Welcome back. What we've seen so far is that when errors are not normally distributed, or not or auto corrected. It can be an indication that your model is incomplete, and that you need additional predictors. You know the words, you need to consider more variables as you analyze your data. By using multiple variables, you can start to consider how multiple factors can affect your sales. By the end of this lesson, you should be able to explain how multiple variables affect your regression equation, and explain how multiple regression can help with marketing mix decisions. The simple linear regression involving only one predictor can be easily generalized to having multiple independent variables. Formally, it means that you go from the equation Y = a + bx + errors to a model where Y = a + b1X1 + b2X2 and so on untill bN + XN, and errors. a here is still going to be the intercept. Meaning that if all the Xs are zeroes, then that will be the baseline level of Y you would have. In the case of marketing and sales, that could be a measure of business sales. X1, for example, is going to be the independent variable 1, and b1 is going to be the effect of this variable on sales. So that's going to be a parameter as well as b2, b3, and so on, and a. Those will be the parameters that we want to estimate through regression analysis. Then the question is, are these numbers statistically significant? And the other question is, what is explanatory power of this regression? What percentage of sales variations or variation of y can we explain with this model? That is a question you want to ask yourself. An important part of marketing is managing the marketing mix. Traditionally, the marketing mix has four components called the four Ps. Price, promotion meaning advertising, product, and place. So reasonable assumption to ask yourself, what are the impacts of all these variables on sales? Consider, for example, a typical packaged goods, like ketchup of coffee. You want to understand how some important variables like price and advertising are going to impact sales. Let's consider this chart. You can observe the blue lines which represents sales over time. And you see that these series has some spikes in it. So the question is what is driving these blue spikes? For example, if you are selling ice creams, could these bags be driven by some random factors like hot weather on the given day? Or could those bags be driven by your marketing actions like whether or not, you feature the product in the local newspaper. So now let's look at these red lines. The red lines represent when the brand is actually featured in the newspaper. And so the question you want to ask yourself is twofolds. First, are these blue lines driven by the red spikes? The other question you want to ask yourself is what is the magnitude of those spikes, those red spikes on the blue spikes? In marketing, it's rare that just one single variable is enough to reason behaviors. And more often than not, you need to include multiple variables to explain marketing outcomes.