108 H1.07: Interpreting the Slope and Intercept
Students frequently have difficulty interpreting the slope and intercept in terms of the variables and units in the problem. Following are additional examples of interpretations and comments.
Example 5. When cigarettes are burned, one by-product in the smoke is carbon monoxide. Data is collected to determine whether the carbon monoxide emission can be predicted by the nicotine level of the cigarette. It is determined that the relationship is approximately linear when we predict carbon monoxide, C, from the nicotine level, N. Both variables are measured in milligrams. The formula for the model is [latex]C=3.0+10.3\cdot{N}[/latex]
Interpret the slope: If the amount of nicotine goes up by 1 mg, then we predict the amount of carbon monoxide in the smoke will increase by 10.3 mg.
Interpret the intercept: If the amount of nicotine is zero, then we predict that the amount of carbon monoxide in the smoke will be about 3.0 mg.
Example 6. Reinforced concrete buildings have steel frames. One of the main factors affecting the durability of these buildings is carbonation of the concrete (caused by a chemical reaction that changes the pH of the concrete) which then corrodes the steel reinforcing the building. Data is collected on specimens of the core taken from such buildings, where the depth, d, of the carbonation, in mm, and the strength, s, of the concrete, in mega-Pascals (MPa,) are measured. It is found that the model is [latex]s=24.5-2.8\cdot{d}[/latex]
Interpretation of the slope: If the depth of the carbonation increases by 1 mm, then the model predicts that the strength of the concrete will decrease by approximately 2.8 MPa.
Interpretation of the intercept: If the depth of the carbonation is 0 mm, then the model predicts that the strength of the concrete is approximately 24.5 MPa.
Comments: In this model, notice that the strength decreases as the carbonation increases, which is shown by the negative slope coefficient. When you interpret a negative slope, notice that you must say that, as the input variable increases, then the output variable decreases.
Notice that it isn’t necessary to fully understand the units in which the variables are measured in order to correctly interpret these coefficients. While it is good to understand data thoroughly, it is also important to understand the structure of linear models and to be able to practice this on applied problems, even if they are not problems in your field.