equals 0.9 we do not know that they are saying that that is true but we want to take our sample and validate whether that is or is not true so you can't take that and the other thing is is it says normally distributed that does not mean we are using our Z or t table um whenever we are talking variance and standard deviation we are always using our Kai Square table so you can't use that piece of information there all right so step one in our process is to determine the population parameter being tested and so we are testing variant I know that it's asking for Sigma but we don't have a test for Sigma so the closest thing and only thing we can use is Sigma squared so we are testing for variance step two is to determine our probability table to use and since we are using variance that means we need our Kai Square table and then next is a set up our hypothesis statements so we're going to have H is Sigma equal to and it wants us to test against Sigma of 0.9 but we need the variance version of that we need Sigma squared so we're going to have to find that 0.9 * 0. n we need to square that and we get 0.81 is the actual variance value we're testing against and then our our alternative hypothesis is going to be Sigma squar greater than 0.81 step two or step four sorry is to set up the test so determine where our reject region is and so I'm going to try to draw a Ki squ distribution over here that was not bad all right um and so we have I'm going to pull out the rest of my variables here while I'm at it so I have Nal to 10 I have S = to 1.2 and Alpha equal to 0.05 and my degrees of freedom is going to be n minus one which is nine so to set up my test I need to determine where my reject region is and so my reject region is going to be based on my Alpha and I have 0.05 for my Alpha and it's a one-sided test upper one-sided test so I need to find the Ki squ value such that I have 0.05 in this tail because that's my upper tail so I'm going to go to my table and I see that that's the direction my table reads so I need to find exactly 0.05 so I'm looking in this column and my degrees of freedom were nine so I need this number right here so 16 919 is my critical value and this is going to be my reject region and this is my do not reject region so now that I have my test set up I can actually compute my test statistic and so that is going to be the kai squar = to n -1 * s^2 / Sigma kn^ 2 so for us n minus one is 9 s^ s is going to be 1.2 squar and sigma not squar is 0.81 plug that into a calculator and we get 16.0 and so now we can compare 16.0 to 16919 remember that this is a number line we start at zero and go up to whatever number um represents our Kai Square table so you'll see as we do more of these examples that this number will change quite drastically so let's take for example down here when we get up to 70 degrees of freedom that number could be 90 instead of 16 here we might have 90 and so we just have to remember that this is a number line down here that we go from zero to whatever our critical value is um and that we need to be mindful of the shading how much space is under the curve at that point we can't rely on this similar to how we do in the standard normal with standard normal we pretty much rely on the zero plus one minus one plus two minus 2 that does not work for K squ you just have to take these numbers and kind of trust them so way back to the problem here we got our test statistics so our K Square test statistic equal to 16 and so 16 is less than 16.9 so it's going to be somewhere over here so I'm G to draw that and say right here this is my 16.0 test statistic value and so what that tells me is this is in my do not reject region so I'm going to draw the conclusion and say that I do not reject H subo that variance equals 0.81 thus there is in sufficient evidence to support the alternative hypothesis and I'm going to change my wording here just to kind of relate it back to the problem better there is insufficient evidence to support that that the standard deviation is greater than 0.9 all right right um and then we have also talked about the other way of doing hypothesis test using our P value so I want to show you how to do the P value for this problem as well so to find our P value we would take our test statistic value that's 16.0 and our degrees of freedom of nine and see what we find so at 16.0 so we are working at nine degrees of freedom so we can only talk about these numbers here and at 16.0 we are somewhere between these two but a little bit closer to this one so we are somewhere between 0.1 and 0.005 but closer to 0.05 so I'm going to guess that to be 0.07 for my P value so I would say P value is approximately equal to 0 07 so then I have my P value is greater than Alpha therefore do not reject and then I would want that full statement again but you can see working with P value you get the same idea and same conclusion all right that's everything on hypothesis testing for variance