AP Statistics 5.5 Least-Squares Regression Study Notes - New Syllabus

5.5.A.1 Least-Squares Regression Line (LSRL)

When the relationship between two quantitative variables appears to be approximately linear, statisticians use a simple linear regression model to describe the relationship and predict values of the response variable.

The regression line is chosen so that it provides the best possible linear fit to the observed data.

This line is called the Least-Squares Regression Line (LSRL) because it is determined by minimizing the sum of the squared residuals.

A residual is the difference between the observed response value and the predicted response value.

\( \mathrm{Residual}=y-\hat{y} \)

Since some residuals are positive and others are negative, simply adding them together could produce a value close to zero. To prevent positive and negative residuals from canceling each other out, each residual is squared before being added.

The Least-Squares Regression Line is the line for which the sum of the squared residuals is as small as possible.

Least-Squares Criterion

\( \mathrm{Minimize}\ \sum (y-\hat{y})^{2} \)

where

• \( y \) = Observed response value
• \( \hat{y} \) = Predicted response value
• \( y-\hat{y} \) = Residual
• \( \sum (y-\hat{y})^{2} \) = Sum of the squared residuals

Because finding the least-squares regression line requires many calculations, the equation is almost always determined using technology, such as a graphing calculator or statistical software.

The equation of the Least-Squares Regression Line is

\( \hat{y}=a+bx \)

where

• \( \hat{y} \) = Predicted response value
• \( a \) = y-intercept of the regression line
• \( b \) = Slope of the regression line
• \( x \) = Explanatory variable

An Important Property of the LSRL

The Least-Squares Regression Line always passes through the point

\( (\bar{x},\bar{y}) \)

where

• \( \bar{x} \) = Mean of the explanatory variable
• \( \bar{y} \) = Mean of the response variable

This means that the average values of the explanatory and response variables always lie on the regression line.

Important AP Exam Notes

• The Least-Squares Regression Line is the best-fitting linear model because it minimizes the sum of the squared residuals.
• The equation of the LSRL is determined using technology; students are not expected to calculate it by hand.
• The regression line always passes through the point \( (\bar{x},\bar{y}) \).
• Minimizing the sum of the squared residuals produces the line with the smallest overall prediction error.

5.5.A.2 Slope of the Least-Squares Regression Line

The slope of the least-squares regression line describes how the predicted response variable changes for each one-unit increase in the explanatory variable.

In the regression equation

\( \hat{y}=a+bx \)

the coefficient \( b \) represents the slope of the regression line.

The slope is calculated using technology, such as a graphing calculator or statistical software.

Interpreting the Slope

The slope represents the predicted change in the response variable for every one-unit increase in the explanatory variable.

Important AP Exam Notes

• The slope is interpreted in the context of the variables.
• The slope describes the change in the predicted response variable, not necessarily the observed value.
• The value of the slope is calculated using technology.

5.5.A.3 y-Intercept of the Least-Squares Regression Line

The y-intercept of the least-squares regression line is the predicted value of the response variable when the explanatory variable is equal to \( 0 \).

In the regression equation 

\( \hat{y}=a+bx \)

the coefficient \( a \) represents the y-intercept.

The y-intercept is calculated using technology.

Interpreting the y-Intercept

The y-intercept represents the predicted response value when

\( x=0 \)

However, the y-intercept should only be interpreted if \( x=0 \) is meaningful in the context of the data.

Important AP Exam Notes

• The y-intercept is not always meaningful.
• If \( x=0 \) is outside the range of the observed data, the y-intercept may have little practical interpretation.
• The y-intercept is calculated using technology.

5.5.A.4 Correlation Coefficient (\( r \))

In simple linear regression, the correlation coefficient, denoted by \( r \), measures the strength and direction of the linear relationship between two quantitative variables.

The correlation coefficient is calculated using technology.

The value of \( r \) always satisfies

\( -1\le r\le1 \)

Important AP Exam Notes

• The value of \( r \) is always calculated using technology.
• The sign of \( r \) indicates the direction of the association.
• The magnitude of \( r \) indicates the strength of the linear association.
• The correlation coefficient has no units.

5.5.A.5 Coefficient of Determination (\( r^2 \))

In simple linear regression, the square of the correlation coefficient, denoted by \( r^2 \), is called the coefficient of determination.

The coefficient of determination measures how well the least-squares regression line explains the variation in the response variable.

Formula

\( r^2=(r)^2 \)

where

• \( r \) = Correlation coefficient
• \( r^2 \) = Coefficient of determination

The value of \( r^2 \) represents the proportion (or percentage) of the variation in the response variable that is explained by the linear relationship with the explanatory variable.

The remaining variation is due to other factors or random variability that is not explained by the regression model.

Important AP Exam Notes

• \( r^2 \) is called the coefficient of determination.
• \( r^2 \) is always between 0 and 1.
• Multiply \( r^2 \) by 100% to express the explained variation as a percentage.
• The remaining variation, \(1-r^2\), represents the proportion of variation not explained by the linear regression model.
• Always interpret \( r^2 \) in the context of the response variable.