Practice Model Fit (Intuition) in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Model fit describes how closely a statistical model's predictions match the observed data — measured by residuals, R2, or loss functions.

Does the model's predictions match reality? Good fit = close match.

Showing a random 20 of 50 problems.

Example 1

medium
Observations {2,4,7,10}, predictions {3,5,6,9}. Compute the residual sum of squares (SSR) and mean absolute residual (MAR).

Example 2

medium
Residual plot for a linear model shows a clear U-shaped pattern. What does this indicate about the model, and what should be done?

Example 3

medium
Observed values {10,12,15}, predictions {11,11,16}. Compute the mean absolute residual.

Example 4

hard
Data (1,2),(2,5),(3,7) with model y^=2.5x. Compute SSR.

Example 5

medium
R2=0.62 and SSR =38 on data with total sum of squares (SST) =100. Verify R2.

Example 6

hard
Data (1,3),(2,5),(3,7) with model y^=2x+1. Compute SSR and R2 (mean of y is 5, SST is 8).

Example 7

medium
A residual plot shows a clear U-shape (curve). What does this tell you about a linear model's fit?

Example 8

easy
A model fits the training data closely but predicts new data poorly. Good fit or misleading fit?

Example 9

hard
Why can R2 be negative when comparing a model against the mean baseline?

Example 10

challenge
You have n=20 points and fit a polynomial of degree 19. Predict the training R2 and explain why test R2 will be catastrophic.

Example 11

medium
Model A: train R2=0.85, test R2=0.80. Model B: train R2=0.99, test R2=0.50. Which generalizes better?

Example 12

challenge
Two models tie on training R2 at 0.90. Model A uses 3 predictors; Model B uses 10. Which would you pick and why?

Example 13

medium
Model A has R2=0.95 with patternless residuals; Model B has R2=0.95 with strongly patterned residuals. Which fits better?

Example 14

challenge
With data points (x,y): (1,3),(2,5) and model y^=ax+b, find a,b giving a perfect fit, and state the SSR.

Example 15

challenge
Data (1,2),(2,4),(3,6) with model y^=2x. Compute SSR, then R2 given total sum of squares (about the mean 4) is 8.

Example 16

medium
Two models: simple line with test error 5, complex curve with test error 8. On generalization, which fits better?

Example 17

easy
A scatter plot of weight vs. height shows points loosely scattered around a line. Two measures of fit are given: R2=0.65 and residuals with SD = 8 kg. Interpret both measures.

Example 18

medium
A scatter plot shows points clustered tightly along a curve, but the linear model's R2=0.30. What does this say?

Example 19

easy
Sum of squared residuals for a model is 0. What does that say about the in-sample fit?

Example 20

easy
A model has R2=0.80. What does this mean?
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