A regression line summarizes the pattern in a scatterplot by giving a predicted value of y for each value of x. In the equation ŷ = a + bx, the slope and intercept are the two numbers that make the line meaningful. The slope tells how the prediction changes as x increases, while the intercept gives the predicted value when x is zero.
Interpreting these values correctly helps connect statistics to real situations, such as studying hours, temperature, cost, or population growth.
The slope b is a rate of change, so its units are the units of y divided by the units of x. The intercept a is a starting prediction at x = 0, but it is only useful if x = 0 makes sense in the context and is near the data. A worked example might use ŷ = 52 + 4.5x to predict test score from hours studied, where each extra hour is associated with 4.5 more points and 52 is the predicted score for 0 hours studied.
The regression line predicts average trends, not exact outcomes for every data point.
Key Facts
- Regression equation: ŷ = a + bx
- Slope formula from two points on the line: b = change in ŷ / change in x
- Slope interpretation: for each 1-unit increase in x, predicted y changes by b units
- Intercept interpretation: a is the predicted value of y when x = 0
- Positive slope means predicted y increases as x increases, while negative slope means predicted y decreases as x increases
- Residual = observed y - predicted ŷ
Vocabulary
- Regression line
- A line that models the average relationship between an explanatory variable x and a response variable y.
- Slope
- The amount the predicted y-value changes for each 1-unit increase in x.
- Intercept
- The predicted y-value when x equals zero.
- Predicted value
- The value ŷ given by the regression equation for a chosen value of x.
- Residual
- The vertical difference between an observed data point and the value predicted by the regression line.
Common Mistakes to Avoid
- Saying the slope is just a number without units is wrong because slope is a rate and must include units of y per unit of x.
- Interpreting the intercept even when x = 0 is unrealistic is wrong because the intercept may have no practical meaning outside the data context.
- Treating ŷ as the exact observed value is wrong because regression gives a prediction, and real data points often differ from the line.
- Using the slope to claim causation is wrong because a regression relationship alone does not prove that changes in x cause changes in y.
Practice Questions
- 1 A regression line predicts weekly savings from hours worked as ŷ = 15 + 8x. Interpret the slope and intercept in context.
- 2 For the regression equation ŷ = 120 - 3.5x, where x is temperature in degrees Celsius and ŷ is predicted hot chocolate sales, find the predicted sales when x = 10 and interpret the slope.
- 3 A model predicts plant height from days since planting as ŷ = 2 + 0.9x. Explain whether the intercept is likely meaningful and what information you would need to decide.