The least-squares regression line and coefficient of determination

What the exam asks

• Calculate or identify the slope $b$, $y$-intercept $a$, correlation $r$, and coefficient of determination $r^2$ from raw data (using technology) or from standard statistical software output.
• Explain the properties of the least-squares regression line: that it minimizes the sum of the squared residuals and always passes through the point of averages $(\bar{x}, \bar{y})$.
• Interpret the slope $b$, the $y$-intercept $a$, and the coefficient of determination $r^2$ in context using precise, non-deterministic language.
• Determine and explain when the $y$-intercept has no reasonable physical meaning in the context of the study.

Core ideas

What makes a line “least-squares”?

When a scatterplot shows a linear form, many lines could be drawn through the data. Each data point $(x, y)$ has a vertical distance from a candidate line $\hat{y} = a + bx$, called a residual ($$). Because some residuals are positive and others are negative, their sum for the best-fitting line is always .

To find the single best line without positive and negative residuals canceling out, we square each residual and add them up: $\sum (y - \hat{y})^2$.

• The least-squares criterion: The least-squares regression line is the unique line that minimizes the sum of the squared residuals.
• The point of averages: Every least-squares regression line is guaranteed to pass directly through the point $(\bar{x}, \bar{y})$, where $\bar{x}$ is the mean of the explanatory variable values and is the mean of the response variable values. Consequently, .

Finding the LSRL with technology and computer output

On the AP Statistics exam, you will never compute $a$, $b$, or $r$ by hand from raw data using sum-of-squares formulas. Instead, you will either enter a small data set into your graphing calculator (or Bluebook’s built-in Desmos calculator) or read a regression table.

Typical regression computer output looks like this (from a hypothetical study of used electric cars relating battery age in years to maximum range in miles):

How to extract the components:

• $y$-intercept ($a$): Next to Constant (or Intercept) under Coef: $a = 285.40$.
• Slope ($b$): Next to the explanatory variable name (Battery Age) under Coef: $b = -14.20$.