Linear regression helps us understand relationships between variables. It's like drawing a line through scattered dots to see patterns. We use this to predict outcomes based on input data.

The regression equation gives us a formula for that line. It shows how one variable changes when another does. This helps us make educated guesses about future trends or unknown values.

The Regression Equation

Least-squares regression line calculation

• Least-squares regression line fits data points best by minimizing sum of squared vertical distances between points and line
• Equation of least-squares regression line: $\hat{y} = b_0 + b_1x$
  • $\hat{y}$: predicted value of response variable (dependent variable)
  • $b_0$: y-intercept of regression line
  • $b_1$: slope of regression line
  • $x$: value of explanatory variable (predictor variable)
• Calculate slope ($b_1$) and y-intercept ($b_0$) using formulas:
  1. $b_1 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2}$
  2. $b_0 = \bar{y} - b_1\bar{x}$
  • $\bar{x}$: mean of explanatory variable
  • $\bar{y}$: mean of response variable
  • $x_i$ and $y_i$: individual values of explanatory and response variables

Interpretation of regression slope

• Slope ($b_1$) represents change in response variable ($y$) for one-unit increase in explanatory variable ($x$)
• Interpretation depends on context of variables studied
  • Salary example: slope of 1,500 indicates employee's salary increases by $1,500 on average for each additional year of experience

Correlation and determination coefficients

• Correlation coefficient ($r$) measures strength and direction of linear relationship between explanatory and response variables
  • $r$ ranges from -1 to 1, values closer to -1 or 1 indicate stronger linear relationship
  • Positive $r$: positive linear relationship
  • Negative $r$: negative linear relationship
• Coefficient of determination ($r^2$) represents proportion of variation in response variable explained by explanatory variable in regression model
  • $r^2$ ranges from 0 to 1, values closer to 1 indicate more variation explained by explanatory variable
  • Example: $r^2 = 0.75$ means 75% of variation in response variable explained by explanatory variable, 25% due to other factors or random variation

Linear Regression Analysis

• Linear regression is a statistical method used to model the relationship between variables
• Scatterplots are used to visualize the relationship between two variables
• The line of best fit (regression line) is determined through regression analysis
• Regression analysis helps identify patterns and make predictions based on the relationship between variables