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# Margin of error 2

## Video transcript

where we left off in the last video I kind of gave you a question find an interval so whether we're reasonably confident we'll talk a little bit more about what to give this kind of kind of vague wording right here reasonably confident there's a 95% chance the true the true population mean the true population mean which is P which is same thing as the mean of the sampling distribution of the sampling mean so there's a 95% chance that the true mean and let me put this here this is also the same thing as the mean of the sampling distribution of the sampling mean is in that interval and to do that let me just throw out a few ideas what is the probability what is the probability the probability that if I take a if I take a sample and I were to take a mean of that sample so if I the probability that a random sample mean is within is within two standard deviations of the sampling mean of our sample mean so what is this probability right over here so let's just look at our actual distribution so this is our distribution this right here is our sampling mean maybe I should do it in blue because that's the number up here that it that's the color up here this is our sampling mean and so what is the probability that a random sampling mean is going to be in two standard deviations well a random sampling mean is a sample from this distribution it is a sample from the sampling distribution of the sample mean so it's literally what is the probability of finding a sample within two standard deviations of the mean that's one standard deviation that's another standard deviation right over there and in general if you haven't committed this to memory already it's not a bad thing to commit to memory is that if you have a normal distribution the probability of taking a sample within two standard deviations is ninety five and if you want to get a little bit a little bit more accurate it's ninety five point four percent but you could say it's roughly it's roughly or maybe I could write this it's roughly 95 percent and really that's all that matters because we have this little funny language here called reasonably confident and we have to estimate the standard deviation any in fact we could say if we want I could say that's going to be exactly equal to ninety five point four percent but in general two standard deviations ninety five percent that's what people equate with each other now this statement is the exact same thing as the probability that the sample mean that the sampling mean or the the mean of the sample not the sample mean the probability of the mean of the sampling distribution is within is within two standard deviations of the sampling distribution of X is also going to be the same number is also going to be equal to ninety five point four percent these are the exact same statement if X is within two standard deviations of this then this then the mean is within two standard deviations of X these are just two ways of phrasing the same thing now we know that the set the mean of the sampling distribution is same as the same thing as the mean of the population distribution which is the same thing as the parameter P the proportion of people who or the proportion of the population that is a1 so this right here is the same thing as the population mean so this statement right here we can switch this with P so the probably probability probability that P is within two standard deviations of the sampling distribution of X is ninety five point four percent now we don't know what this number right here is but we have estimated it remember our best estimate of this is the true standard or it is the true standard deviation of the population divided by ten we can estimate the true standard deviation of the population with our sampling standard deviation which was 0.5 0.5 divided by 10 our best estimate of the standard deviation of the sampling distribution of the sample mean is point zero five so now we can say and I'll switch colors the probability that the parameter P the proportion of the population saying one is within is within two times remember our best estimate of this right here is 0.05 of a sample mean that we take is equal to 95.4% and so we could say the probability probability that P is within 2 times 0.05 is going to be equal to 2 point 0 is going to be zero point 100 of our mean is equal to 95 well and actually let me be a little careful here I can't say the equal now because over here if we knew this if we knew this parameter of the sampling distribution of the sample mean we could say that it is 95.4% we don't know it we are just trying to find our best estimator for it so actually what I'm going to do here is actually just say is roughly and I'm just to show that we don't even have that level of accuracy I'm going to say roughly 95 percent we're reasonably confident that it's about 95 percent because we're using this estimator that came out of our sample and if the sample is really skewed this is going to be a really weird number so this is why we just have to be a little bit more exact about what we're doing but this is a tool for at least saying how good is our result and so this is going to be about 95 percent or we could say that the probability the probability that P is within is within 0.10 of our sample mean that we actually got so what was the sample mean that we actually got it was 0.4 3 so if we're within 0.1 of 0.4 3 that means we are within 0.4 3 plus or minus 0.1 plus or minus 0.1 is also roughly where reasonably confident about 95% and I want to be very clear everything that I started all the way from pure and brown to yellow to all this magenta I'm just restating the same thing inside of this it became a little bit more loosey-goosey once I went from the exact standard deviation of the sampling distribution to an estimator for it and that's why this is just becoming you know I kind of put the squiggly equal signs there to say this is we're reasonably confident I even got rid rid of some of the precision but we just found our interval an interval that we can be reasonably confident that there's a 95% problem the D that P is within that is going to be 0.4 3 plus or minus 0.1 or an interval of we have a confidence interval we have a 95% confidence interval confidence interval interval of and we could say 0.4 3 minus 0.1 minus 0.1 is 0.33 if we write that as a percent we could say 33% 2 and if we add the point 1 point 4 3 plus 0.1 we get 53 percent to 53 percent so we are 95% confident confident so we don't we're not saying kind of precisely that the probability of the actual proportion is 95% but we're 95% confident that the true proportion is between is between 33% and 55% that P is in this range over here or another way and you'll see this in a lot of surveys that have been done people will say we did a survey we did a survey and we got 43% 43% we'll vote for number one and number one in that kid this case is candidate B it for candidate B candidate B and then the other side since everyone else voted for candidate 57% will vote for a and then they're going to put a margin of error and you'll see this in any survey that you see on TV they'll put a margin margin of error and the margin of error is just another way of describing this confidence interval and they'll say that the margin of error in this case is 10 percent is 10 percent which means that there's a 95% confidence interval if you go plus or minus 10% from that value right over there and I really want to emphasize you can't say with certainty that there is a 95% chance that the true result will be within 10% of this because we had to estimate the standard deviation of the sampling mean but I just wanted but this is the best measure we can with the information you have if you're going to do a survey of 100 people this is the best kind of confidence that we can get and this number is actually fairly big so if you were to look at this you would say you know roughly there's a 95% chance that the true value of this number is between 33 and 53 percent so there's actually still a chance that candidate B can win even though only 43 percent of your hundred are going to vote for them if you wanted to make if you want to make a little bit more precise you want it you would want to take more samples you can imagine if you instead of taking a hundred samples instead of n being a hundred if you made any Qin then you would take this number over here you take this number here and divided by the square root of a thousand instead of the square root of 100 so you you know you'd be dividing by 33 or whatever and so then your margin of the number the size of the standard deviation of your of your sampling distribution will go down and so the distance of two standard deviations will be a smaller number and so then you will have a smaller margin of error and maybe you want to get the smaller margin area the margins are small enough so that you can figure out decisively who's going to win the election