Linear Regression and Correlation

Scatter Plots Review

A scatter plot shows the relationship between two quantitative variables.

Direction: Positive, negative, or none Form: Linear or nonlinear Strength: Strong, moderate, or weak

Correlation Coefficient ($r$)

Measures the strength and direction of a linear relationship:

• $r = 1$: Perfect positive linear
• $r = -1$: Perfect negative linear
• $r = 0$: No linear relationship
• $|r|$ close to 1: Strong

Line of Best Fit (Regression Line)

The line that best represents the trend in the data: $\hat{y} = mx + b$

Making Predictions

Use the regression equation: If $\hat{y} = 2.5x + 10$ and $x = 8$: $\hat{y} = 2.5(8) + 10 = 30$

Residuals

$\text{Residual} = \text{Actual} - \text{Predicted} = y - \hat{y}$

• Positive residual: actual is above the line
• Negative residual: actual is below the line

Interpreting the Slope

"For every 1-unit increase in $x$, the predicted $y$ increases/decreases by $m$ units."

Interpreting the Y-Intercept

"When $x = 0$, the predicted $y$ is $b$." (May not always make practical sense.)

Cautions

1. Correlation ≠ Causation
2. Don't extrapolate beyond the data range
3. Outliers can strongly affect the regression line
4. $r$ only measures linear relationships

Example interpretation: "There is a strong positive linear relationship ($r = 0.92$) between hours studied and test score. For each additional hour of studying, the predicted test score increases by about 5 points."