So let’s make some predictions! Making predictions is easy. I predict none of us will report to work on Sunday. This is probably a good prediction, since Sunday is not part of the typical work week. I predict the average price of a gallon of gas in the US will be $100.00 on January 1. This is far less likely to be a good prediction. We don’t want to just make predictions, we want to make good predictions. How? We can use a function. This is a linear prediction function which includes a random error term. Y is the dependent variable, what we want to predict. Beta zero is a constant term, the intercept. It represents the predicted value of y when x=0. What is x? X is the independent variable, what you are using to make the prediction. Going back now to beta one, this term is the slope of the line. The predicted change in y for each one unit change in x. Finally at the end of the function, epsilon. This is the random error term I mentioned before. Without the random error term, as a function, y can only represent an exact linear relationship between x. Including the random error term allows us to take into account any deviation of the actual y values which exist in our dataset from our predictions. To put it another way, the random error term accounts for other unpredictable factors. Why are we using a linear function? Our topic is linear regression right? It turns out lines can do a pretty good job predicting dependent variables as long as the slope of the line doesn’t change as x changes. If this happened, it wouldn’t really be a line.