Analyzing Departures from Linearity - AP Statistics
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Identify the formula for calculating a residual.
Identify the formula for calculating a residual.
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$e_i = y_i - \bar{y}_i$. Where $e_i$ is residual, $y_i$ is observed, and $\bar{y}_i$ is predicted value.
$e_i = y_i - \bar{y}_i$. Where $e_i$ is residual, $y_i$ is observed, and $\bar{y}_i$ is predicted value.
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What does a residual plot display?
What does a residual plot display?
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Residual plot displays residuals vs. fitted values. Shows errors on y-axis and predicted values on x-axis to check assumptions.
Residual plot displays residuals vs. fitted values. Shows errors on y-axis and predicted values on x-axis to check assumptions.
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What does a random pattern in a residual plot indicate?
What does a random pattern in a residual plot indicate?
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Random pattern indicates a good fit. No systematic patterns suggest linear model assumptions are met.
Random pattern indicates a good fit. No systematic patterns suggest linear model assumptions are met.
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What does a curved pattern in a residual plot suggest?
What does a curved pattern in a residual plot suggest?
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Curved pattern suggests non-linearity. Indicates the relationship is not linear and needs a different model form.
Curved pattern suggests non-linearity. Indicates the relationship is not linear and needs a different model form.
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Which transformation can stabilize variance in a dataset?
Which transformation can stabilize variance in a dataset?
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Logarithmic transformation. Reduces variability when variance increases with the mean.
Logarithmic transformation. Reduces variability when variance increases with the mean.
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Identify the role of transformations in linear regression.
Identify the role of transformations in linear regression.
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To linearize relationships and stabilize variance. Creates linear relationships and meets regression assumptions.
To linearize relationships and stabilize variance. Creates linear relationships and meets regression assumptions.
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What does a residual plot with increasing spread indicate?
What does a residual plot with increasing spread indicate?
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Non-constant variance. Spread increases with fitted values, violating homoscedasticity assumption.
Non-constant variance. Spread increases with fitted values, violating homoscedasticity assumption.
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What is the impact of non-linearity on predictions?
What is the impact of non-linearity on predictions?
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Leads to inaccurate predictions. Linear model cannot capture true relationship, causing prediction errors.
Leads to inaccurate predictions. Linear model cannot capture true relationship, causing prediction errors.
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Identify a method for testing linearity in regression analysis.
Identify a method for testing linearity in regression analysis.
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Examine the residuals for patterns. Look for systematic patterns that indicate assumption violations.
Examine the residuals for patterns. Look for systematic patterns that indicate assumption violations.
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What does heteroscedasticity in a residual plot indicate?
What does heteroscedasticity in a residual plot indicate?
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Indicates non-constant variance. Error variance changes across fitted values, violating equal variance assumption.
Indicates non-constant variance. Error variance changes across fitted values, violating equal variance assumption.
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How do you identify an influential point in regression?
How do you identify an influential point in regression?
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Influential points significantly affect the slope. Points with high leverage and large residuals change regression coefficients substantially.
Influential points significantly affect the slope. Points with high leverage and large residuals change regression coefficients substantially.
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What graphical tool is used to check linearity assumptions?
What graphical tool is used to check linearity assumptions?
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Residual plot. Primary diagnostic tool for evaluating whether linear model assumptions hold.
Residual plot. Primary diagnostic tool for evaluating whether linear model assumptions hold.
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Which assumption is violated if residuals show a funnel shape?
Which assumption is violated if residuals show a funnel shape?
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Homoscedasticity is violated. Funnel shape indicates variance increases with fitted values.
Homoscedasticity is violated. Funnel shape indicates variance increases with fitted values.
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What transformation is used for positive skewness?
What transformation is used for positive skewness?
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Log transformation. Compresses large values more than small ones, reducing right tail.
Log transformation. Compresses large values more than small ones, reducing right tail.
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Which pattern in residuals suggests a transformation is needed?
