4.3 Fitting Linear Models to Data

Linear models help us understand relationships between variables using data. Scatter plots visualize these relationships, while lines of best fit summarize them mathematically. By analyzing patterns and calculating equations, we can make predictions and draw insights.

Regression analysis takes this further, allowing us to create predictive models. However, it's important to distinguish between linear and nonlinear relationships, and to be aware of limitations like extrapolation and the influence of outliers on our results.

Fitting Linear Models to Data

Scatter plots for variable relationships

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• Scatter plots visualize relationships between two quantitative variables
  • Each data point represents a pair of values $(x, y)$
  • Independent variable (explanatory) plotted on x-axis (time studying)
  • Dependent variable (response) plotted on y-axis (exam score)
• Analyzing scatter plots reveals correlation patterns
  • Positive correlation: As x increases, y tends to increase (height and weight)
  • Negative correlation: As x increases, y tends to decrease (price and demand)
  • No correlation: No apparent relationship between x and y (shoe size and IQ)
  • Outliers deviate significantly from overall pattern (anomalous data points)

Line of best fit interpretation

• Line of best fit (regression line) represents relationship between two variables
  • Straight line that best fits the data points
  • Minimizes sum of squared vertical distances between points and line
• Calculating line of best fit using technology (graphing calculator, spreadsheet software)
  • Equation in form $y = mx + b$, where $m$ is slope and $b$ is y-intercept
• Interpreting line of best fit components
  • Slope $(m)$: Change in y for one-unit increase in x (rate of change)
  • Y-intercept $(b)$: Predicted y-value when x is zero (starting point)
  • Correlation coefficient $(r)$: Strength and direction of linear relationship ($-1 \leq r \leq 1$)
• Residuals: Differences between observed values and predicted values on the line of best fit

Linear vs nonlinear relationships

• Linear relationships have data points following straight-line pattern
  • Constant rate of change (slope) between variables (income and expenses)
• Nonlinear relationships have data points not following straight-line pattern
  • Rate of change varies across range of independent variable
  • Examples: Exponential (population growth), quadratic (projectile motion), logarithmic (pH scale)
• Identifying relationship type by visually inspecting scatter plot
  • Analyze residual plot (differences between observed and predicted values)
    • Random distribution of residuals suggests linear relationship
    • Non-random distribution of residuals suggests nonlinear relationship

Regression analysis for predictive modeling

• Regression analysis models relationship between variables using mathematical equations
• Simple linear regression models relationship between two variables using linear equation
  • Makes predictions about dependent variable based on independent variable (sales based on advertising)
• Steps in constructing predictive model:
  1. Collect and organize data
  2. Create scatter plot to visualize relationship
  3. Calculate line of best fit using technology
  4. Assess goodness of fit using correlation coefficient $(r)$ and coefficient of determination $(r^2)$
  5. Use regression equation to make predictions
• Limitations and considerations when applying regression analysis
  • Extrapolation: Cautious when predicting outside range of observed data (future trends)
  • Correlation does not imply causation: Other factors may influence relationship (confounding variables)
  • Outliers and influential points can significantly affect regression line (skewed results)