# Statistics – Introduction to Confidence Intervals | Margin of Error | Standard Error | Proportions Hi this is Christopher Ferbrache and I’m going to review confidence intervals and t distributions Okay we’ll get started so this section it covers the t distribution and confidence intervals and they are frequently put together and that’s um because of a couple reasons but we’ll go over it um we’ll go over in the notes primarily it’s because we are to the phase in statistics where we’re now starting to look at other similar distributions to the z distribution when we have let’s say data that is not necessarily as normalized or we’re a little bit unsure about but with certain types of adjustments we can still apply some decision making to it and one way that we can do that is with confidence intervals so with confidence intervals it’s one way that we can look at a range of the what the uh mean should be or you know the the probability that the mean will be around a certain number and that’s what a confidence interval essentially is and we’ll talk about it more Okay so this says uh the student’s t distribution okay the t distribution takes its name from Will William C Lee Gossett in his 1908 paper Biometrica pinned under the pseudo named student so the deal was he was a um I believe he’s a oh here it says he worked in a brewery well he um he did a lot of sampling in the brewery and he figured out this this t-distribution but he didn’t want to get in trouble so he wrote this paper under the name of student and later they figured out who wrote it but that’s why it’s labeled student it was a way of being anonymous so here’s the the some of the main parts of why we use the t-distribution and it says the t-distribution is used when the population standard deviation is un unknown or so the population standard deviation is unknown okay so right off the bat so we’re unsure about the population standard deviation so we’re going to have to make some some adjustments and we’re going to probably lose some precision just based on that statement then the other one is the sample size is small meaning less than 30. Okay so it’s less than 30 as we’ve taught as we we’ve we described before in uh one of my other uh ones that let’s see with sampling distributions we talked about normalized data and how usually around 30 um the data is considered normalized to that point now 30 isn’t the magic number cut off 100 but it’s a number where it’s around the range where people believe that the data is normalized like I said in the other video if it was if the sample size was 29 that might not be a problem either it’s just 30 is is the the point where most people have decided that it’s probably normalized at that point and then the third one it is unknown if the sample uh comes from a normalized distribution. So one of the the um the deals uh with the sampling distribution and we talked about the theorems and and some of the rules there it was if the sample comes from a normalized distribution then we assume that the sample is normalized and those are the those are probably the the general rules so in this case the t-distribution it is essentially what I would say all of the reasons why we would use the t-distribution and not the z-distribution so the t-distribution is the the uh the prime of that it’s all of the the other times when we don’t use the z we use the t and that’s um that’s kind of the way way this is so it says characteristics of the t distribution so it says it has fat tails so if you take a look at this it says the red curve in the image on the left is a t distribution compared to the normal distribution in blue so what happens is um if you think about a normal distribution we have um it’s technically a perfect normal distribution would be very pointy so the data would be very grouped towards the middle and it makes sense because with everything with the frequency of the data it’s going to group towards the middle and very rare events end up being on the far edges the the tails so when we are less sure if the the data is normalized then um what happens is the the point of the distribution meaning the and this is the red one the point will come down and the tails get wider and that’s because some of the probability that our our uh our actual in this case our actual say mean of the um the point that we’re looking at that comes from say a population but in this case because we don’t have as good of information it ends up that some of that probability of where that point lies can end up in the tail and so the tails get larger and that’s just that’s I guess the side effect of of uh of the distribution and what it stands for and it stands for um something that’s that’s very similar to the z distribution but it is a little bit more distributed throughout the distribution and it’s not quite as um it’s not quite as centralized in its grouping it’s not as tight in its grouping and um that’s one of the precision issues of uh of the t distribution it’s not as precise in the way that we can do some of our measurements with it we can still come up to conclusions but it but it’s quite possibly not not as accurate because of these things such as the population standard deviation is unknown the sample size is small it’s unknown if the sample comes from a normalized distribution. Okay the next point so um a lot of things I said there but uh if you take a look at this the red line that is the t distribution curve that’s what that represents so t distributions have fatter tails and then as the sample size increases it becomes closer to the z distribution and that’s because as our samples get close larger we actually have better data to work off and it’s a it’s a way that the distribution works that it gets closer um and then the more real let’s see oh the t distribution is more realistic since true population means are rare so I’ve mentioned it a couple times in this course especially when we first started doing continuous distribution calculations the the continuous distribution calculation is not as uh realistic of an example and it’s because we don’t have population data all the time so once we start looking at the sampling distribution and then we start looking at the t distribution the t