Least Squares Regression - AP Statistics

Perfect explanatory power in the model. Model explains all variation in $y$ perfectly.

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$. Standardized covariance measuring linear association strength.

Can significantly alter slope and intercept. Outliers pull the line toward themselves disproportionately.

Residual plot. Plots residuals vs. fitted values or explanatory variable.

$a = 4$. Using formula $a = 10 - (2)(3) = 4$.

Minimizing the sum of squared residuals. Method minimizes sum of squared prediction errors.

Can disproportionately influence the regression line. Points far from $\bar{x}$ have greater impact.

Normality of residuals. Tests if residuals follow normal distribution.

$SS_{total} = \sum (y_i - \bar{y})^2$. Total variation in $y$ around its mean.

$SS_{reg} = \sum (\hat{y}_i - \bar{y})^2$. Variation in predicted values around mean of $y$.

$SS_{res} = \sum (y_i - \hat{y}_i)^2$. Variation not explained by the regression model.

Constant variance of residuals. Residual spread remains constant across all fitted values.

Errors are normally distributed. Required for valid inference and prediction intervals.

Non-constant variance of residuals. Residual spread changes systematically with fitted values.

Predicting outside the range of the data. Risky because relationships may change outside observed range.

85% of variance is explained by the model. Strong model fit with good predictive power.

Slope of the regression line. Rate of change in $y$ per unit change in $x$.

Y-intercept of the regression line. Value of $y$ when $x = 0$.

No explanatory power in the model. Model explains none of the variation in $y$.

Correlation coefficient. Ranges from -1 to +1, denoted by $r$.

$b = 2$. Using formula $b = \frac{10}{5} = 2$.

$a = \bar{y} - b\bar{x}$. Ensures line passes through point $(\bar{x}, \bar{y})$.

Model may not be appropriate. Patterns suggest violations of linearity assumption.

Proportion of variance explained by the model. Coefficient of determination; ranges from 0 to 1.

Positive relationship between variables. As $x$ increases, $y$ tends to increase.

Negative relationship between variables. As $x$ increases, $y$ tends to decrease.

Minimize the sum of squared residuals. Finds line that best fits data by minimizing prediction errors.

Linear relationship between variables. Assumes straight-line relationship exists.

$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$. Covariance divided by variance of $x$.