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Studying for a test? Prepare with these 6 lessons on Exploring bivariate numerical data.
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In the last several videos, we did some fairly hairy mathematics. And you might have even skipped them. But we got to a pretty neat result. We got to a formula for the slope and y-intercept of the best fitting regression line when you measure the error by the squared distance to that line. And our formula is, and I'll just rewrite it here just so we have something neat to look at. So the slope of that line is going to be the mean of x's times the mean of the y's minus the mean of the xy's. And don't worry, this seems really confusing, we're going to do an example of this actually in a few seconds. Divided by the mean of x squared minus the mean of the x squareds. And if this looks a little different than what you see in your statistics class or your textbook, you might see this swapped around. If you multiply both the numerator and denominator by negative 1, you could see this written as the mean of the xy's minus the mean of x times the mean of the y's. All of that over the mean of the x squareds minus the mean of the x's squared. These are obviously the same thing. You're just multiplying the numerator and denominator by negative 1, which is same thing as multiplying the whole thing by 1. And of course, whatever you get for m, you can then just substitute back in this to get your b. Your b is going to be equal to the mean of the y's minus your m. Let me write that in yellow so it's very clear. You solved for the m value. Minus m times the mean of the x's. And this is all you need. So let's actually put that into practice. So let's say I have three points, and I'm going to make sure that these points aren't colinear. Because, otherwise, it wouldn't be interesting. So let me draw three points over here. Let's say that to one point is the point 1 comma 2. So this 1, 2. And then we also have the point 2 comma 1. And then, let's say we also have the point, let's do something a little bit crazy, 4 comma 3. So this is 4, 3. So those are our three points. And what we want to do is find it the best fitting regression line, which we suspect is going to look something like that. We'll see what it actually looks like using our formulas, which we have proven. So a good place to start is just to calculate these things ahead of time, and then to substitute them back in the equation. So what's the mean of our x's? The mean of our x's is going to be 1 plus 2 plus 4 divided by 3. And what's this going to be? 1 plus 2 is 3, plus 4 is 7 divided by 3. It is equal to 7/3. Now, what is the mean of our y's? The mean of our y's is equal to 2 plus 1 plus 3. All of that over 3. So this is 2 plus 1 is 3. Plus 3 is 6. Divided by 3 is equal to 2. This is 6 divided by 3 is equal to 2. Now, what is the mean of our xy's? So our first xy over here is 1 times 2. Plus 2 times 1 plus 4 times 3. And we have three of these xy's. So divided by 3. So what's this going to be equal to? We have 2 plus 2, which is 4. 4 plus 12, which is 16. So it's going to be 16/3. And then the last one we have to calculate is the mean of the x squareds. So what's the mean of the x squareds? The first x squared is just going to be 1 squared. Plus this 2 squared, plus this 4 squared. And we have three data points again. So this is 1 plus 4, which is 5. Plus 16. Is equal to 21/3, which is equal to 7. So that worked out to a pretty neat number. So let's actually find our m's and our b's. So our slope, our optimal slope for our regression line, the mean of the x's is going to be 7/3. Times the mean of the y's. The mean of the y's is 2. Minus the mean of the xy's. Well, that's 16/3. And then, all of that over the mean of the x's. The mean of the x's is 7/3 squared. Minus the mean of the x squareds. So it's going to be minus this 7 right over here. And we just have to do a little bit of mathematics. I'm tempted to get out my calculator, but i'll resist the temptation. It's nice to keep things as fractions. Let's see if I can calculate this. This is 14/3 minus 16/3. All of that over, this is 49/9. And then minus 7. If I wanted to express that as something over 9, that's the same thing as 63/9. So in our numerator, we get negative 2/3. And then in our denominator, what's 49 minus 63? That's negative 14/9. And this is the same thing as negative 2/3 times negative 9/ 14. Divide numerator and denominator by 3. Well, the negatives are going to cancel out first of all. You divide by 3. That becomes a 1. That becomes a 3. Divide by 2. Becomes a 1. That becomes a 7. So our slope is 3/7. Not too bad. Now, we can go back and figure out our y-intercept. So let's figure out our y-intercept using this right over here. So our y-intercept, b, is going to be equal to the mean of the y's, the mean of the y's is 2, minus our slope. We just figured out our slope to be 3/7. Times the mean of the x's, which is 7/3. These just are the reciprocal of each other, so they cancel out. That just becomes 1. So our y-intercept is literally just 2 minus 1. So it equals 1. So we have the equation for our line. Our regression line is going to be y is equal to-- We figured out m. m is 3/7. y is equal to 3/7 x plus, our y-intercept is 1. And we are done. So let's actually try to graph this. So our y-intercept is going to be 1. It's going to be right over there. And the slope of our line is 3/7. So for every 7 we run, we rise 3. Or another way to think of it, for every 3.5 we run, we rise 1.5. So we're going to go 1.5 right over here. So this line, if you were to graph it, and obviously I'm hand drawing it, so it's not going to be that exact, is going to look like that right over there. And it actually won't go directly through that line. So I don't want to give you that impression. So it might look something like this. And this line, we have shown, that this formula minimizes the squared distances from each of these points to that line. Anyway, that was, at least in my mind, pretty neat.