8.2.3 - Interpreting the Coefficients

Once we have the estimates for the slope and intercept, we need to interpret them. For Bob’s data, the estimate for the slope is 10.417 and the estimate for the intercept (constant) is 49.542. Recall from the beginning of the Lesson what the slope of a line means algebraically. If the slope is denoted as \(m\), then

\(m=\dfrac{\text{change in y}}{\text{change in x}}\)

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, \(\hat{\beta}_0\) 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. In Bob’s example, \(X = 0\) or 49.542 would be a plot of land with no critical areas. This might be of interest in establishing a baseline value, but specifically, in looking at land that HAS critical areas, this might not be of much interest to Bob.

As we already noted, 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. In Bob’s example, for every one unit change in critical areas, the cost of development increases by 10.417.

Interpreting the slope of the regression equation, \(\hat{\beta}_1\)

\(\hat{\beta}_1\) represents the estimated change in Y per unit increase in X

Note that the change may be negative which is reflected when \(\hat{\beta}_1\) is negative or positive when \(\hat{\beta}_1\) is positive.

If the slope of the line is positive, as it is in Bob’s example, then there is a positive linear relationship, i.e., as one increases, the other increases. If the slope is negative, then there is a negative linear relationship, i.e., as one increases the other variable decreases. If the slope is 0, then as one increases, the other remains constant, i.e., no predictive relationship.

Therefore, we are interested in testing the following hypotheses:

\(H_0\colon \beta_1=0\\H_a\colon \beta_1\ne0\)

Let’s take a closer look at the hypothesis test for the estimate of the slope. A similar test for the population intercept, \(\beta_0\), is not discussed in this class because it is not typically of interest.