hi everyone in this video I'm showing how to compute critical values for some of the major interval estimates that we'll be doing using Excel so there's just a little review of what a critical value is or kind of what we're looking for we're looking at some kind of distribution in this case this is the normal distribution but we will be looking at different distributions depending upon what we're trying to estimate but we're looking at the kind of distribution of values and the idea of a critical value is that we're kind of trying to identify in this picture this labeled critical region these would correspond to significantly high or significantly low values based on a given sample when we're looking at trying to estimate population parameters so in this case they're showing that the critical values for this distribution are Z equals 1.96 z equals negative 1.96 we will have different values depending upon what kind of size of critical region we want so as a first example say we're finding critical values to estimate a population proportion there are calculations for estimating proportions use the normal distribution so I'm imagining that normal bell curve and I want to find critical values for a 95% confidence level in this in this case and this corresponds to an alpha value of 0.05 or essentially in the picture here I want 95% of the data to be in this central white region here and then this the 5% will fall into these two little blue areas so one thing you always have to be thinking about when you're finding critical values is you know how much area is there in each part of the graph and I definitely recommend drawing a little picture so in this case we're using the normal distribution and we've got out an alpha value of 0.05 which means that point zero five is getting divided up into these two little tails that means that each tail has an area of 1/2 of point zero five so in this case point zero two five so I'm imagining point zero two five area in each of these tails now we would have to come up with what is the Z value you know it could be 1.96 but not necessarily in this case that corresponds with those areas you should already be pretty familiar with this using the normal distribution we're going to use our normal inverse function so our normal inverse function takes a probability or an area to the left of the value that we want now keep in mind that for interval estimates there's going to always be two critical values but with a normal distribution they're basically the same value but once positive and ones negative so if I have to put in the area to the left I can either put in the area to the left of this point which would be the area in this blue tail in that case that would be 0.025 or I could also find this one by finding the area to the left of this now that would include the whole middle area which was 0.95 or that 95% area as well as this extra tail that takes just a little bit of extra computation so typically I would start by finding the one on the left so using this point zero two five as my area or my probability and then the mean and standard deviation inputs I'm just finding this for a generic normal distribution so all of these calculations are going to be based on that standard normal not given any particular mean or standard deviation given in the problem so our standard normal distribution has mean 0 standard deviation 1 so I'll plug those in and I get in this case actually this value negative 1.96 so negative one point nine five nine nine six and then the other one would just be the positive version and I know that from symmetry but I could compute that directly if I did instead of the area just to the left of this guy the area to the left of this right-hand critical value which in this case would be twenty nine five plus point zero two five I can have Excel compute this this would be point nine seven five if I wanted to just plug that in right so with my mean and standard deviation I'm expecting just the positive number of the same a positive version of the same number and there you go so this would be finding critical values for a normal distribution but not all of our problems are going to use a normal distribution so say we're doing an interval estimate for a mean if you talked about this in class already you know that there are actually two different types of calculations for the mean one actually uses a normal distribution as well and that would be when the standard deviation of the population is known or that Sigma known case you may have not even talked about that case in class because it's very uncommon usually we don't know the population standard deviation and so in those cases we use a different distribution called the student T distribution it does actually look a lot like the normal distribution but it's slightly varies in shape depending upon the sample so the formula we use for this looks like this so it's got a tee indicating that student T distribution and then dot inv that's the inverse just like we had the inverse for the normal distribution and then I'm using this version with two T on the end and what that 2t is telling us is that I want to chop off two tails so in my picture I'm chopping off area from different sides there are cases in statistics where you would find a critical value for just one side or the other instead of having the area for both but in interval estimates we've always got these two tails sort of the significantly high and the significantly low so using this function the first input we're going to put in is basically just the alpha value this is one minus the confidence level so if we have a 95% confidence level that would be point zero five notice that that is not the area in one tail that's actually the area in both tails and that's important because we're using this two tails function the second input is what's called the degrees of freedom in in the context of interval estimates degrees of freedom is always one less than the sample size or n minus one so say we've got this