distribution is really practical because anybody can kind of go out and do this where you don’t always have the population standard deviation if you think about it in your life as you’ll see in this you can go and you can you can do the t-distribution studies a little bit easier than you can the population the population distribution the z-distribution type of studies because we don’t have population standard deviations usually let’s see this other other one down here is used to compare extremes in sample means okay so we can use this to compare extremes and sample means. Okay will be used in a large number of calculations including confidence intervals and hypothesis testing so as I mentioned we’re going to talk about confidence intervals and this actually will be the first part of this video will be the theory and then it will go into the application the application of the these types of calculations with the confidence intervals they’re not they’re not complicated um they’re they’re I always say at all but they’re they’re they’re pretty much not complicated at all compared to some of the things we do so we don’t do hypothesis testing in this video but we will um next week when I make that video Okay so the big deal here to today is we’re going to learn a new table it’s the t t distribution um which has a table and the table is quite a bit different than the z distribution so let’s see the t distribution consists of a distribution of curves so there are many many t distribution curves and that’s the big part of why we have a table like this it says the individual curves relate to the degrees of freedom so um and the degrees of freedom are usually represented as in minus one now um I’ll discuss it next but the reason why we have the minus 1 is because we have an inaccuracy in inherent in this calculation that we’re going to do and because we are estimating the population standard deviation we have to go and account for that and theoretically the best way to account for that is decreasing the n value by one and so it’s an adjustment because of an inaccuracy built into it okay discuss next and the significance okay says for t distribution confidence level calculations use the two-tailed t-test and select the significance the significance level from the table um that added to the confidence level equals one okay um oh so this is how yeah this works the confidence level is is the beta plus the two-tailed test of significance which is alpha equals one okay so example uh calculating the 98 confidence interval for the t distribution requires one to find the t score if the sample size is eight meaning n equals eight and a two percent significance level uh for the two-tailed distribution will be used as well as the appropriate degrees of freedom to locate the related score from the table okay so for this we have to know the sample size so before we can use this t table we have to look at the sample size and we have to know the significance level now before we were looking at um we’ve looked at the uh essentially we’ve looked at the percentages and the way that um well we haven’t gotten into it too much yet so for for these uh two percent significance level is essentially and this is what it talks about up here the inverse of it is um we’re looking for the 98 area on the inside of the curve so when we say two percent significance level what we’re actually doing is we’re looking at two percent out on the edges and 98 in between the edges or the tail and that’s how that works if it was a two-tailed distribution let’s see and then we have to um look it up with the degrees of freedom okay so we have the degrees of freedom and our n in this case is eight so we’re going to have eight subtract one so we have right here 8 subtract 1 equals 7. so our degrees of freedom are going to be 7 and then this is a little way that we write out the what the inputs are that goes into the uh the numbers that make up the the t uh the t score and so we write them down here like like subscript so it’s like t um two percent and then seven so that’s the two percent significance level and the um the seven for the degrees of freedom and so we look at about on the table we have to have both of these so what we do is we come down here and this is a little little snapshot of what the table at least the table that I use in my classes looks like is um we go down to the degrees of freedom and make sure you remember it’s degrees of freedom not just the n value so it’s seven and then you go over and if we’re looking for the two-tailed test you have to find the right one right here it says two-tailed test two percent and then we go and we take a look at where these intersect and it says 2.998 this is our t-score from the table let’s see it is not a probability it is just like the z-score for the normal distribution so the big deal here is that we use this table to find t scores which are just like z scores so if we think about this where with the z distribution we go and we look up with a z score and we get this area under the curve value so the the actual t table does not give us an area under the curve value the t distribution table putting in those inputs gets us to the t score so it’s it’s different than the z distribution table the z distribution table actually is very dynamic and there’s about three different ways to use it and i haven’t shown them all yet but the big deal here is is the the t table we put in the degrees of freedom and then our significance level based on whether we’re doing a two-tail or one-tail test and we get the t-score okay and we’ll pull up the full t-table in a little bit let’s see so this talks about the degrees of freedom and the main thing which i spoke about is um when we have an inaccuracy there’s a couple different ways to look at this thing fundamentally the issue is because we are estimating our population parameter because we don’t have a population the population parameter meaning the population standard deviation because we don’t have a population standard deviation we’re estimating it and because we’re estimating it we are introducing some inaccuracy and lack of precision into our calculations so we have to decrease our in by one and that is essentially one of the