# 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.

$$\text{weight }=-222.5 +5.49\text{ height }$$

1. Interpret the slope of the regression equation.
2. Does the intercept have a meaningful interpretation? If so, interpret the value.
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.
$$H_0\colon \beta_1=0$$
$$H_a\colon \beta_1\ne0$$