example here find a critical value for a 95% confidence level if our sample size is 25 okay so I'm assuming that we're using a student T distribution here we've got our confidence level or sample size with a confidence level of 0.9 five or 95% our alpha value would be point zero five and then the degrees of freedom would be one less than the sample size so in this case that would be twenty four so our critical values I'm going to use the TI NV dot two t I'm going to put in this is saying probability this would be my alpha value again that's the area in both tails in the two tails so point zero five in this case and then my degrees of freedom I'm saying twenty four this gives a positive number the picture that you should be thinking about is actually still very similar to the normal distribution and that there be a positive and a negative side to this the shape will be a little bit different than the normal distribution but very similar so there are actually two different critical values but in each of these cases were typically only going to use the positive one in our calculations okay there's a third type of critical values for interval estimates this would be when you're you doing an interval estimate for a standard deviation or for a variance all the calculations are actually based on the variance and hopefully you'll talk about that in class but in this case we're going to be using what's called the chi-squared distribution and sometimes this is written with the Greek letter Chi which looks like a capital x squared and the chi-square distribution really looks different here's a picture of the chi-squared distribution with a couple of different degrees of freedom and both of these pictures this one and the critical region these are coming from open source textbooks that I found online I will link both of these textbooks below in the description in case you want to check them out as resources but I just found some handy pictures here so you'll notice that this really doesn't have a bell curve shape it's very kind of leaned to one side very skew right and they will look a little bit different with different degrees of freedom in fact up above you have a degrees of freedom of two and it looks quite a bit different but this picture is kind of what you want to be thinking about now with the critical values I'm still imagining cutting off tails on both sides and I'm going to cut off tails of the same size so they would have the same area but the critical values you know maybe on this side I'm getting something like two but on this side I'm getting something like 12 because of the scale here so this is a different distribution which means it different needs a different function the function that we're going to use from Excel there's actually two different versions but it's very similar to the previous ones so it starts with ch is Q so this is an abbreviation of chi-squared CH I is just the spelling of that Greek letter Chi and then dot inverse just like before and there's two different versions the regular inverse and the RT or right version of this so I put a note at the bottom of the screen the first of these formulas uses the area to the left as this probability input this is just what we're used to with the normal distribution function the other equation this probability is going to indicate an area to the right so they really function the same way it's just are we talking about area to the right or are we talking about area to the left and you get to choose which one makes more sense based on the scenario that you're looking at just like with the t-distribution example the second argument is the degrees of freedom and it will also be n minus 1 or 1 less than the sample size okay so say we're going to find the critical values for a 95% confidence level using standard deviation or estimating standard deviation when we have a sample size of 38 we know that the confidence level is 95 which would mean that the area in the two tails put together would be point zero five that would be the Alpha value just like we had in the student T different distribution but in this case I want to look at the area in each tail which would be point zero two five or half of that point zero five value the degrees of freedom is going to be thirty seven so one less than the sample size and rather than getting plus and minus the same values for my critical values I'm going to have two distinct numbers so I'm going to do the left and then the right so for the left critical value I know that in my picture even though it's not symmetric like this the left critical value is going to have an area to the left of 0.025 so I'm going to use the chi-square function the left version so the gist dot inverse not the dark RT and I'm going to put in this probability and put that area to the left so point zero to five is my area to the left for that left-hand side value and then my degrees of freedom was 37 and then on the right hand side I'm going to choose the dot RT version so that I can put in the area to the right the area to the right of the right-hand critical value is also going to be point zero two five so by using the left and the right hand versions I really kind of take the extra thinking out of you know trying to figure out how much area is where so you get two numbers notice these are very different from the kinds of values that we see from the normal distribution or the student-t distribution they tend to be much larger although it does depend on your sample size but notice that the left one is definitely a smaller number and the right one is a bigger number so if you imagine one of these graphs kind of spread out as the degrees of freedom gets bigger the smaller number is over here on the left side of the graph and then that bigger 55 number would be over on the right so those are the three functions that we'll be using or the three types of functions that we'll be using for computing critical values thanks for watching

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