APEX AP Stats 3.2.1 The Least-Squares Regression Line Study Guide

What is a scatterplot?

A graph with x and y axes used to find relationships between variables.

What do the y-values represent in a scatterplot?

The response or dependent variable.

What do the x-values represent in a scatterplot?

The explanatory or independent variable.

What is the Line of Best Fit?

A straight line that is closest to including all data points, used for making predictions.

What does the line of best fit allow us to do?

make estimations/predictions of the y-value when we know the x-values

What does y-hat (ŷ) represent?

Predicted y: the predicted value of the dependent variable based on the line of best fit.

Residual

distance between the predicted and observed y-values

How do you calculate a residual?

Residual = observed y-value - predicted y-value (y - ŷ).

What indicates a positive residual?

The observed value is larger than the predicted value.

Why do we get negative residuals?

because the variables may have a non-linear relationship

What is the goal of minimizing the squares in regression analysis?

To make the area of the squared residuals as small as possible.

What is the Least-Squares Regression Line?

The line that minimizes the sum of the squares of the residuals.

What is the least squares regression line formula?

ŷ=b0+b1x ŷ=predicted value of y b0=y-intercept b1=slope of the regression line/regression coefficient x=observed x value

What does the regression coefficient indicate?

The change in the dependent variable for a one-unit increase in the independent variable.

What does a positive slope in a regression indicate?

There is a positive association between the two variables.

What is extrapolation in the context of regression analysis?

Predicting a y-value beyond the range of observed x-values.

What does a negative residual indicate?

The observed value is less than the predicted value.

Why might negative residuals occur?

Because the variables may have a non-linear relationship.

What is the significance of squaring residuals in regression analysis?

It helps to assess the goodness of fit by emphasizing larger errors.