ways of explaining why we subtract why we have n minus one and if you look at the standard deviation calculations you’ll see the n minus one so it’s the same exact thing and it’s the same reason because with the um sample standard db when we do the sample standard deviations we are introducing a little bit of a lack of precision in the calculation confidence intervals okay so in inferential statistics we have two primary um uh methods of coming to a conclusion type of thing or or uh making decisions one of them is hypothesis testing and that that is a it produces a point estimate so with a hypothesis test we say if we were to do something what is the likelihood that something will be greater than or less than or equal to something and that’s the single point part and with confidence intervals um it says confidence intervals resulting in a large in a uh not large resulting in a range of values above and below the population parameter so with the confidence interval what we’re saying is we are coming to a conclusion or we’re trying to figure out what the probability is based on we say what is this interval what would be the interval um to a certain degree of certainty that the actual population mean lies within um you know accuracy of of uh say five percent type of thing and so we have and then we get these numbers and so it’s based on trying to come up with a range of where that actual mean is that’s what confidence intervals are people confuse that but it’s it’s our confidence level of a range of um of where the actual mean is and that’s what it is now we’re either wrong or where we’re right so either the confidence interval falls within that range or it doesn’t but it’s not it’s not both so that’s that’s something to think about um this is a lot of the ways this is laid out you’ll see that a lot of this we’ve already seen so um let’s see student t distribution confidence interval related so this is sample mean formulas i say sample mean because we have proportions too okay confidence interval for mu with the standard deviation population standard deviation unknown so we use the t distribution so you’ll start to see this where i say you know is we we’re looking for mu but we have an unknown population standard deviation or you’ll see the size of the sample and then it will say whether the sigma is known or unknown and so in this case unknown population standard deviation and we’re looking for mu so we’re going to use the t distribution and this is the formula for this for the confidence interval and so this shouldn’t look too bad you know we’ve already done this this right here we’ve changed up maybe the values just slightly because before we actually had the standard deviation up here that was the population one in this case I’m gonna I’m gonna zoom in just a little bit in this case we have a sample standard deviation up here so it’s the x bar plus and minus we see plus and minus it means we run we we produce the calculation for it being plus and then we run it for it being minus and then this is our t score that we’re going to get from the table and it’s the t score multiplied by the sample standard deviation over the root of n and that’s that’s all it is these calculations like I said are not bad the standard error which is when the um the population standard deviation is unknown this is the standard error part and this is what this what this does and the standard error is essentially saying that is an estimation of the population standard deviation and then the margin of error for mu with an unknown standard deviation population standard deviation this is the actual margin of error value right there to actually write out the margin then you would um you would actually do the the plus or minus what whatever the x bar is but this is actually the margin of error so you would take this value and you would add it or subtract it to get that margin to actually get the the interval range let’s see for the z distribution very similar it says confidence interval for mu with standard deviation known so in this case we do the same type of setup except we have a z-score and then we have the standard deviation over uh the root of n and then very similar except right here it is the population standard deviation over the root of n and then same thing down here margin of error for mu with population standard deviation known so same type of scenario so we see that z that’s just the z score. Okay and then for proportions same type of scenario we’ve already done these so some of these have the little squiggly lines I think in some of my notes I had the the p bar same it’s the same type of deal it’s um if it’s a different different hat don’t worry about that with this so the same type of deal we have the proportion and then in this case plus or minus and then the z times times this and then uh standard error it’s this whole part and then margin of error is the z times this part and we’ll look at how this works Okay this is the background theory of the confidence intervals and talks about you know that we do confidence intervals to provide an answer of how to deal with uncertainty and it says results derive from data that are randomly selected a subset of population due to the possibility of misunderstanding where the actual population parameter lies the proper method of describing the interval is widely debated so people don’t say you don’t supposed to say the confidence interval there is a 95 technically you also say there’s 95 percent um there’s the uh ninety-five percent of the time it’s between a certain range what you say is um within a ninety-five percent level of certainty it’s within this data what that means is five percent of times you have a five percent chance you’re just not in there and that’s the way it works and this is a little diagram that shows it says um if you look at these different ranges of where a value is um some of these confidence intervals are not going to be where mu actually is and that’s that’s what this is representing um so it says the after all if one calculates the 95 percent interval approximately five percent of the of our time the ninety-five percent interval will be um will be wrong so one hundred percent so um five percent of time our