Analyzing Departures from Linearity - AP Statistics

What is a residual in the context of linear regression?

Residual = Observed value - Predicted value. The difference between what actually occurred and what the model predicted.
Which transformation can address right skewness?

Square root transformation. Reduces the effect of large values in right-skewed distributions.
How do you identify an influential point in regression?

Influential points significantly affect the slope. Points with high leverage and large residuals change regression coefficients substantially.
What transformation is used for positive skewness?

Log transformation. Compresses large values more than small ones, reducing right tail.
Identify a method for testing linearity in regression analysis.

Examine the residuals for patterns. Look for systematic patterns that indicate assumption violations.
What does a residual plot with increasing spread indicate?

Non-constant variance. Spread increases with fitted values, violating homoscedasticity assumption.
What does a lack of randomness in residuals suggest?

Model assumptions are violated. Systematic patterns indicate linear model is inappropriate for the data.
What is the effect of non-linearity on model accuracy?

Reduces model accuracy. Linear model cannot capture true relationship, leading to systematic errors.
Which assumption is violated if residuals show a funnel shape?

Homoscedasticity is violated. Funnel shape indicates variance increases with fitted values.
Identify the effect of outliers on a regression line.

Outliers can skew the regression line. Extreme values pull the line away from the true relationship.
What does a pattern in residuals indicate?

Indicates a model misfit. Systematic patterns suggest model assumptions are violated.
Which plot can help identify non-linearity in data?

Residual plot. Curved or systematic patterns in residuals reveal non-linear relationships.
Identify the result of high leverage points in regression.

Can disproportionately influence the model. Points far from center of predictors can heavily influence fitted line.
Identify a common method to address non-linearity.

Using polynomial regression. Adds curved terms to capture non-linear relationships in data.
What method can address the issue of non-linearity?

Transformation of variables. Mathematical functions applied to create linear relationships from non-linear data.
What pattern in residuals suggests a better fit with a quadratic model?

Curved pattern. U-shaped residuals indicate relationship has quadratic component.
What is examined to ensure the linear model is appropriate?

Residual plot. Checks whether linear relationship assumption is reasonable for data.
What is the effect of random residuals on a regression model?

Indicates a good fit. No patterns suggest linear model assumptions are satisfied.
Which transformation can stabilize variance in a dataset?

Logarithmic transformation. Reduces variability when variance increases with the mean.
What is a sign of a good regression model in a residual plot?

No discernible pattern in residuals. Random scatter indicates model captures the relationship appropriately.
What does a residual plot with no apparent pattern indicate?

Assumptions likely satisfied. Random scatter confirms linear model is appropriate for the data.
Which pattern in residuals suggests the need for a polynomial model?

Curved pattern. U-shaped pattern indicates quadratic relationship exists in data.
What is the impact of non-linearity on predictions?

Leads to inaccurate predictions. Linear model cannot capture true relationship, causing prediction errors.
What does heteroscedasticity in a residual plot indicate?

Indicates non-constant variance. Error variance changes across fitted values, violating equal variance assumption.
Identify a technique to improve model fit with non-linear data.

Use transformations. Mathematical functions applied to variables to create linear relationships.
Identify the role of transformations in linear regression.

To linearize relationships and stabilize variance. Creates linear relationships and meets regression assumptions.
What is the role of residual analysis in regression?

To check model assumptions. Evaluates whether model meets required assumptions for valid inference.
What does a curved pattern in a residual plot suggest?

Curved pattern suggests non-linearity. Indicates the relationship is not linear and needs a different model form.
What does a linear pattern in residuals suggest about the model?

Model is not appropriate. Linear trend in residuals indicates model structure is wrong.
Which pattern in residuals suggests a transformation is needed?

Non-random pattern. Systematic structure indicates linear model is inadequate.
What does a pattern in a residual plot reveal about a model?

Model may not be appropriate. Patterns indicate linear assumptions are violated and model needs modification.
Identify the formula for calculating a residual.

$e_i = y_i - \bar{y}_i$ . Where $e_i$ is residual, $y_i$ is observed, and $\bar{y}_i$ is predicted value.
What graphical tool is used to check linearity assumptions?

Residual plot. Primary diagnostic tool for evaluating whether linear model assumptions hold.
What is a common issue if residuals increase with fitted values?

Heteroscedasticity. Increasing variance violates constant variance assumption of linear regression.
What is the purpose of a normal probability plot of residuals?

To assess normality of residuals. Checks if errors follow normal distribution as required by assumptions.
What does a random pattern in a residual plot indicate?

Random pattern indicates a good fit. No systematic patterns suggest linear model assumptions are met.
What does a residual plot display?

Residual plot displays residuals vs. fitted values. Shows errors on y-axis and predicted values on x-axis to check assumptions.
What does a residual of zero indicate?

Observed value equals predicted value. Perfect prediction with no error between observed and fitted values.

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