# Simple Linear Regression

In an earlier example we considered accumulated savings over time. The correlation coefficient indicates how closely these observations conform to a linear equation. The slope (b1) is the steepness of the regression line, indicating the average or expected change in Y for each unit change in X. In the illustration above the slope is 307.5, so the average savings per week is \$307.50.

A useful analogy for slope is to think about steps. In the image below each step has a width of 1 and a rise of 2, so for each horizontal (X) increment of 1 step, there is a 2 unit increase in the vertical measurement (Y). If each 1 year increase in age is associated with a 2 lb. weight gain on average, then the slope for this relationship is 2. Predicted weight gain is 2 lb. per year.

A slope of 0 means that changes in the independent variable on the X-axis are not associated with changes in the dependent variable on the Y-axis, that is, there is no association. Consider the following example. Height and age were recorded for a group of children 4 to 15 years old, and a regression analysis indicated the following relationship:

Height (inches) = 31.5 + 2.3(Age in years)

If two children differ in age by 5 years, how much do you expect their height to differ?

The slope is 2.3, so if the children differ in age by 5 years, we would predict that the difference in height is 5 x 2.3 = 11.5 inches.