Lesson 3
The simple regression equation
Big question
How do we read the simple regression equation in plain language?
Lesson progress
Complete checkpoints as you learn
Learning objectives
- Explain the simple regression equation in plain language.
- Use regression equation correctly in an interpretation.
- Connect the lesson idea to a formula, graph, Python result, or real example.
Simple explanation
The simple regression equation has three core pieces: an intercept, a slope, and an error term. The intercept anchors the line, the slope gives the direction and steepness, and the error term represents influences not included in x.
Key terms
- Regression equation
- A compact model that links y to x, an intercept, a slope, and an error term.
- Intercept
- The model value of y when x equals zero.
- Slope
- The predicted change in y when x increases by one unit.
- Error term
- The part of y not explained by x in the population model.
Population regression model
beta0 and beta1 are population parameters. The data help us estimate them.
Example
A store might estimate sales = beta0 + beta1 advertising + u. The slope tells how predicted sales change when advertising increases by one unit.
Interactive visual
Equation pieces
Each symbol has a job: outcome, intercept, slope, explanatory variable, and error term.
y
Dependent variable: the outcome being explained.
beta0
Intercept: the line's anchor when x equals zero.
beta1
Slope: the predicted change in y for one more x.
x
Explanatory variable: the measured predictor.
u
Error term: other influences on y.
Interactive activity
Formula diagram
Click each part of the model to see what it means.
y: outcome
The dependent variable we want to explain.
Equation builder
Select the pieces in order to build the simple regression model.
Build the equation in this order: outcome, intercept, slope times x, error term.
Try it yourself
Write one plain-English sentence explaining the main idea from this lesson.
Common mistakes
Check these before you move on.
A regression coefficient describes a pattern unless the assumptions or research design support a causal interpretation.
Quick quiz
What does beta1 represent in the simple regression equation?
Quick quiz
What does the error term u represent?
Key takeaway
The simple regression equation is easiest to read when each symbol is translated into a role.