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, R2R^2, 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}\{2, 4, 7, 10\}, predictions {3,5,6,9}\{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}\{10, 12, 15\}, predictions {11,11,16}\{11, 11, 16\}. Compute the mean absolute residual.

Example 4

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

Example 5

medium
R2=0.62R^2 = 0.62 and SSR =38= 38 on data with total sum of squares (SST) =100= 100. Verify R2R^2.

Example 6

hard
Data (1,3),(2,5),(3,7)(1, 3), (2, 5), (3, 7) with model y^=2x+1\hat{y} = 2x + 1. Compute SSR and R2R^2 (mean of yy is 55, SST is 88).

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 R2R^2 be negative when comparing a model against the mean baseline?

Example 10

challenge
You have n=20n=20 points and fit a polynomial of degree 1919. Predict the training R2R^2 and explain why test R2R^2 will be catastrophic.

Example 11

medium
Model A: train R2=0.85R^2=0.85, test R2=0.80R^2=0.80. Model B: train R2=0.99R^2=0.99, test R2=0.50R^2=0.50. Which generalizes better?

Example 12

challenge
Two models tie on training R2R^2 at 0.900.90. Model A uses 33 predictors; Model B uses 1010. Which would you pick and why?

Example 13

medium
Model A has R2=0.95R^2=0.95 with patternless residuals; Model B has R2=0.95R^2=0.95 with strongly patterned residuals. Which fits better?

Example 14

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

Example 15

challenge
Data (1,2),(2,4),(3,6)(1,2),(2,4),(3,6) with model y^=2x\hat{y}=2x. Compute SSR, then R2R^2 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=0.65R^2 = 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=0.30R^2 = 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=0.80R^2 = 0.80. What does this mean?
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