Statistics Flashcards: Evaluate Model Fit With Residuals

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This deck focuses on Evaluate Model Fit With Residuals, giving you a quick way to review the definitions, rules, and examples that matter most for Statistics.

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Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the residual if $y=7$ and $\hat{y}=10$.

Answer: $-3$. Calculate $7-10=-3$ since residual equals observed minus predicted.

Flashcard 2: Identify the observed value $y$ if the residual is $-2$ and $\hat{y}=11$.

Answer: $9$. Since residual $=-2$ and $\hat{y}=11$, then $y=11+(-2)=9$.

Flashcard 3: What is the reference line drawn on a residual plot to judge model fit?

Answer: The horizontal line $y=0$. Zero line helps identify if residuals are balanced above and below.

Flashcard 4: Identify the better fit if Model A residuals are mostly within $\pm 1$ and Model B within $\pm 5$.

Answer: Model A. Smaller residuals indicate better fit.

Flashcard 5: Which residual plot pattern most supports that a linear model is appropriate?

Answer: Random scatter around $0$ with no pattern. Good linear fit shows residuals randomly distributed above and below zero.

Flashcard 6: Identify the issue if residuals fan in (spread decreases) as $x$ increases.

Answer: Non-constant variance (heteroscedasticity). Decreasing spread violates constant variance assumption.

Flashcard 7: What is the residual for $y=18$ when the model predicts $\hat{y}=20$?

Answer: $-2$. Calculate $18-20=-2$.

Flashcard 8: What is the residual for a data point with observed $y$ and predicted \hat{y}?

Answer: $y-\hat{y}$. Residual measures the vertical distance from observed to predicted value.

Flashcard 9: Which option indicates the model tends to overpredict: residuals mostly $>0$ or mostly $<0$?

Answer: Mostly $<0$. Negative residuals mean predictions exceed observed values.

Flashcard 10: Identify the model issue if residual spread increases as $x$ increases (a widening band).

Answer: Nonconstant variance (heteroscedasticity). Variance should be constant; increasing spread violates this.

Flashcard 11: Identify the model issue if residuals are positive for small $x$ and negative for large $x$ in a smooth trend.

Answer: Nonlinear pattern; the model form is wrong. Systematic sign changes suggest a linear model misses curvature.

Flashcard 12: What is the predicted value $\hat{y}$ if $y=25$ and the residual is $-3$?

Answer: $28$. Since $y-\hat{y}=-3$, then $\hat{y}=y+3=25+3=28$.

Flashcard 13: Which option best indicates a good fit: residuals centered at $0$ or centered far from $0$?

Answer: Centered at $0$. Residuals balanced around zero indicate unbiased predictions.

Flashcard 14: What is plotted on the vertical axis of a residual plot for predicting $y$ from $x$?

Answer: Residuals $y-\hat{y}$. Vertical axis shows differences between observed and predicted values.

Flashcard 15: Identify the better fit if residuals for Model 1 show random scatter and Model 2 show a U-shape.

Answer: Model 1. Random pattern beats systematic pattern.

Flashcard 16: What does a residual plot with many points exactly on $0$ indicate about predictions?

Answer: Many data points are predicted exactly by the model. Zero residuals mean perfect predictions for those points.

Flashcard 17: What does a residual plot with points clustered far from $0$ suggest about the model?

Answer: Large typical errors; the fit is poor. Large residuals indicate poor predictions overall.

Flashcard 18: What is typically plotted on the horizontal axis of a residual plot for a model $\hat{y}=f(x)$?

Answer: The explanatory variable $x$. Horizontal axis shows the predictor variable to check patterns across $x$.

Flashcard 19: Which residual plot feature indicates possible outliers in the $y$-direction?

Answer: One or more unusually large $|y-\hat{y}|$ values. Points far from zero line have large prediction errors.

Flashcard 20: What does a positive residual $y-\hat{y}>0$ mean about the model prediction?

Answer: The model underpredicts the actual $y$ value. Positive residual means observed exceeds predicted.

Flashcard 21: Choose the correct residual: observed $y=6$, predicted $\hat{y}=10$; is it $4$ or $-4$?

Answer: $-4$. Calculate $6-10=-4$.

Flashcard 22: Which option indicates the model tends to underpredict: residuals mostly $>0$ or mostly $<0$?

Answer: Mostly $>0$. Positive residuals mean observed values exceed predictions.

Flashcard 23: Identify the issue if residuals fan out (spread increases) as $x$ increases.

Answer: Non-constant variance (heteroscedasticity). Increasing spread violates constant variance assumption.

Flashcard 24: What is the residual for a data point with observed $y$ and predicted value $\hat{y}$?

Answer: $y-\hat{y}$. Residual measures the vertical distance from observed to predicted.

Flashcard 25: Identify the predicted value $\hat{y}$ if $y=15$ and the residual is $4$.

Answer: $11$. Since $y=15$ and residual $=4$, then $\hat{y}=15-4=11$.

Flashcard 26: Identify the residual if $y=12$ and $\hat{y}=9$.

Answer: $3$. Calculate $12-9=3$ since residual equals observed minus predicted.

Flashcard 27: Choose the correct residual: observed $y=10$, predicted $\hat{y}=6$; is it $4$ or $-4$?

Answer: $4$. Calculate $10-6=4$.

Flashcard 28: Identify the conclusion if residuals show a curved pattern around $0$ as $x$ increases.

Answer: The model form is wrong; a nonlinear model may fit better. Curved residual pattern suggests linear model is inappropriate.

Flashcard 29: Which residual plot pattern suggests nonlinearity (a curved relationship) in the data?

Answer: A systematic curve (U-shape or S-shape). Curved patterns in residuals indicate the relationship isn't linear.

Flashcard 30: Which residual plot pattern suggests nonconstant variance (heteroscedasticity)?

Answer: A fan or funnel shape in residual spread. Changing spread violates the constant variance assumption.

Flashcard 31: What is the observed value $y$ if $\hat{y}=50$ and the residual is $6$?

Answer: $56$. Since $y-\hat{y}=6$, then $y=\hat{y}+6=50+6=56$.

Flashcard 32: What does a negative residual $y-\hat{y}<0$ mean about the model prediction?

Answer: The model overpredicts the actual $y$ value. Negative residual means predicted exceeds observed.

Flashcard 33: What is the residual for $y=42$ when the model predicts $\hat{y}=39$?

Answer: $3$. Calculate $42-39=3$.

Flashcard 34: Identify whether the model overpredicts or underpredicts when the residual is $+7$.

Answer: Underpredicts. Positive residual means $y>\hat{y}$.

Flashcard 35: Which option is the correct interpretation: small $|y-\hat{y}|$ or large $|y-\hat{y}|$ indicates better fit?

Answer: Small $|y-\hat{y}|$ indicates better fit. Smaller absolute residuals mean predictions are closer to observed.

Flashcard 36: Which residual-plot pattern indicates a good fit: random scatter around $0$ or a clear curve?

Answer: Random scatter around $0$. No pattern in residuals indicates the model fits well.

Flashcard 37: Identify whether the model overpredicts or underpredicts when the residual is $-5$.

Answer: Overpredicts. Negative residual means $\hat{y}>y$.

Flashcard 38: Which plot is used to assess model fit by graphing residuals versus the explanatory variable $x$?

Answer: A residual plot (residuals vs. $x$). Plots residuals on y-axis against x-values to reveal patterns.