 In this lecture, we review some basic high school math concepts that you'll find that you'll need in our statistics course. Okay, in this slide, you see that the fraction three tenths is the same as point three point 30. And which is also the same as 30%. Okay, to get three tenths, just divide the ten into three. And that equals point 30. So that's how you get a fraction into a decimal. And now if you want to get it as a percentage, take point 30 times 100%. You get 30%. So these are all equivalent three tenths point three 30%. Another measure of variation that we're going to learn when we look at the descriptives lecture in the statistics course is the coefficient of variation. It's a measure of variation that we were going to use for comparing different data sets. And the way we calculate it is a converting a fraction to a percentage. We take the standard deviation of the sample s divided by the average, the mean of the sample X bar and then take that fraction and multiply it by 100%. The coefficient of variation is used to express the standard deviation as a percentage of the mean. So what we do is we're eliminating the units. The standard deviation and the mean are both in the same units as the original data. And when we take this to the fraction, the standard deviation over the mean, we're canceling the units as well. So we have a pure number. We multiply it by 100%. And that's what we call the coefficient of variation. Example one, the standard deviation was four days. The sample mean was 10 days. And the coefficient variation is four over 10 times 100% of 40%. In other words, the standard deviation is 40% of the mean. Example two, using the same logic, you could see that the coefficient of variation is 5%. The standard deviation is 5% of the mean. On your calculator, you'll probably notice there's a key with an exclamation point. If you've never used it before, that's a factorial. Any number, any integer, we can get the factorial of any integer as the product of all of the integers between that number and one. Easiest to see with an example. Take a look at the examples. 6 factorial is equal to 6 times 5 times 4 times 3 times 2 times 1. And if you do all of those multiplications, you have 720. And just as a little item, if you take 8 factorial is, yes, it's 8 times 7 times 6 times 5 times 4 times 3 times 2 times 1. But notice that inside of 8 factorial, we have 6 factorial. So if we already computed 6 factorial, all we need to do is take 8 times 7, which is 56, and multiply it by 6 factorial, which we know to be 720. And you get 40,320. Notice, factorials blow up really quickly. They get larger really, really fast. And just so we have it as a starting point, some formulas need it. Zero factorial, even if it doesn't make any sense, is the equivalent to 1. If you ever see zero factorial, that's equal to 1. Okay, here we have one equation, one unknown. Okay, 2 equals x minus 3 divided by 4. What you got to do is multiply both sides by 4. Now you'll have 8 equals x minus 3. Now you got to bring the 3 over to the other side. So you add 3 to both sides. So I have 8 plus 3 equals x minus 3 plus 3, which is equal to 11. Okay, in this problem, we have minus 5 equals x minus 6 divided by 7. Okay, multiply both sides by 7. Now you have x minus 6 on the right and minus 35 on the left. Add 6 to both sides. So now you have minus 29 equals x. One of the things you might worry about in looking ahead to your statistics course is plugging numbers into a formula and calculating the result. Is it going to be different if you do the intermediate calculations in the wrong order? You're right to be worried about it because actually it will. And if you do the calculations in the wrong order, you're going to get the wrong answer and very often you're not going to be able to get credit for your work. Very important to know what you're doing. If you have a large complicated formula, what do you do? Well, if you see parentheses, that comes first. And after that, if you have exponents like x squared, that goes next. If you have square roots, that's at that level. And after that, multiplication and division, after that addition and subtraction. Inside of any level, it doesn't matter what goes first and second and third. But you really have to be careful of the order, especially parentheses. Here's an example of how we do this. The first example at the top is just a simple average of three numbers, 10 plus 3 plus 27. Add that up, divide by 3. You get 30 over 3 or 10. The example at the top here is a very, very simple example of what we're talking about with order of operations. We know if we take an average, we want to add all the numbers up first and then divide by 3 if we have three numbers. So in this case, 10 plus 3 plus 17 divided by 3 comes out to 10. 30 over 3 is 10. We know we want to do all the numbers to add up all the numbers in the numerator first because we know we have to do the parentheses first. In the second example, we have z equals, and by the way, you're going to come back to that z over and over again. Once we start the statistics course and have it get it rolling. But in this particular example, z is equal to 120 minus 110 on the numerator. And imagine there's parentheses around that because it's in the numerator. It's going to be done first. And then in the denominator, we have a bunch of stuff, two fractions under a square root. And each of the fractions has a square in the numerator. That's kind of complicated. How are we going to do that? Well, first we have to say, okay, the numerator gets done first. Imagine it was in parentheses, 120 minus 110 gives you 10. And then in the denominator, we have 10 squared over 100. That's the first fraction. The 10 squared gets done first. We take that 10 squared, which is 100, divide by 100. That gives us 1. The other fraction, we still didn't do the square root because we have to do the other fraction first. We have to do everything under the square root first. 20 squared, we do the 20 squared, and we divide that by 80. We end up with a 5. Finally, 1 plus 5 is 6. We put that under the square root. Now we can do the square root. The square root of 6 is 2.45. Now we can finally do the original fraction. Z equals 10 over 2.45, and it's equal to 4.08. Here's another example just to give you more practice. If you need more practice, I'm sure you can find examples in your high school algebra book. T is equal to 200 minus 180, so the numerator is 20. The denominator has a square root. It has 400 multiplied by then something inside of parentheses. Inside of parentheses, what do we have? Two fractions that we have to add up. That gets done first. You get 0.15 multiplied by 400. Put that under a square root. Then you can finally take 20 divided by 7.75. Your solution is that T computes to 2.58. One other thing I want to point out that this doesn't always happen on a calculator. If you're writing out the formula by hand and plugging numbers in, make sure you write things neatly because if you forget that something's in parentheses or if you forget that something is part of the numerator, you could mess up the entire operation. Be very, very careful here. What does a straight line mean? In this example, we see our study that's x and y is the grade on a certain quiz. This actually turns out to be a straight line. In a moment, you'll see how we know it's a straight line, but we plot it for you on the right and you see these five points are exactly on a straight line. Once we know something's a straight line, we can answer questions like, what would y be the grade if x, our study, is 6? What would the grade be if you study 0 hours? That's why we want to be able to plot a straight line. Of course, we need to know what is the equation of the straight line. A straight line is defined by a slope. That's a change in y of a change in x. It shows you how much y changes as x changes. In the previous problem, the grade y changed by 10 points every time hours studied x changed by 1. So we say the slope is plus 10. Again, it's the change in y, which is the change in the grade over change in x, change in hours studied. Now the A, that's called the intercept, the y intercept. That's the value of y when x is 0. If you didn't study at all, 0 hours, what do we predict your grade would be? In this problem, we see it would be 25. So now we can actually write out the equation. Y, notice that little symbol on top of it, the hat, y hat equals 25 plus 10x. Every straight line is defined by an intercept term and by a slope. Why do we use the symbol y hat? Because we don't want to confuse it with the y, which is your original data points, your actual data points. In order not to confuse it, we want you to know that y hat is the actual line. So we say y hat equals 25 plus 10x. Here's a second example just to give you some more exercise. We have x and y, we're not even giving you a story about it. I don't know what x is and what y is. But notice here, the numbers are going in the other direction. They're going down instead of going up the way the numbers in the first example are going. Let's see what happens. We see that as x increases by 1 each time, y decreases by 5. So the slope is negative 5. This is an inverse relationship between x and y. And obviously when x is 0, the y intercept would be 50 if we extend it. And the equation ends up being y hat is equal to 50 minus 5x.