So what do we see here? On the horizontal axis we have price. On the vertical axis we have demand in units. In addition to the points that you were seeing earlier, we also see a straight line. That's your regression line. That's basically capturing how price is impacting demand. In particular if you notice, the coefficient in front of price is -0.9. What does that tell you? As price goes up by one dollar, sales is expected to come down by 0.9 units. The fact that it's negative makes a lot of sense. Why? As price goes up, sales is expected to come down. What else do we see here? What the regression line is predicting as well, is pretty much, if your price is at zero, close to free, about ten units would be sold. So that's what we get with the regression line. What else do we see? We also see something called an R squared. That's a number that's from zero to one. It captures basically how good is your regression. The closer that number is to one, the regression is doing a good job of capturing how demand is changing the price. The closer that number is zero, unfortunately the regression is not doing a very good job. What do we see here? R square is quite high. It's about 89%. That's a very high R square. Typical R squares, 70 to 80%. That's reasonably high. What we can do from here, is now that we're convinced that the regression is doing a good job of understanding how demand is weighting the price, we can start using this regression to make predictions. That's what we are after. What we are after is how do we make demand predictions at different prices? That's what we will do next. So for doing that, we can take multiple steps. We start out by looking at the regression line, which is what we had shown earlier, and start inserting different prices. We can first take the prices that are already in our data set and compare how our predicted regression line, or making predictions from the regression, is comparing with actual data. That's what is shown first. What we see here is actual data and the predictive regression line are quite close to each other. It's not surprising R square of the regression remember which measures how good that regression is, is quite high. What we can also do is to go one step further. And that's the power of regression. We can also start looking at demand predictions at prices that were not there in the dataset. And that's what I do next. Look at the prices on the top right. When you look at those prices, these are prices that are not there in the dataset. These are future prices. This is when a manager can start thinking about how to use regression. Make predictions of demand for prices that he has not tested as yet. What we see here, is that regression can make predictions for demand at those prices as well. So that's the beauty of regression. Looking at some data, you can start understanding how quantitatively demand relates to prices. Look at the fit of the regression, that's 2r square. Once you're convinced that the regression is doing a good job, you can start using that regression to make demand predictions. Once you can make demand predictions, optimal prices are just one step away. And the intuition is the following. If we can make predictions at different prices, we can start looking at the revenue and profit at those different prices. Once you can do that, you can then start understanding what price should be charged to make the maximum revenue or maximum profit, if you know your underlying costs. So the beauty of regression is, we start from overall data. We start quantifying the relationship between variables. Once we have that, we start looking at how to make demand predictions. Next you can start thinking about optimal prices and weighting prices to figure out what your demand is going to be. What we just showed you was a simple example using one dependent variable one independent variable. But of course, regression can be expanded to cover multiple independent variables. The general idea is the following. Again on the left hand side we have Y, which is your dependent variable. On the right hand side we have the Xs which are your independent variable. In the bottom of this page, you see an example. One can speculate that sales is a function of price, but price might not drive everything. It may be advertising. It may be promotions. Regression can handle as many independent variables as you want. The idea there is exactly that same. What you're trying to do is to look at a combination of variables on the right hand side. Prices, advertising, promotions. And look at the relative impact of each one to see how much do they contribute to overall sales. Now regression, as I mentioned, is just one example of making demand predictions. It's a great tool for understanding drivers of demand, making demand predictions, talking about optimal prices. But there are many, many other examples. So regression is one way we can start thinking about things like cart. You can start thinking about things like networks. All of those predictive analytic tools that Pete is gonna cover later on in this course.