Which pattern in residuals suggests a transformation is needed?
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Non-random pattern. Systematic structure indicates linear model is inadequate.
Non-random pattern. Systematic structure indicates linear model is inadequate.
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What is the effect of non-linearity on model accuracy?
What is the effect of non-linearity on model accuracy?
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Reduces model accuracy. Linear model cannot capture true relationship, leading to systematic errors.
Reduces model accuracy. Linear model cannot capture true relationship, leading to systematic errors.
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Identify a common method to address non-linearity.
Identify a common method to address non-linearity.
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Using polynomial regression. Adds curved terms to capture non-linear relationships in data.
Using polynomial regression. Adds curved terms to capture non-linear relationships in data.
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What does a pattern in residuals indicate?
What does a pattern in residuals indicate?
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Indicates a model misfit. Systematic patterns suggest model assumptions are violated.
Indicates a model misfit. Systematic patterns suggest model assumptions are violated.
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Which plot can help identify non-linearity in data?
Which plot can help identify non-linearity in data?
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Residual plot. Curved or systematic patterns in residuals reveal non-linear relationships.
Residual plot. Curved or systematic patterns in residuals reveal non-linear relationships.
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What does a linear pattern in residuals suggest about the model?
What does a linear pattern in residuals suggest about the model?
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Model is not appropriate. Linear trend in residuals indicates model structure is wrong.
Model is not appropriate. Linear trend in residuals indicates model structure is wrong.
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What does a residual of zero indicate?
What does a residual of zero indicate?
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Observed value equals predicted value. Perfect prediction with no error between observed and fitted values.
Observed value equals predicted value. Perfect prediction with no error between observed and fitted values.
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What is a sign of a good regression model in a residual plot?
What is a sign of a good regression model in a residual plot?
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No discernible pattern in residuals. Random scatter indicates model captures the relationship appropriately.
No discernible pattern in residuals. Random scatter indicates model captures the relationship appropriately.
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Which transformation can address right skewness?
Which transformation can address right skewness?
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Square root transformation. Reduces the effect of large values in right-skewed distributions.
Square root transformation. Reduces the effect of large values in right-skewed distributions.
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Identify the result of high leverage points in regression.
Identify the result of high leverage points in regression.
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Can disproportionately influence the model. Points far from center of predictors can heavily influence fitted line.
Can disproportionately influence the model. Points far from center of predictors can heavily influence fitted line.
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What is the purpose of a normal probability plot of residuals?
What is the purpose of a normal probability plot of residuals?
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To assess normality of residuals. Checks if errors follow normal distribution as required by assumptions.
To assess normality of residuals. Checks if errors follow normal distribution as required by assumptions.
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What does a residual plot with no apparent pattern indicate?
What does a residual plot with no apparent pattern indicate?
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Assumptions likely satisfied. Random scatter confirms linear model is appropriate for the data.
Assumptions likely satisfied. Random scatter confirms linear model is appropriate for the data.
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What is examined to ensure the linear model is appropriate?
What is examined to ensure the linear model is appropriate?
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Residual plot. Checks whether linear relationship assumption is reasonable for data.
Residual plot. Checks whether linear relationship assumption is reasonable for data.
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What is the role of residual analysis in regression?
What is the role of residual analysis in regression?
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To check model assumptions. Evaluates whether model meets required assumptions for valid inference.
To check model assumptions. Evaluates whether model meets required assumptions for valid inference.
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Which pattern in residuals suggests the need for a polynomial model?
Which pattern in residuals suggests the need for a polynomial model?
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Curved pattern. U-shaped pattern indicates quadratic relationship exists in data.
Curved pattern. U-shaped pattern indicates quadratic relationship exists in data.
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What method can address the issue of non-linearity?
What method can address the issue of non-linearity?
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Transformation of variables. Mathematical functions applied to create linear relationships from non-linear data.
Transformation of variables. Mathematical functions applied to create linear relationships from non-linear data.
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