ninety-five percent confidence interval will just be wrong and that’s because that’s the way this works it talks about also like sampling distributions the larger the sample the closer one will be to the actual population value in regards to confidence intervals the larger the sample the narrower the confidence interval will be so the larger our our sample of data the more accurate we can be to how how wide that interval is or how narrow the interval is when we don’t have much data that interval has to be really wide because we are not as sure as we would be if we had a lot of data and it makes it makes sense we’ll we’ll look at that too um let’s see I think I okay I think I mentioned a little bit of this if one is to calculate the plus or minus z value for the calculation one must find the correct z score by dividing the confidence interval level uh desire desired by two the result will be half the probability of the confidence interval for normal distribution when the probability probability values from zero to point one is okay okay we’ll look at this and this is the way this works as we um as we increase our confidence interval the actual interval the um the likelihood that we are wrong so as we increase the beta value part which is our confidence intervals as they get larger the what happens is as they get larger our actual confidence level will decrease and that it’ll be more precise now the issue is if we said oh we want a confidence interval down to um you know 99 confidence interval that we’re going to be right between a certain ratio it’s going to be really wide and in the when I said the 99 um so the the the beta part will be the 99 and then the alpha part will be the the um the essentially 0.5 on each side but that actual level will be really wide and that’s how this works mathematically the relationship for the confidence interval is one minus alpha for example the significance level of 10 alpha equals 0.1 is the opposite of 1 subtract 0.1 which equals 90 confidence interval while this relationship exists between confidence intervals and significance levels they are used differently what is important is that one understands how they are related okay okay to make it even easier an annotated z table is included um with these notes too so many of the common values are available let’s see the annotations link the value to the z table probability under the curve to the context of which they are frequently used and examples listed below I’m going to zoom in but i will show you how this works so what it is is if somebody said okay i want a confidence interval and this is this is all z score confidence interval because with the t distribution it’s much easier but when when we say um we want a 90 confidence interval then you would go and you would say oh well if we just take a look at this blue one right here it says confidence interval ninety percent and then we get a z score of 1.65 and that’s the confidence interval for that and it shows what the actual values and under the curve um go into making that 1.65 and and and what it would be um and then we have the one tail hypo test that’s not we’re not going to have to use this for for this video in the two-tail hypotest okay and the hypothesis is what that means um so there’s one of these for each one and I’ll show you how this works of of course there are only certain ones that I chose to make this for because these are the more common ones that come up I said this is only the z distribution part like i said the t table is much easier okay so this is the z table let’s see it says find the z score for 95 confidence interval calculation well this example shows the procedure for obtaining the z score it is similar to other confidence interval calculations um let’s see 95 divided by two so confidence intervals always have two sides so with hypothesis testing they don’t but with confidence interval it’s an interval it’s plus or minus so we have two sides um with the mean being in the middle so we have 95 divided by two equals .475 so this is how we would do this normally just with a table um like i said most of the time this is going to be easier because you’re going to use that little annotated box i gave you so it says okay 95 divided by 2 equals 0.475 now we look up using the z table the probability value from 0 to 0.5 on the table and we’re actually looking for this one right here 0.475 so it says  that makes it easy for us like i said we can look that up I’ll pull up the z table so on the z table this is the way it looks the z table I give you it looks messy but there’s a reason why it’s laid out like this so what you would do is you would go this is a really common one so you’d probably remember it after a while you would go where is that one uh where’s that 95 percent so this is 98 90 this one’s 95 and then we have 99 80 96 so it says 95 percent confidence interval 1.96 and then it shows point four uh 475 plus 475. now if we follow this up it has uh 0.475 and then if we go out it says 1.9 0.06 so this shows you how this goes together okay so that’s that’s that’s how the table is laid out like that okay I’m gonna back this okay so let’s see we have the z-score then we would just do the math right here which we did with the sampling distribution same type of math the proportion sampling distribution we would get the answer there and then you just do calculate this for the negative and calculate it for the positive and we have our answers and this is our standard error over here okay that’s our standard error and then um let’s see how do we do this in the calculator so we do the if we do this in the calculator we do the TI-83/84 calculator we do one prop z and and I believe this is i don’t have a calculator in front of me it’s either in the um the the DIST or the um the stats key I’d have to take a look that most of our calculations are in the the second DIST or the stats key button and um in there I’m not positive which one it is some of them are easy to remember where they’re at um let’s see the x is the number of success in the sample now um so 0.35 and so so the number of success in the sample would be in this case 0.35 because it’s 0.35 and then 80 times 80. but here’s the deal so that whatever value