1. Chapter 2 – Simple Linear Regression - How

2. Here is a perfect scenario of what we want reality to look like for simple linear regression. Our two variables are not perfectly related, as we can see, but nonetheless there is a relationship. The means of each distribution is connected by a straight line. Y X

3. Y X In order to discover what the actual equation for the straight line is we need to sample from the population.

4. Y X And as far as we can tell there is a scatter of ordered pairs. From these ordered pairs we need to determine the equation of the line. (a, b)ab

5. Y X How do we determine the actual equation of the line? The formula used to determine the actual line will not be very informative. So instead I will tell you what the formula will achieve.

6. Y X The error is defined as the difference between the observed y-value (blue dots are represented using the observed y) and the predicted y-value (the predicted y-value is located on the line). y1y1 (x 1, y 1 ) Calculate the error for every observed y-value. Take the square of all the results and add them up. The least squares regression line has the property that no other line will have a smaller squared sum of the errors. Observations

7. Y X y1y1 (x 1, y 1 ) Keep in mind that since we are going to attempt to find the linear equation using a sample from our population, then the linear equation that we calculate is an approximation of the actual linear equation. In other words the slope and y-intercept are estimates of the actual slope and y-intercept.