Regression Analysis

A scatterplot shows paired values of two quantitative variables, with the explanatory variable on the x-axis and the response variable on the y-axis. Describe its form (linear or not), direction (positive or negative), strength (strong, moderate or weak) and any unusual features, such as clusters or points that don't fit the pattern.

The correlation coefficient r is a unit-free number from −1 to 1 that measures the direction and strength of a linear relationship: values near −1 or 1 are strong, and r = 0 means no linear relationship. A strong r doesn't prove a line is the right model, and correlation on its own doesn't show that one variable causes changes in the other.

A linear regression model, ŷ = a + bx, uses the explanatory variable x to predict the response, where b is the slope and a is the y-intercept. Predicting for an x inside the range of the data is interpolation; predicting outside that range is extrapolation, and it gets less reliable the farther out you go.

A residual is the observed value minus the predicted value, y − ŷ: a positive residual means the line underestimated, and a negative one means it overestimated. A residual plot with no clear pattern supports a linear model, while a curved pattern suggests a line isn't the best choice.

The least-squares regression line (LSRL) is the line that makes the sum of the squared residuals as small as possible; it always passes through (x̄, ȳ), and you find its slope, y-intercept and r with technology. Interpret the slope as the predicted change in y for each one-unit increase in x, the y-intercept as the predicted y when x = 0 (which may make no sense if x = 0 is far outside the data), and r² as the proportion of the variation in y explained by the linear model.