that would be you can actually put it in just like that and as you move to the next line it’ll calculate it now if um if it comes up where it’s not a integer you have to round it to the nearest integer you’ll get an error if you leave it with a decimal in this part because the success cannot be a fractional success it doesn’t work that way um now let’s see it says the easiest way to obtain this value is to multiply the easiest way to do it is just to which is what this is n times the um sample proportion hat and like I said right here you just do right there and then you put that in like I said if you get an error this has to be an integer and then you put in the 95 level and you are good you have the answer okay so let’s take a look at another one calculate the 90 confidence interval estimate for the population proportion of interest based on p-hat of 0.45 and n of 60 in addition to the confidence interval what is the margin of error so in the other one this was the standard error and it’s also known as it’s also known as the air based mean this part right here okay so standard error is also known as the air based mean which is kind of saying that this is the standard error that we get now it is not the margin of error because the margin of error is actually the standard error or airbase mean multiplied by the um t-score or the z-score because that’s actually the margin that’s the range and if you think about it when we hear the word margin we’re thinking of a range because the standard deviation is involved okay so for this one same type of scenario you would look up your 90 confidence interval confidence interval of 90 so you look up 90 so we go and we find 90 percent like i said if we’re going to do this backwards we would take like without having this stuff annotated you would do 90 divided by 2 and you get 0.45 so i see this is right in front of me but we would go up and we’d find the closest one and it would be well right between this and this and what I have found is most people use this one the 1.65 and not this one go figure I don’t know why um technically it’d probably be more accurate being 1.645 but this is what most people go by so it is 1.65 so we have 1.65 right there we do the math on this one okay so this is our kind of our range our range within that um range proportion figure we can say um so we we do this math right here and we we get this and then we just go and we multiply it by our z-score figure and we get the margin of error so the margin of error is is all of this together and then we have the the negative side the the low side and the high side we do the math and we get that so then on this one we do the same thing so just put in the 0.45 and then 60. like i said if this is comes out to a decimal you need to make it an integer so round up to an integer I believe you usually round up but um if it’s you may want to try it two different ways and then 60 and the confidence level they say confidence level but actually means confidence interval of 90 and then you get your answers Okay so this one is a okay this is a non-proportion type of one and this one says calculate the 96 confidence interval estimate based on so sample mean of 78 population standard deviation of 20 and n is 40. same scenario this is this is the goal getting all the pieces for that so the 96 percent um the the 96 confidence interval the the z score for the 96 percent is 2.05 and so we would look at this and we’d say what is which one is the 96 percent and we find that it’s over here on the orange this is 96 confidence interval 2.05 and it actually is made up with these values right here these are the closest so it’s a two point so it’s 2.0 okay that’s how that works okay so 2.05 and then we do the math right here and we get that value we multiply these together and that’s our margin of error so that’s the error part multiplied by the standard score part in this case it’s the z-score now when we do this in the calculator it says use the given standard deviation stated in the problem not the calculated so in your in your calculator there are times where we use the normal normal district the the normalCDF function and the normalCDF function is based on working with populations so in the sampling distributions I always tell students if you’re doing sampling distributions and using the normalCDF part of the calculator you have to put in the sampling distribution of the standard deviation of the sampling distribution okay for those problems so you’re putting the calculated standard deviation in for the z intervals or or the t interval you have to put in the given and that’s because your calculator because this is the interval your calculator knows that it is essentially um not the it knows how to do the math for you so it’s it’s inherent in there they know that they are working with what would be more of imperfect data so they actually do the math um and what happens is you put in the standard deviation part and you put in the 40 and it will calculate this for you so it actually does that for you in the calculator whereas like I was saying with the sampling distributions you have to put in this that the result of this value in as that one so this one just put in what they give you and so that is a change from my prior lesson where I told people to put in the calculated one this one you don’t you put in the stated one so you have to play with these things so you put in our values and there we go and if we are a little bit off it’s not a big problem remember we’re using the tables and the tables sometimes we have to pick which one to use type of deal and the calculator is usually more accurate z interval okay so this is another z interval one so I will probably skip this one for time for the video i want to so this is just like the one before this one’s a 90. Okay and the standard error so this is says am what’s the standard error the standard errors this part remember standard error is just that the margin of error is that part All right the margin of error is that part actual confidence interval the whole thing the high and the low both of these so confidence interval the whole thing low number high number margin of error just the part that goes above and below the mean just