Normal distribution problems: Empirical rule

let's do another problem from the normal distribution section of ck-12 dot orgs AP statistics book and I'm using this because it's open source it's actually quite a good book the problems are I think good practice for us so let's see number three number two you could go to their sidemen I think you can download the book assume that the mean weight of one year-old girls in the US is a normally distributed or is normally distributed with a mean of about nine point five grams that's got to be kilograms I have a ten month old son and he weighs about 20 pounds which is about nine kilograms not nine point to the nine point five grams is nothing this would be talking about like like mice or something now this has got to be kilograms but anyway it's about nine point five kilograms with a standard deviation of approximately 1.1 grams so the mean is equal to nine point five kilograms I'm assuming and the standard deviation is equal to one point one grams without using a calculator so that's an interesting clue estimate the percentage of one-year-old girls in the US that meet the following condition so when they say that without a calculator estimate that's a big clue or a big giveaway that we're supposed to use the empirical rule empirical empirical rule sometimes called the 6895 99.7 rule and this is actually if you remember this is the name of the rule you've essentially remembered the rule what that tells us if we have a normal distribution I'll do a bit of a review here before we jump into this problem if we have a normal distribution let me draw a normal distribution so it looks like that that's my normal distribution I didn't draw it perfectly but you have the idea it should be symmetrical this is our mean right there that's our mean if we go one standard deviation above the mean and one standard deviation below the mean so this is one stance so this is our mean plus one standard deviation this is our mean minus one standard deviation the probability of finding a result if we're dealing with a perfect normal distribution that's between one standard deviation below the mean and once temptation above the mean that would be this area and it would be you could guess 68 percent 68 percent 68 percent chance you're going to get something within one standard deviation of the mean we their standard yishun below or above or anywhere in between now if we're talking bout two standard deviations around the mean so if we go down another standard deviation so we go down another standard deviation in that direction and another standard deviation above the mean and we were to ask ourselves what's the probability of finding something within those two or within that range then it's you could guess it 95 percent and that includes this middle area right here so it's a the sixty-eight percent is a subset of that ninety-five percent and I think you know where this is going if we go three standard deviations below the mean and above the mean the empirical rule or the 68 95 99 point seven rule tells us that there is a ninety-nine point seven percent chance ninety-nine point seven percent chance of finding a result in a normal distribution that is within three standard deviations of the mean so above three standard deviations below the mean and below three standard deviations above the mean that's what the empirical rule tells us now let's see if we can apply it to this problem so they gave us the mean and the standard deviation let me draw that out let me draw my axes first as best as I can that's my axis let me draw my bell curve let me draw the bell curve it's about as good as a bell curve is you can expect a freehand draw to do and the mean here is nine point and this should be symmetric this height should be the same as that height there I think you get the idea I'm not a computer nine point five is the mean I won't write the units it's all in kilograms one standard deviation above the mean so one standard deviation above the mean we should add one point one to that because they told us a standard deviation is one point one that's going to be ten point six if we go let me just draw a little dotted line there one standard iation below the mean one standard deviation below the mean we're going to subtract one point one from 9.5 and so that would be that would be what eight point four point four if we go two standard deviations above the mean we would add another standard deviation here right we went one standard deviation two standard deviations that would get us to eleven point seven and if we were to go three standard deviations we'd add one point one again that would get us to twelve point eight doing it on the other side one standard deviation below the mean is eight point four two standard deviations below the mean subtract one point one again would be seven point three and then three standard deviations below the mean if you're right there would be six point two kilograms so that's our set up for the problem so what's the probability that we would find a one-year-old girl in the US that has that weighs less than eight point four kilograms or maybe I should say whose mass is less than eight point four kilograms so if we look here the probability of finding a baby or female baby is one years old with a mass or weight of less than eight point four kilograms that's this area right here I said mask because kilograms is actually a unit of mass but most people use it as weight as well so that's that area right there so how can we figure out that area under this normal distribution using the empirical rule well we know we know what this area is we know what this area between minus one standard deviation and plus one standard deviation is we know that that is 68% and if that's 68% then that means in the parts that aren't in that middle region you have 32% because the area under the entire normal distribution is 100 or a hundred percent or one depending on how you want to think about it because you can't have well you know all of the possibilities combined there you know you can only add up to one it you can't have it more than a hundred percent there so if you add up this leg and this leg so this Plus that leg is going to be the remainder so 100 minus 68 that's 32 32% 32% is if you add up this left leg and this right leg over here and this is a perfect normal they told us it's normally distributed so it's going to be perfectly symmetrical so this side and that side add up to 32 but they're both symmetrical meaning they have the exact same area then this side right here do it in pink this side right here I end up looking more like purple would be 16% and this side right here would be 16% so your probability of getting a result more than one standard deviation above the mean so that's this right hand side would be 16% or the probability of having results less than one standard deviation below the mean that's this right here 16% so they want to know the probability of having a baby or at one-years-old less than eight point four kilograms less than eight point four kilograms is this area right here and that's 16 percent so that's 16 percent or Part A let's do Part B between seven point three and eleven point seven kilograms so between seven point three that's right there that's two standard deviations below the mean and eleven point seven one two standard deviations above the mean so they're essentially asking us what's the probability of getting result but within two standard deviations of the mean right this was the mean right here this is to stamp duty two standard deviations below this is two standard deviations above well that's pretty straightforward the empirical rule tells us between two standard deviations you have a 95 percent chance of getting that result or a 95 percent chance of getting a result that is within two standard deviations so the empirical rule just gives us that answer and then finally Part C the probability of having a one-year-old u.s. baby girl more than twelve point eight kilograms so twelve point eight kilograms is three standard deviations above the mean so we want to know the probability of having more than three standard of a result more than three standard deviations above the mean so that is this area this area way out there that I drew in orange maybe I should do it in a different color to really contrast it so it's this long tail out here this little small area so what is that probability so let's turn back to our empirical rule well we know the probability we know this area we know the area between minus three standard deviations and plus three standard deviations we know this I can sis's the last problem I can color the whole thing is we know this area right here between minus 3 and plus 3 that is 99.7% the bulk of the results fall under there I mean almost all of them so what do we have left over for the two tails remember there are two tails this is one of them and then you have the results that are less than three standard deviations below the mean this tail right there so that tells us that this less than three standard deviations below the mean and more than three standard deviations above the mean combined have to be the rest well the rest there's only it's only 0.3% for the rest or the rest and these two things are symmetrical they're going to be equal so this right here is has to be half of this or 0.15 percent and this right here is going to be 0.15 percent so the probability of having a one-year-old baby girl in the US that is more than twelve point eight kilograms if you assume a perfect normal distribution is the area under this curve the area that is more than three standard deviations above the mean and that is 0.15 percent point one five percent anyway hope you found that useful