Normal distributions and the empirical rule
Normal distribution problems: Empirical rule
Let's do another problem from the normal distribution section of ck12.org's 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 can go to their site, and I think you can download the book. Assume that the mean weight of one-year-old girls in the US is normally distributed with a mean of about 9.5 grams. That's got to be kilograms. I have a 10-month-old son, and he weighs about 20 pounds, which is about 9 kilograms. 9.5 grams is nothing. This would be if we were talking about like mice or something. This has got to be kilograms. But anyway. It's about 9.5 kilograms with a standard deviation of approximately 1.1 grams. So the mean is equal to 9.5 kilograms, I'm assuming, and the standard deviation is equal to 1.1 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, sometimes called the 68, 95, 99.7 rule. And if you remember, this is the name of the rule. You've essentially remembered the rule. What that tells us is 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 get 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 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 one standard deviation above the mean, that would be this area. And it would be-- you could guess-- 68%, 68% chance you're going to get something within one standard deviation of the mean, either a standard deviation below or above or anywhere in between. Now if we're talking about 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%. And that includes this middle area right here. So the 68% is a subset of 95%. 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, 99.7 rule tells us that there is a 99.7% 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 axis first, as best as I can. That's my axis. Let me draw my bell curve. Let me draw the bell curve. That's about as good of a bell curve as you can expect a freehand drawer to do. And the mean here is-- 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. 9.5 is the mean. I won't write the units. It's all in kilograms. One standard deviation above the mean, we should add 1.1 to that. Because they told us the standard deviation is 1.1. That's going to be 10.6. Let me just draw a little dotted line there. Once standard deviation below the mean, we're going to subtract 1.1 from 9.5. And so that would be, what? 8.4. If we go two standard deviations above the mean, we would add another standard deviation here. We went one standard deviation, two standard deviations. That one goes to 11.7. And if we were to go three standard deviations, we'd add 1.1 again. That would get us to 12.8. Doing that on the other side-- one standard deviation below the mean is 8.4. Two standard deviations below the mean, subtract 1.1 again, would be 7.3. And then three standard deviations below the mean, it would be right there, would be 6.2 kilograms. So that's our setup for the problem. So what's the probability that we would find a one-year-old girl in the US that weighs less than 8.4 kilograms? Or maybe I should say whose mass is less than 8.4 kilograms. So if we look here, the probability of finding a baby or a female baby that's one-years-old with a mass or a weight of less than 8.4 kilograms, that's this area right here. I said mass because kilograms is actually a unit of mass. But most people use it as weight, as well. So that's in that area right there. So how can we figure out that area under this normal distribution using the empirical rule? Well, 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 100%, or 1, depending on how you want to think about it. Because you can't have-- well, all the possibilities combined can only add up to 1. You can't have more than 100% 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%. And 32% is if you add up this left leg and this right leg over here. And this is a perfect normal distribution. They told us it's normally distributed. So it's going to be perfectly symmetrical. So if 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-- it ended 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 a result 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, at one-years-old, less than 8.4 kilograms. Less than 8.4 kilograms is this area right here, and that's 16%. So that's 16% for Part A. Let's do Part B. Between 7.3 and 11.7 kilograms-- so between 7.3, that's right there. That's two standard deviations below the mean. And 11.7-- it's two standard deviations above the mean. So they're essentially asking us what's the probability of getting a result within two standard deviations of the mean. This was the mean, right here. This is 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% chance of getting bad results, or a 95% 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 US baby girl more than 12.8 kilograms. So 12.8 kilograms is three standard deviations above the mean. So we want to know the probability of having a result more than three standard deviations above the mean. So that is 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 this area. We know the area between minus three standard deviations and plus three standard deviations. We know this. Since this is the last problem, I can color the whole thing in. 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-- it's only 0.3%. for the rest. And these two things are symmetrical. They're going to be equal. So this right here it has to be half of this, or 0.15%, and this, right here, is going to be 0.15%. So the probability of having a one-year-old baby girl in the US that is more than 12.8 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%. Anyway, hope you found that useful.