Chapter 7: Linear Regression and Correlation

Learn to model relationships and make predictions with confidence

Key Concepts You'll Master

Correlation

Strength of linear relationship

\(\rho\)

Regression Line

Best prediction equation

\(\hat{y} = b_0 + b_1x\)

R-squared

Variation explained by model

\(R^2\)

Standard Error

Typical prediction error

\(s_e\)

What is Linear Regression?

Linear regression reveals relationships between variables, enabling prediction and understanding. From fuel quality to stock prices, master the tools to model real-world relationships. You'll learn to quantify relationships, make predictions, and understand the uncertainty in your models.

Chapter 7: Linear Regression and Correlation - Learning Hub

Master Linear Regression and Correlation

Master prediction and relationship analysis

All sections are accessible - follow the numbered sequence for the best learning experience

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Interactive Formula Builder

Build formulas step-by-step

Master linear regression formulas by building them interactively. Click on each part to understand why it's there and how it works.

Key topics:

  • Regression line: ŷ = b₀ + b₁x
  • Correlation coefficient: r formula
  • Coefficient of determination: R²

7.1 Correlation Coefficient

Measuring relationship strength

How strong is the linear relationship between two variables? Learn to calculate and interpret correlation.

Key topics:

  • Calculate Pearson correlation coefficient
  • Interpret correlation values (-1 to +1)
  • Understand correlation vs causation
Complete prerequisites first

7.2 Simple Linear Regression

Finding the best-fit line

Discover how to find the line that best predicts one variable from another using least squares.

Key topics:

  • Derive the least squares equations
  • Calculate regression coefficients
  • Interpret slope and intercept
Recommended after prerequisites
Complete prerequisites first

7.3 Hypothesis Testing in Regression

Testing slope significance

Is the relationship real or just random noise? Test whether your regression slope is statistically significant.

Key topics:

  • Test if slope equals zero
  • Calculate t-statistics for regression
  • Construct confidence intervals for slope
Recommended after prerequisites
Complete prerequisites first

7.4 Confidence & Prediction Intervals

Quantifying uncertainty

Learn the crucial difference between confidence intervals for the mean and prediction intervals for individual values.

Key topics:

  • Distinguish confidence vs prediction intervals
  • Calculate both types of intervals
  • Visualize uncertainty bands
Recommended after prerequisites
Complete prerequisites first

7.5 Analysis of Variance (ANOVA)

Decomposing variation

Break down total variation into explained and unexplained parts. Understand the F-test for overall model significance.

Key topics:

  • Decompose sum of squares
  • Understand SST, SSR, and SSE
  • Perform F-test for regression
Recommended after prerequisites
Complete prerequisites first

7.6 Coefficient of Determination

Model goodness-of-fit

Master R-squared and adjusted R-squared. Learn what they really mean and their limitations.

Key topics:

  • Calculate and interpret R²
  • Understand adjusted R²
  • Recognize R² limitations
Recommended after prerequisites

Learning Tips

Build Intuition First

Start with fundamentals to understand the core concepts

Practice Real Scenarios

Apply concepts to real-world data to reinforce understanding

Master the Concepts

Learn when and how to apply each concept effectively