# High School: Statistics and Probability

### Interpreting Categorical and Quantitative Data HSS-ID.C.7

7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in context of the data.

The fun part about being a statistician or math teacher is getting to say things like, "I love working with models," and, "My model is better than yours." Let other people assume what they will about your "models," but your students should know that a model is a way to use math to approximate a data set. After all, if statistics had beautiful men and women walking around, we bet it would be a much more popular subject.

One of the most common models used in statistics is the linear model. You may know this better as a linear equation or a best fit line. We can call it whatever we want, but it still makes estimates of the dependent variable, which means it's a model.

Students should know that all linear models take the form y = mx + b where m is the slope and b is the y-intercept. Hopefully it looks familiar—too familiar—to them. In fact, they probably think there's nothing more they could possibly learn about linear equations.

They might be right, but it's worth repeating.

The important part in using linear equations as models is the context. Students should be able to associate the independent variable with x and the dependent variable with y. Students should also be aware of what the slope and y-intercept represent in terms of the context and whether these values make sense or not. For example, it wouldn't make sense for a model describing a tree's yearly growth to have a negative slope.

Graphically, the slope of the line affects how "steep" the line is. The following plot shows the effect of varying slopes.

Graphically, the y-intercept of the line affects how "high" the line is. The following plot shows the effect of varying intercepts.

#### Drills

1. What is the slope of the linear model y = 2.0x – 12.0?

2.0

The slope is the coefficient in front of the x. So, in this case, the slope is 2.0.

2. What are the slope and intercept of the linear model y = x?

m = 1; b = 0

Wait...what? This is a model? Yes. The form is a little bit unusual, but it's still a model. In this case, we could also rewrite the model as y = 1x + 0, which means the slope m is 1 and the intercept b is 0.

3. Which of the following is the best description of a mathematical model?

A way to approximate a data set

Perhaps there is a model out there who does love math and could technically be described as a mathematical model, but the best answer in this case is (B). A mathematical model is a way to approximate a data set, not just any old function.

4. Which of the following equations is a linear model?

y = 2x + 40

A linear model always takes the form y = mx + b. Always. For something to be a linear model, the independent variable cannot be squared or have a sine or cosine function. That eliminates all answers except for (C).

5. What does the intercept of a linear model represent?

The value of the dependent variable when the independent variable equals zero

The intercept is the location where the line intersects with the y-axis. The y-axis is the location where x (the independent variable) equals 0. So the intercept is the value of the dependent variable when the independent variable is 0. We can also think of this as y = mx + b. Only when x = 0 does y reach b, the y-intercept.

6. What does the slope of a linear model represent?

How quickly the independent variable changes with respect to the dependent variable

The slope describes the change in y with respect to the change in x. This means the change in the dependent variable with respect to the independent variable. Both (A) and (D) are incorrect because they describe intercepts, and (C) describes the reciprocal of the slope.

7. A linear model describing the height of a river with respect to the rainfall total for the previous month suggests that for each inch of rainfall, the river rises five inches. What is the slope for this linear model?

5

The slope describes how the independent variable changes with the dependent variable. In this case, the problem description says that the height of the river depends on the rainfall. So x must be the rainfall and y must be the height of the river in inches. Since the change of y is 5 and the change of x is 1, our slope is 5/1 or 5. Both (A) and (C) are incorrect because they're the wrong values, and (D) would suggest that the river decreases by 5 inches for every inch of rainfall.

8. A linear model describing the height of a river with respect to the rainfall total for the previous month suggests that for each inch of rainfall, the river rises five inches. The linear model also suggests that as the amount of rainfall approaches zero, the height of the river is about 24 inches. What is the intercept for this linear model?

24

The intercept of a linear model is the value of the independent variable when the dependent variable approaches or equals zero. In this case, the dependent variable is the height of the river and it is dependent on the amount of rainfall. When there is no additional rainfall, the height of the river is 24 inches. The intercept is also 24 inches.

9. In the following plot, which of the lines has an intercept equal to 5?

Line d

The line that crosses the y-axis at y = 5 is labeled as Line D in the figure. None of the other lines have intercepts at y = 5 because they don't cross the y-axis at that point.

10. In the following plot, which of the linear models has the highest slope?

Line a