Let’s do another problem from the normal distribution section of CK12.org’s AP Statistics book. And I’m using theirs because it’s open source, and it’s actually quite a good book. The problems are, I think, good practice for us. So let’s see, number three. Number three. 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 U.S. is a normally distributed, or 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 nine kilograms. 9.5 grams is nothing. This would be if we’re talking about like mice or something. Now this has got to be kilograms. But anyway, it’s about 9.5 kilograms with the 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 U.S. that meet the following conditions. 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 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, 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. Say it looks like that. That’s my normal distribution. I didn’t draw 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%. 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 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 that 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, 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 as a bell curve, as you can expect a free hand drawer to do. 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. So 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.
If we go, let me just draw a little dotted line there, one standard deviation below the mean, one standard deviation below the mean, we’re going to subtract 1.1 from 9.5. And so that would be, that would be what? 8.4. 8.4. 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 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 it 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, it would be right there, it 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 U.S. that weighs less than 8.4 kilograms. Or maybe I should say who’s mass is less than 8.4 kilograms. So if we look here the probably of finding a baby or a female baby, that’s one year 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 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 one, depending on how you want to think about it, because you can’t have well, all of the possibilities combined there can only add, add up to one. You can’t have it 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. 32%. 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, 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 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 or 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 one, two standard deviations above the mean. So they’re essentially asking us what’s the probability of getting result within two standard deviations of the mean, right? This is 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 that result, 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 U.S. 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 drew an 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. Since this is the last problem, I can color the whole thing in.
We know this area right here between minus three and plus three, 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. For the rest. And these two things are symmetrical. They’re going to be equal. So this right here 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 U.S. 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%, 0.15%. Anyway, hope you found that useful.