Module 12: Linear Regression and Correlation

# Linear Equations

Barbara Illowsky & OpenStax et al.

Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:

where *a* and *b* are constant numbers.

The variable ** x is the independent variable, and y is the dependent variable.** Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.

### try it

*y* = –0.125 – 3.5*x*

The graph of a linear equation of the form *y* = *a* + *bx* is a **straight line**. Any line that is not vertical can be described by this equation.

Graph the equation *y* = –1 + 2*x*.

### try it

- Is the following an example of a linear equation? Why or why not?

Aaron’s Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.

Find the equation that expresses the **total cost** in terms of the **number of hours**required to complete the job.

### try it

# Slope and *Y*-Intercept of a Linear Equation

For the linear equation *y* = *a* + *bx*, *b* = slope and *a* = *y*-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the *y*-intercept is the *y* coordinate of the point (0, *a*) where the line crosses the *y*-axis.

*y*= 25 + 15

*x*.

What are the independent and dependent variables? What is the *y*-intercept and what is the slope? Interpret them using complete sentences.

### try it

Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is *y* = 25 + 20*x*.

What are the independent and dependent variables? What is the *y*-intercept and what is the slope? Interpret them using complete sentences.

# References

Data from the Centers for Disease Control and Prevention.

Data from the National Center for HIV, STD, and TB Prevention.

# Concept Review

The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form ** y = mx + b**, where

**and**

*m***are constants,**

*b***is the independent variable,**

*x***is the dependent variable. In a statistical context, a linear equation is written in the form**

*y***, where**

*y = a + bx***and**

*a***are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation**

*b**y = a + bx*, the constant

*b*that multiplies the

**variable (**

*x**b*is called a coefficient) is called as the

**slope**. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation

*y = a + bx*, the constant a is called as the

*y*-intercept. Graphically, the

*y*-intercept is the

*y*coordinate of the point where the graph of the line crosses the

*y*axis. At this point

*x*= 0.

The **slope of a line** is a value that describes the rate of change between the independent and dependent variables. The **slope** tells us how the dependent variable (*y*) changes for every one unit increase in the independent (*x*) variable, on average. The ** y-intercept** is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.

# Formula Review

*y* = *a* + *bx* where *a* is the *y*-intercept and *b* is the slope. The variable *x* is the independent variable and*y* is the dependent variable.