Once we have the estimates for the slope and intercept, we need to interpret them. Recall from the beginning of the Lesson what the slope of a line means algebraically. If the slope is denoted as \(m\), then
\(m=\dfrac{\text{change in y}}{\text{change in x}}\)
In other words, the slope of a line is the change in the y variable over the change in the x variable. If the change in the x variable is one, then the slope is:
\(m=\dfrac{\text{change in y}}{1}\)
The slope is interpreted as the change of y for a one unit increase in x. This is the same idea for the interpretation of the slope of the regression line.
Interpreting the slope of the regression equation, \(\hat{\beta}_1\)
\(\hat{\beta}_1\) represents the estimated increase in Y per unit increase in X. Note that the increase may be negative which is reflected when \(\hat{\beta}_1\) is negative.
Again going back to algebra, the intercept is the value of y when \(x = 0\). It has the same interpretation in statistics.
Interpreting the intercept of the regression equation, \(\hat{\beta}_0\)
\(\hat{\beta}_0\) is the \(Y\)-intercept of the regression line. When \(X = 0\) is within the scope of observation, \(\hat{\beta}_0\) is the estimated value of Y when \(X = 0\).
Note, however, when \(X = 0\) is not within the scope of the observation, the Y-intercept is usually not of interest.
