Evaluate Model Fit with Residuals - Statistics and Probability

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

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

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

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

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

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

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

   Model A. Smaller residuals indicate better fit.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Model 1. Random pattern beats systematic pattern.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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