Regression equations help us understand relationships between variables. They use data to find the best-fitting line, showing how one thing changes when another does. This powerful tool lets us predict outcomes and analyze trends.

The equation has two key parts: slope and y-intercept. The slope shows how much y changes when x increases by one. The y-intercept is where the line crosses the y-axis. Together, they paint a picture of the data's pattern.

The Regression Equation

Least-squares regression line calculation

• Least-squares regression line best fits data points minimizes sum of squared residuals
  • Residuals represent vertical distances between data points and regression line ($y_i - \hat{y}_i$)
• Regression equation in form $\hat{y} = b_0 + b_1x$
  • $\hat{y}$ represents predicted value of y for given x
  • $b_0$ represents y-intercept, value of y when x is 0 (height at origin)
  • $b_1$ represents slope, change in y for one-unit increase in x (rise over run)
• Calculate slope ($b_1$) using formula: $b_1 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n}(x_i - \bar{x})^2}$
  • $x_i$ and $y_i$ represent individual data points (coordinates)
  • $\bar{x}$ and $\bar{y}$ represent means of x and y, respectively (averages)
• Calculate y-intercept ($b_0$) using formula: $b_0 = \bar{y} - b_1\bar{x}$
  • Substitute slope ($b_1$) and means ($\bar{x}$ and $\bar{y}$) into equation

Interpretation of slope and y-intercept

• Slope ($b_1$) represents change in response variable (y) for one-unit increase in predictor variable (x)
  • Positive slope indicates positive linear relationship between x and y (direct)
  • Negative slope indicates negative linear relationship between x and y (inverse)
• Y-intercept ($b_0$) represents value of response variable (y) when predictor variable (x) is 0
  • Y-intercept may not have meaningful interpretation if x cannot realistically be 0 (extrapolation)
• Interpret slope and y-intercept using context of data and units of variables
  • Slope units: units of y per unit of x (mph per year)
  • Y-intercept units: same as units of y (starting salary in dollars)

Strength of linear relationships

• Correlation coefficient (r) measures strength and direction of linear relationship between two variables
  • r ranges from -1 to 1
    1. r = 1 indicates perfect positive linear relationship (straight line increasing)
    2. r = -1 indicates perfect negative linear relationship (straight line decreasing)
    3. r = 0 indicates no linear relationship (scattered points)
  • Stronger linear relationship as r approaches -1 or 1 (tighter clustering around line)
  • Calculate r using formula: $r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2}\sqrt{\sum_{i=1}^{n}(y_i - \bar{y})^2}}$
• Coefficient of determination ($r^2$) represents proportion of variation in response variable (y) explained by predictor variable (x)
  • $r^2$ ranges from 0 to 1
    1. $r^2 = 1$ indicates all variation in y explained by x (perfect fit)
    2. $r^2 = 0$ indicates none of variation in y explained by x (no relationship)
  • Calculate $r^2$ by squaring correlation coefficient (r)
  • $r^2$ often expressed as percentage (50% of variation explained)
• Visualize relationship between variables using a scatter plot

Assessing model fit and reliability

• Examine residuals for patterns or trends to assess model assumptions
  • Check for homoscedasticity (constant variance of residuals across predictor values)
• Identify potential outliers that may influence regression results
• Standard error of estimate measures average deviation of observed values from predicted values
• Calculate confidence intervals for slope and intercept to assess precision of estimates