Practice Prediction in Math

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

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Quick Recap

A prediction is a model-based estimate of an unknown or future value, accompanied by a measure of confidence or uncertainty.

Every prediction uses patterns from the past to extrapolate forward — good predictions come with explicit uncertainty bounds, not false precision.

Showing a random 20 of 50 problems.

Example 1

medium
A regression gives prediction y^=20\hat{y}=20 with standard error 44. Give an approximate 95% prediction interval (use ±2\pm 2 SE).

Example 2

easy
A model trained on data from 2010-2020 is used to forecast 2050. This is an example of ___.

Example 3

hard
A regression on years 2000-2020 gives global average temperature y^=0.02x+14\hat{y}=0.02x+14. Predict the temperature in year 2100 and discuss two reasons to distrust the result.

Example 4

medium
A model predicts y^=42\hat{y}=42 with 80% interval [38,46][38,46]. Customer requires 90% certainty. Will the 90% interval be narrower or wider, and why?

Example 5

easy
A linear model predicts y=3x+2y=3x+2. What is the predicted yy when x=4x=4?

Example 6

easy
Using the model y^=3x−5\hat{y} = 3x - 5, predict yy when x=4x = 4 and x=0x = 0. Then find xx when y^=25\hat{y} = 25.

Example 7

easy
For the linear model y^=−2x+30\hat{y}=-2x+30, predict yy at x=4x=4.

Example 8

medium
A model predicts ice cream sales from temperature with high accuracy. Can we conclude temperature causes the sales pattern? Why or why not?

Example 9

hard
You have two unbiased models for the same quantity, with variances 99 and 44. Combining them with weights inversely proportional to variance gives optimal weights w1=4/13w_1=4/13 and w2=9/13w_2=9/13. Find the combined variance.

Example 10

medium
A trend grew linearly for 5 years. A naive model predicts the same growth for 50 more years. What is the main danger?

Example 11

hard
A model predicts sales: y^=15x+200\hat{y}=15x+200. Actual values for x=10,20,30x=10,20,30 are 360,510,660360, 510, 660. Compute the predictions, residuals, and discuss whether the model fits well.

Example 12

hard
Why is extrapolation (predicting outside the observed range) dangerous? Give an example where extrapolating from a linear model would give a clearly unreasonable prediction.

Example 13

medium
A model predicts test scores with a residual standard deviation of 66. About 95% of actual scores should fall within how many points of the predicted value?

Example 14

medium
The same line y=2x+5y=2x+5 is used to predict yy at x=50x=50. Compute it and flag the risk.

Example 15

easy
A prediction interval that is very wide tells us the model has ___ confidence.

Example 16

medium
In a weather model: P(rain)=0.7P(\text{rain})=0.7 today, but it does not rain. Does this prove the model wrong?

Example 17

medium
A line y=2x+5y=2x+5 fits data for x∈[0,10]x\in[0,10]. Predict yy at x=8x=8 and state whether it is interpolation or extrapolation.

Example 18

hard
A model predicts house prices. In-sample R2=0.92R^2 = 0.92, but out-of-sample R2=0.45R^2 = 0.45. Explain what this means and identify the problem with the model.

Example 19

medium
A model achieves R2=0.99R^2=0.99 on the data it was fit on but R2=0.20R^2=0.20 on new data. The most likely cause is ___.

Example 20

challenge
A nonlinear model fit on x∈[0,10]x\in[0,10] gives y^=x2\hat{y}=x^2. The relationship beyond x=10x=10 is actually y^=10x\hat{y}=10x (linear). At x=20x=20 the model predicts 400400, but reality is 200200. Compute the relative error.
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