So, what we're going to do now is relate these two other numbers, a and the b values, and ultimately, tie it back to something we saw before in algebra, which was the idea of slope and y-intercept. Because at the end of the day, these are still just lines like we've seen them before. And ultimately, here's the thing: what we've talked about in this particular class is the idea of "trend." We said that a trend could be positive as the scatter plot, the blue dots, make their way upwards. And we said that the trend can be negative as the trend of the dots move downwards. And then, if you look at the red line, we can see that it is moving upwards, meaning the slope would be positive. And then, if you look at that red line, we can see the line is moving downwards, telling us the slope is negative. And so, what am I ultimately trying to push here? Ultimately, what I'm trying to push is the fact that the sign of the trend is going to be the same as the sign of your slope, meaning when your slope is positive, then your association will be positive. And remember, that's directly related to the sign of r. In the same way, if you have a negative slope, meaning you have a negative trend, meaning you'll have a negative "r" value. What's the thing I want you just to take away from this? Pictures here is ultimately to emphasize slope and r will have the same sign. They're going to have the same sign. However, if you go back to your calculator and you look at the value of r and you look at the value of b (b, b being the slope), notice they both are positive, definitely solidifying that idea. However, notice how they're different numbers. Notice how they are different numbers. And so, while slope and r will have the same sign, meaning that they are going to go in the same direction, the actual value of slope and the value of r is different, emphasizing the fact that slope is going to have its own meaning. Alright, slope will have its own meaning. So, let's talk about the meaning of slope here. For starter, slope will always be the coefficient of x. Slope will be that number right next to x. And by definition of slope, slope is going to be the comparison of the average y-value as we have a one-unit change in the x-value. Now, I completely, completely understand that that definition seems a little vague. So, how am I going to interpret it? Well, you guessed it, guys. I'm gonna always give you a template, right? I'm always going to give you a template where in this template, there's going to be three main things you need to identify: First, what is your X variable in units? Second, what is your y-variable in units? And then, third, you need to use the value of slope to decide if you're going to pick the word "increasing" or "decreasing" and then actually give you the value of slope. So, when you look at this interpretation for slope, I want you to see that there are three major components: What's your X variable? What's your y-value variable? And what's the slope? So, instead of talking about this template very hypothetically, let's do a practical example. So, in this case, let's look at example three. When using a person's height in inches to predict their shoe size, came up with the following regression: one. And what I want you guys to see here is we are using a person's height. So, right off the bat, we can see here that the X variable, the X variable is going to be height. What we are using is height and that what we want to predict, what we want to predict is shoe size. So, shoe size is then going to be my y-variable, right? And so, what I want us to do then is interpret the slope. So, looking at the remaining two numbers in this equation, which of the numbers is the slope here? Can you guys give me a hand here? Which of the two numbers here is the slope? Yeah, perfect. Slope is going to be that positive, and I'm going to emphasize positive, it's that positive 0.570. That is going to be our slope. Okay, guys, so now let's interpret the slope. So, for starters, what we want to begin our template with is describing the fact we're looking at a one-unit change for my X variable. Well, what is my X variable? It's height. Height in inches. So, for every 1 inch increase in height, that's what the first part of the sentence will emphasize. It's emphasizing for every 1 inch increase in height. For every one unit increase in my X variable. For every one inch increase in height. We are then going to follow that up with a sentence describing my y-variable. Remember, my y-variable is shoe size. Alright, shoe size. But from there, that is when we are going to look at the slope. Alright, the slope then is going to dictate if we're looking at an increase or decrease in shoe size. Now, given that my slope is positive, given that my slope is positive, do you think we're going to use the word increase or decrease? Given that slope is positive, do you think we're going to use increase or decrease? Yeah, we're going to use increase. And again, the why for that is because the slope is positive. It's continuing that same positive trend. Alright, so in this case, we have that since the slope is positive, we then say, for every 1 inch increase in height, then the shoe size will increase by an average, and then you just take the actual value of your slope, that 0.570. And then, what's my unit for shoe size? There really isn't one. So, we'll just say "size". So, in this case, for every 1 inch increase in height, the shoe size will increase by an average of 0.570 size. Notice there's still another number there. There's another number here that we need to interpret, and this number is what we call the y-intercept. The y-intercept, which is that constant "a" all by itself, is ultimately telling us the predicted average value when X is zero. The predicted average value when X is zero. Now, once again, to give you a template, of course, of course, give you a template to interpret the Y value or, again, the first half of the sentence is going to be what is "x" in the Y-intercept. Well, again, "x" is zero. Whenever you're looking at the Y-intercept, "x" is zero. And so, we begin this sentence by emphasizing when my X variable is zero, the predicted Y value is going to be that Y-intercept. So, let's take a look at this. Let's go back to my height example, and let's take a look at this. So, in this case, we see here my Y-intercept is -29.361. So, in this case, if we are looking at this as a Y-intercept, we're saying that the Y value is -29.361, and my X value is zero. Whenever you see Y-intercept, it's emphasizing that's the Y number and that X will be zero. But here's the rub when it comes to interpreting Y-intercept is that before you even plug and chug this into the template, you need to ask the ever-important question: does this even make sense? What do I mean? Well, let's think about "x". "X" is representing height. "X" is representing height. So, that means "x" equaling zero is a height of 0. You need to ask the question: does this make sense? So, guys, tell me, does a height of a person being zero inches make sense? No, not at all. That is super, super weird. No one is zero inches. So, this height of zero inches does not make sense. Uh-oh, sad face. You can also check if your Y value even makes sense. Again, why? What is "y" representing? It's representing shoe size. It's representing a shoe size. And in this case, can you have a shoe size of -29.361? Does that make sense? No, not at all. Again, this does not make sense. So, why am I emphasizing this? It's emphasizing the fact that sometimes you won't be able to interpret the Y-intercept. Actually, for that matter, quite often, the Y-intercept is not something that's meaningful. The only way you can interpret the Y-intercept is, first and foremost, the "x" value of zero needs to make sense. Second, the "y" value needs to make sense. And so, when one or both of the "x" and/or "y" value of the Y-intercept does not make sense, then it just simply means we don't make an interpretation.