8.8 Scatter Plots, Correlation, and Regression Lines

Scatter plots help us visualize relationships between two variables. By plotting points on a graph, we can see patterns and trends in data. This visual representation is crucial for understanding how different factors might be connected.

Correlation coefficients and regression lines take scatter plots a step further. These tools let us quantify relationships and make predictions based on data. Understanding these concepts helps us interpret real-world information and make informed decisions.

Scatter Plots and Correlation

Creation of scatter plots

• Visualize relationships between two quantitative variables by plotting data points on a coordinate plane
  • Each point represents a single observation (student's height and weight)
  • Independent variable plotted on x-axis (hours studied)
  • Dependent variable plotted on y-axis (exam score)
• Construct scatter plots by hand or using technology (Excel, graphing calculator)
• Choose appropriate scales for x and y axes to accurately represent data
• Label axes clearly with variable names and units (time in minutes, distance in kilometers)

Interpretation of correlation coefficients

• Correlation coefficient ($r$) quantifies strength and direction of linear relationships between variables
  • $r$ ranges from -1 to 1
    • $r = 1$: perfect positive linear relationship (income and education level)
    • $r = -1$: perfect negative linear relationship (car's value and age)
    • $r = 0$: no linear relationship (shoe size and IQ)
  • Stronger linear relationships indicated by $|r|$ values closer to 1 (0.9 vs 0.2)
• Positive $r$: variables increase together (hours of exercise and cardiovascular health)
• Negative $r$: one variable increases as the other decreases (product price and demand)
• Coefficient of determination ($r^2$) is proportion of variation in dependent variable explained by independent variable
  • $r^2$ ranges from 0 to 1
  • $r^2 = 0.81$: 81% of variation in test scores explained by study time
• Correlation does not imply causation; other factors may influence the relationship

Regression Lines

Regression lines for predictions

• Regression line (least-squares line) best fits data points in scatter plot
  • Minimizes sum of squared vertical distances between points and line
• Equation of regression line: $\hat{y} = mx + b$
  • $\hat{y}$: predicted value of dependent variable
  • $m$: slope, change in $\hat{y}$ per one-unit increase in $x$
  • $b$: y-intercept, $\hat{y}$ value when $x = 0$
• Make predictions by substituting $x$ value into equation and solving for $\hat{y}$
  • Predict test score ($\hat{y}$) for 5 hours of studying ($x$): $\hat{y} = 10 + 5(5) = 35$
• Interpret slope in context of problem
  • Slope of 1.5 with dependent variable of sales (thousands) and independent variable of advertising expenditure (thousands): each $1,000 increase in advertising associated with$1,500 increase in sales

Analyzing Regression Models

• Interpolation: Making predictions within the range of observed data
• Extrapolation: Making predictions outside the range of observed data (less reliable)
• Residual: Difference between observed and predicted values, used to assess model fit
• Variance: Measure of spread in data points, affects reliability of regression model