Unit 2: Linear Regression Models and Analysis

Least-Squares Regression Line (LSRL)

A unique line that minimizes the sum of the squared vertical distances between data points and the line.

Residual

The difference between the actual observed value ($y$) and the value predicted by the regression line ($\hat{y}$).

Residual Plot

A scatterplot of the explanatory variable ($x$) against the residuals that helps assess the appropriateness of a linear model.

Slope ($b_1$)

The rate of change of the response variable with respect to the explanatory variable, calculated using $b1 = r \frac{sy}{s_x}$.

Y-Intercept ($b_0$)

The predicted value of the response variable when the explanatory variable is 0, calculated as $b0 = \bar{y} - b1\bar{x}$.

Extrapolation

Using a regression model to predict a value outside the range of the data used to create the model.

Coefficient of Determination ($r^2$)

The proportion of variation in $y$ explained by the variation in $x$.

Actual Value ($y$)

The observed value of the response variable.

Predicted Value ($\hat{y}$)

The value predicted by the regression line for a given explanatory variable $x$.

High-Leverage Points

Data points whose $x$-value is far from the mean of the x-values, potentially influencing the regression line significantly.

Outliers in Regression

Points with large residuals that fit the data pattern poorly, located far above or below the regression line.

Influential Points

Data points that, if removed, substantially change the slope, y-intercept, or correlation of the regression model.

Standard Deviation of the Residuals ($s$)

Measures the typical distance between actual data points and the regression line.

Interpreting Slope

Describes how the predicted response variable changes with a one-unit increase in the explanatory variable.

Interpreting Y-Intercept

Tells the predicted value of the response variable when the explanatory variable is 0, if contextually appropriate.

Random Scatter in Residual Plot

Indicates that the linear model is appropriate if it shows no clear pattern.

Curved Pattern in Residual Plot

Indicates that a linear model is inappropriate if the residuals show a clear curve.

Fanning in Residual Plot

Indicates increasing variability in predictions, potentially making them less reliable.

AP Statistics

Advanced Placement Statistics, a high school course focused on the principles of statistics.

Summation of Residuals

The sum of all residuals for a least-squares regression line is always 0.

Prediction Equation

The equation used to make predictions in linear regression is $\hat{y} = b0 + b1x$.

Correlation ($r$)

A measure of the strength and direction of the linear relationship between two variables.

Standard Deviation of Y ($s_y$)

Measures the spread of the response variable values.

Standard Deviation of X ($s_x$)

Measures the spread of the explanatory variable values.

Interpretation of $r^2$

Describes how much of the variation in the response variable is explained by the relationship with the explanatory variable.

Common Mistake: Forgetting the 'Hat'

A critical error in regression where $\hat{y}$ is mistakenly written as $y$, implying a perfect fit.

Departures from Linearity

When data points or residual patterns indicate that a linear model is not fitting the data well.