okay in the moments that we have that are remaining I want to make a comment about the margin of error okay so um as as in response to Stacy's uh question before I said well what's the margin of error the margin of error is what's after the plus minus part uh so for for the purpose of uh so in the case of the normal distribution that's Z * Sigma xar but let's unwrap that that just was uh Sigma X bar was just Sigma over theun of n so it's just Z * Sigma / the RO TK of n um is the margin of error and the margin of error ER so it's called capital E it's what sorry it's called what e all right for well and if you have a t distribution how do you find it it's exactly it becomes T * FID theun of n so e is margin of error okay CU e stands for error in this case we ever need to for uh no we're about I'm about to say something important about it um which is the last thing I'll say today okay so what's the relationship between the so so so far we've been talking about the confence interval we said it's a lower value and a higher value well what's the effect of the margin of error on the on the confidence interval what happens if your error is bigger what happens to your confidence interval what happens to the if the yeah it's a bigger range of values because uh think back think back to the formula if your this part is your margin of error so if your margin of error is bigger what are you doing you're going to subtract a big number and then add a big number so it's going to make your cut off values at the ends further out not bigger or smaller but just further out now is that good or bad to have a bigger confidence interval is that good or bad that's bad that's bad would you rather say that the that the mean is between uh 59 and 61 or if it's between 20 and 120 right so that's called being precise you'd rather be more precise you'd rather have a smaller range uh yes but what's the problem with having a smaller range let's look at what happened in the last problem if I can get them on I can't get them on the screen at the same time okay uh all right let me drop down notice that in the last question what we did was we said let's find a 98% confidence interval and then let's I said let's find a 95% confidence presentable so what happened when I changed the level of confidence so when the tales get bigger what else happens oh wait the the confidence the confidence gets smaller so you can so you can make you can say hey let's make a smaller range of values and that's good good but then what's bad that happens you you have less confidence so there's a direct tradeoff there you can have you can have more Precision a smaller interval but with less confidence okay so is that clear okay but there's one thing we haven't looked at sure to me well to me the EAS to me the easiest way to is look at the picture if you if you want to make the interval smaller just just move the move the cut off values in but when you do that what happens it changes your confidence so instead of so for example in this problem instead of having 098 in the middle what do we have you have 095 in the middle so you have a you have a better you have a better interval but you have lower confidence so it's when confence right here's another way to say it um I'm going to say that the mean is uh 36.8 I have 0% confidence that that that's true but I'm going to be very precise right that's a useless statement because you have 0% confidence this is more subtle there's a tradeoff between how precise you are and how much confidence you have but there's one thing we didn't look at we did not look at well let's look in that formul there's the Z there's the sigma and there's the N changing the confidence is the same as changing the zcore does that make sense cuz when you move the NS in that's the same as making the zv value smaller okay so we solve the effect of changing Z can we change Sigma no we you can't Sigma is what it is you can't change it so what's left n do we have control over n yeah we can do a bigger study which is in fact what we were talking about earlier okay so let's see all right uh I'm going to come back and I'm going to do this problem uh next time when we have more time but I do want to show you one thing suppose we want to specify what e has to be in other words we don't just want to calculate the confidence interval and then say uh hey by the way what was the margin of error what we can do is we can say what if you want e to be a certain value what if you want to have a certain margin of error what you're going to have to do is you're going to have to change your n so we need a way to figure out what does n have to be but we have a formula for what e is what is the formula for E Z * Sigma ided by the square root of n now you can either plug in values and solve it out or as we've seen we can just use algebra and we can say if the question is what does n have to be we want an equation that says n equals so that's what we're going to do and we're going to do some algebra so let's do some algebra and find out what n has to be so from our original formula for E the first thing is I don't like fractions so let's cross multiply that gives e * the < TK of n = z * Sigma next I want to get n by itself so I want to get rid of Eide e now I want to have I want to have just n so what do I have to do to get rid of the square root you square it so there's the formula for E excuse me there's the formula for n and remember this is this answers the question if I if I say ahead of time what I want e the margin of error to be what do I have to make n uh conf confidence or yeah exactly right and because you're given the confidence you can find Z okay so uh at the very least I've equipped you with this formula so if it just says what is n what is n uh in the homework you not know how to answer that question just use this formula okay ask you that after you answer