um okay so again so these are called confidence interval questions so now uh let's see how to do some of these things well okay so what is the 98% confidence interval for the average age of college students so you do a survey you get X bar is 21.9 take Sigma to be 6.1 and N to be 120 so one thing you should check is to make sure that you have a normal distribution do we have a normal distribution are we going to be using a normal distribution here yes why have Sig have Sigma and and N is over 30 so you're good okay so that's the first Che um right because remember you don't need to well once we know that it's normal there's one other thing we should write down notice that we're not going to write down mu for xar because we're not going to say anything about mu that's part of our final statements but what what are we going to need to to work out this problem Sig you're going to need Sigma for xar so now let's find that uh okay over theare < TK of 120 and then we just need to say what that is oops okay 5569 55 up to rounding okay okay so now to simplify things um now look I all the so I I we spent about like 20 minutes talking about the justification and where things come from and all that so now I'm going to shorten up this entire procedure by just saying look there's a formula for all this okay so uh um well we already have actually I already gave you the formula okay so that's the formula that's it xar uh and then Z * Sigma xar now in the formula in the real formula is going to be plus or minus why does it say plus or minus cuz there's a positive one and a negative one and that's it so remember that how many how many values are we actually trying to get two so one from the plus one from the minus and that's it you um I would appreci actually it's good to it's good to draw the picture and we're going to see why okay so I'm going to I'm I'm going to insist that you draw the picture and you'll see why all right so first off yes let's draw the picture okay actually this is interesting so why should we draw the picture well so the formula for a confidence interval and by the way confidence interval I'll always I'll almost always abbreviate CI so I hope that's relatively straightforward okay draw a picture exactly all right so what is it that we do have and what is it that we don't have in the formula do we have xar yes plus plus or minus do we have Z no do we have Sigma xar yes and what is that. 5569 so the thing is we need to find Z Now how do we find Z and this is where the picture helps exactly okay so let's draw the picture okay so what goes on the picture 21.9 and two tails okay and uh actually just for I mean it doesn't it doesn't really matter but just to be consistent uh there okay and now what else there's one important thing that we should put on the picture 098 and that's the middle okay and 0.01 on each side so this is why the picture helps the picture with when you draw the picture it becomes immediately obvious 098 goes in the middle there's 02 split between the two tails so each tail is 01 is that I I just said it was obvious that doesn't make it obvious is it obvious okay okay so now once you know that what do you do now we can find the zcore okay so just for practice let's do this with stat disk so with stat disk is there a difference between 0.1 and 0.01 yes okay question answered all right normal distribution so so so when we have this interface what do we type in are to the area to the left which is 01 and what do we need from there the zcore so negative uh 2. 326 3 64 yeah 2. 2.32 64 and now we have the Z score um and that's it now the rest is arithmetic so 21 1.9 plus or minus so we have to multiply that 23264 times 5569 so that's 1. 2956 okay and now I can use my little two arrow notation okay so 21.9 minus 1. 2956 so the lower value is going to be 20. 6044 and then the upper value plus 1.29 56 and that gives 23956 okay is the 98% confidence interval for you so the 20 and of course to fill things out uh we can we can put these values in on the picture you don't have to but I like to have a com I like to do it just for a sense of completion okay okay should be way to do it I thought that was pretty easy okay form you know like the number right right it's um yeah actually this is this is sort of like a a more advanced version of that if you think if you think of the five number summary where it gave that sort of approximate breakdown yeah this is a little more this is more advanced but it's like that okay so and so at this point it should be clear that when you're doing a confidence interval you you're always working with two tails uh you've got a lower value and upper value so there's always a tail on each side okay so so let me make another comment that another way to write the confidence interval so so you've got all these different notations as we went over last time um we're 98% confident that the population mean is between these two values or the 98% confidence interval is such and such or or I'll give you a new way to write this out you can you can leave it in terms of the plus minus notation that is another uh way of writing the confidence interval so you can say the 98% confidence interval is 21.9 plus or minus 1.29 [Music] that's plus or minus so um so what do the so let me talk a little bit about these two different know two different ways of writing it down so if we write it down the first way like this which values are we giving we're giving uh well we're always giving xar values but which values are we giving we're giving the cut off values the one on the left and the one on the right now if we do it this way with the plus minus what values are we giv well what's the first value the first can't you say can't you say mean anymore right the first value is xar all right now what does xar represent in a sense we're saying we did a study and my best guess from me is xar my best guess from you is 21.9 if I had to pick one value that's my guess so philosophically that's what that value represents what's the second part called right so from the formula it's the zcore times the standard deviation but the whole thing together has a name it's called the margin of error the margin of error that's what 1.95 is the 1. 295 is called the margin of error so if you go back to the formula just like Stacy said the Marg the formula well when you're dealing with X bars and Z's and all that the formula for the margin of error is z * Sigma xar [Music] um well we're going to be seeing a lot a whole lot more of it okay um so xar so in the sense that xar is sort of like our our one best guess from mu the name for that is X bar is called a point estimate for Mu estimate obviously an estimate and what is a point point just means one value or one spot I mean we're still sorry it's still xar I'm just saying in the context of doing confidence intervals xar is called the point estimate from you problem now it may ask what is your what is the point estimate um yeah it may or may not if it comes up now you know okay so I want to do this problem again but this time I'm asking for a 95% confidence interval so for 95% confidence interval what do you have to do draw the picture and exactly and uh as someone just said 02 that immediately means there's 025 in each tail so what do we have to do find the zcore what's the zcore yeah we did this already and what was it9 1.96 exactly now will you ever get a proportion yes and we'll be working on that shortly okay so right so so notice that you can you can fill in the data values on the picture um for the purposes of doing the problem it's sort of just it's uh it's extra um cuz what we what the the main purpose that the picture serves is to say to find the zcore um you can't fill in the data values it's not necessary because now you just go back to saying the CI is xar plus or minus Z * Sigma xar xar was 21.9 our Z is 1.96 and our Sigma xar uh same problem. 5569 so now have to work this out 1.96 time. 5569 and that's 1.09 one5 so if we split this up into upper and lower value we get ph. minus 1.09 one5 and plus so that gives you these two values 20. 8085 and 22.9 915 okay and that's a 95% confidence interval for me okay um all right so let me say that uh hopefully uh you guys Now understand how to do these problems uh with the normal distribution uh for the and hopefully you now understand how to do the sorts of problems that we've just uh gone over uh any questions about these problems yes need to5 uh it's I would prefer that and the reason is I want you to remember what it is that you're saying instead of just writing down two numbers they should write down you know from this number to this number represents the 95% confidence interval from you that's what it represents um and then of course when we get to proportion problems we'll be finding a conf interval for p so just to keep in mind what it is that we're doing can you just say um you can write it like this uh 20885 22915 is a 95% compal from you or you can write it in any of the p in any of the previous ways so again there are would maybe about three or four different ways to write down the answer that are all equivalent and all acceptable so some people may prefer to write down we are 98% confident that cuz we've been doing that for so long um and that's acceptable or you can simply write down the 95% confident interal is that's fine too I like writing the last way cuz I can use abbreviations and I can write it so short okay