9.2.2 - Interpreting the Coefficients

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.

Example 9-3: Student height and weight (Interpreting the coefficients) Section

Suppose we found the following regression equation for weight vs. height.

\(\hat{\text{weight }}=-222.5 +5.49\text{ height }\)

  1. Interpret the slope of the regression equation.
  2. Does the intercept have a meaningful interpretation? If so, interpret the value.

  1. A slope of 5.49 represents the estimated change in weight (in pounds) for every increase of one inch of height.
  2. A height of zero, or \(X = 0\) is not within the scope of the observation since no one has a height of 0. The value \(\hat{\beta}_0\) by itself is not of much interest other than being the constant term for the regression line.

If the slope of the line is positive, 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:

\(H_0\colon \beta_1=0\)

\(H_a\colon \beta_1\ne0\)

There are some assumptions we need to check (other than the general form) to make inferences for the population parameters based on the sample values. We will discuss these topics in the next section.