APEX AP Stats 3.2 Overview The Least-Squares Regression Line Study Guide

Cause and Effect

one factor causes another.

Explanatory Variable

independent variable

Least-Squares Regression Line

single identified line that best fits the linear pattern present in a set of bivariate data.

How is the least-squares regression line expressed as and what data pairs will be found on this line?

1. All data pairs (observed x-value, predicted y-value) will be found on this line in a scatterplot

2. line is expressed as a linear equation.

What does the least-squares regression line allow you to predict?

allows you to predict values of the response variable given observed values of the explanatory variable

What is the least-squares regression line formula?

y-hat = b1x + b0

y-hat=the predicted value of y

b0=the y-intercept b1=the slope; aka regression coefficient

x=observed value of x

LinReg (a+bx) On Your Graphing Calculator

name of the function for computing a least-squares regression line using a graphing calculator. On the calculator, there are two types of LinReg; in statistics, it's better to use LinReg(a + bx).

Negative Association

negative correlation; one increases as the other decreases.

Positive Association

positive correlation; direct relationship.

Observed Y-Values

actual values of the response variable found in a set of data.

Predicted Y-Values

values of a response variable that are predicted using a least-squares regression line equation, given observed values of the explanatory variable;

Why are the predicted and observed y values different?

predicted y-values are different from observed y-values as they're computed using the regression equation.

Regression Coefficient

slope of the least-squares regression line, symbolized by b1.

What is the regression coefficient equivalent to?

This value is equivalent to the change in the predicted value of the response variable associated with a one-unit change in the value of the explanatory variable.

RESID On Your Graphing Calculator

list of residuals generated when you run a linear regression analysis using a graphing calculator. You can graph this information along with the observed x-values or predicted y-values in a residual plot.

Residual

equivalent to the distance between the observed y-value and predicted y-value for any given value of the explanatory variable; the observed minus the predicted value.

Residual Plot

type of plot in which the residuals associated with a computed least-squares regression line are shown on the y-axis, and the x-values, or actual data are shown on the x-axis

What is the residual plot used to evaluate?

used to evaluate whether there is a linear association between the 2 variables

What does it mean if the residuals appear to be randomly scattered around y=0?

If the residuals appear randomly scattered around the line y = 0, then it's most likely linear.

Response Variable

dependent variable.

Scatterplot

two-dimensional graph used to display a set of bivariate data on a single graph.

Slope

slope is equivalent to the regression coefficient. In general, the slope is the change in the y-variable associated with a one-unit change in the x-variable.

Sum of Squared Residuals

formula is where yi is the i-th observed value of y and y^i is the predicted value of yi^.

Sum of squared residuals formula

summation(yi-y-hat i)^2

yi=the i-th observed value of y

y-hat=the predicted value of y-hat i

Verifying Linear Models

process in which residual plots are used to evaluate whether there's a linear association between two variables.

What does it mean if the residuals appear to have a noticeable pattern in the residual plot?

then a linear model may not be appropriate.