that segment and then the standard error just that part okay okay let’s look at the t distribution we need to look at look at this so focus on this a little bit calculate the 99 confidence interval estimate based on a sample mean of 3.13 the sample standard deviation of 1.2 and n of 20. This is what I usually do usually with t t distribution problems I will give you an n that’s less than 30 and that will kind of key you in oh this is and I’ll give you an s not a sigma so this is a sampling standard deviation okay so this but this math should look really similar from what we’ve been doing remember degrees of freedom we use degrees of freedom with the t distribution so it’s 20 subtract 1 equals 19, and then um this is the little thing where we write out you don’t have to do it like this but for your own sake you probably should you write out the inputs that get you the the t score and you do that so that when you go back and you if you make a mistake you can see what inputs you put in otherwise it’s just where did you get that t-score you don’t know so 99 and then 19 and we’re going to look these things up now remember this is 99 and 19. just remember remember that the way that the table is laid out is it’s a little different so that was 99 percent 99 and 19. Okay and it’s two-tailed because it’s confidence interval and we have both sides that’ll come up different lesson okay 19 this one’s easy we go to the 19 first I’m going to zoom in so we go to this range so this one’s super easy to find I’m gonna try to highlight these going across okay now the question is which which column do we pick and we go to two-tail two-tail now the questions confidence intervals are usually written in the um the large space the beta space the big parts underneath the curve but the way that the t table is laid out it’s laid out by making you select the alpha portion which is the little tail part so if if we’re looking for 99 we’re gonna look for a two-tail one percent two-tail one percent and that’s because we have 99 which is our beta in this case our alpha is one percent two-tail it’s actually split um but but one percent and so we’re going to use the one percent significance level on this table and our answer for our t score is going to be 2.861 2.861 okay like i said this gets us the t score this does not get us the areas under the curve like the z table does okay that’s a big difference is that we do all of this just to get the t-score where with the z-distribution the t-score is easy to get right we can go further with the z-table so was uh 2.86 okay it has 2.861 that’s fine 2.86 so boy those that that almost looks like it’s similar okay so this is our uh t-score and then we go and we get the um the other component which is this part right here and we put these together and then we um so we would multiply these and then we um do the subtraction do the addition we have the low side we have the high side okay now if we do this on the t interval like I said it’s either in second DIST I don’t have one of the calculators in front of me second DIST or it’s in the stats key of the calculator let’s see and then what we do is let’s see oh the Sx that represents the sample standard deviation so that’s that’s what that means and this points out again to use the given standard deviation stated in the problem not the calculated and it’s because in this case with the t intervals they know that they’re estimates and so they they know kind of what you’re doing with it you’re not putting in really good population data in this they know that what what you’re putting in and so it kind of does some of the math in here so it’s a it’s a higher level program than the other ones are or the other ones are a little bit more versatile I guess and then for the c interval you put in 0.99 and it gives you the answer it should be pretty close not perfect not perfect but close and that’s because the calculator is just more accurate okay let’s do one more of these and then I have more videos on this so if you’re watching this video I have other videos where you can practice this and I’ll go over the k I pull up the calculator too calculate the 90 oh I use the 99 again that’s too bad I should have picked a different one oh but the um the uh the n is different so our degrees of freedom is different we’ll get a different score okay calculate the 99 confidence interval based on so x bar of 5 sample standard deviation of 3 and n of 10 okay and here’s our calculation our degrees of freedom are 9 and we’re doing the 99 again but because our degrees of freedom are different we should come up with this one three point two four so nine and ninety nine nine nine and 99 so we do the two tail right here and nine so it is the 3.25 oh so this one says 3.2498 so um it’s possible I did get this from a different example or um this I’m suspecting it’s more accurate so um don’t get too concerned if you’re like hey this isn’t the same this is very very similar I get examples from different sources sometimes and then um what we just do is we calculate the the other component to it the three over and three three seems these numbers are pretty small um so three over the root of 10 so we do that and um then we have the plus side and the I’m sorry the the low side the subtraction the high side is the plus side and we have our answer okay I just want to go back and reinforce one thing real quick and that is just going to go back so there’s there are a couple terms here that I just want to reinforce and that is the so the air based mean is this this um right most part okay so it’s not the z score it’s not the score part it’s the um it’s the part that deals with the um the ins the the um the the sample standard deviation over the um or the population standard deviation over the root of the n so that’s the airbase mean and it’s also known as the standard error. Okay so the airbase mean standard error same thing and then when we put them all together we have the margin of error because this is kind of the range of that margin now this isn’t actually the actual confidence interval the confidence interval is the whole thing it’s the low and the high okay maybe at a later date I’ll I’ll point that out a little bit more clearly. Okay thank you