Prediction Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Prediction.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept 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.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: A prediction is an estimate of an unknown value that always carries how sure you are about it.
Common stuck point: The procedure for prediction is the easy part; the trap is reporting a prediction as an exact number with no range. Asking "Am I stating a value I have not observed, with a sense of how uncertain it is?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint: Ask: Am I stating a value I have not observed, with a sense of how uncertain it is?
Worked Examples
Example 1
mediumA linear regression model gives where = hours studied and = test score. Predict the score for a student who studies 8 hours, and explain why this is a prediction, not a guarantee.
Answer
for hours. This is the average predicted score, not an individual guarantee.
First step
1
Substitute : ... wait, let's check context. More realistic: (assuming intercept 50 for a score scale)
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Example 2
hardA model predicts house prices. In-sample , but out-of-sample . Explain what this means and identify the problem with the model.
Example 3
mediumA regression was fit on . (a) Predict at . (b) State whether and are interpolation or extrapolation.
Example 4
mediumA model gives prediction with standard error . Give a rough 95% prediction interval using the SE rule.
Example 5
mediumA linear model fit to (1,3), (2,5), (3,7) gives . Predict at and identify whether the prediction is interpolation or extrapolation.
Example 6
hardYou average two independent unbiased predictions with variances and . What is the variance of the averaged prediction, and what does the reduction tell you about combining models?
Example 7
hardA model predicts sales: . Actual values for are . Compute the predictions, residuals, and discuss whether the model fits well.
Example 8
challengeA new model and an old model both make 100 predictions. New model: mean absolute error (MAE) , 95% interval coverage . Old model: MAE , coverage . Which is preferred and why?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyUsing the model , predict when and . Then find when .
Example 2
hardWhy is extrapolation (predicting outside the observed range) dangerous? Give an example where extrapolating from a linear model would give a clearly unreasonable prediction.
Example 3
easyA good prediction reports an estimate together with a measure of its ___.
Example 4
easyA model fit on ages 10-18 is used to predict height at age 50. What error is this?
Example 5
easyA linear model predicts . What is the predicted when ?
Example 6
easyA model predicts sales of '500 units, 95% interval [450, 550].' What does the interval represent?
Example 7
easyA model predicts well but cannot explain why X causes Y. Prediction does not require ___.
Example 8
easyEvery prediction uses patterns from the ___ to estimate the future.
Example 9
easyWhich is a more honest prediction: 'exactly 1,000 sales' or '1,000 sales, give or take 100'?
Example 10
easyA weather model outputs '80% chance of rain.' Is this a prediction with uncertainty?
Example 11
mediumA line fits data for . Predict at and state whether it is interpolation or extrapolation.
Example 12
mediumThe same line is used to predict at . Compute it and flag the risk.
Example 13
mediumTwo predictions for the same value: A says , B says . If both are unbiased, which is more useful and why?
Example 14
mediumA model predicts ice cream sales from temperature with high accuracy. Can we conclude temperature causes the sales pattern? Why or why not?
Example 15
mediumA trend grew linearly for 5 years. A naive model predicts the same growth for 50 more years. What is the main danger?
Example 16
mediumA regression gives prediction with standard error . Give an approximate 95% prediction interval (use SE).
Example 17
mediumWhy does prediction uncertainty typically grow as you predict further from the center of the data?
Example 18
mediumA model predicts house price where is square footage. Predict the price for a 1200 sq ft house.
Example 19
mediumA prediction says '20 units expected, 90% interval [12, 28].' If actual sales are 26, was the outcome within the predicted uncertainty?
Example 20
challengeA model was trained on . Predict at and ; state which prediction you trust and why.
Example 21
challengeTwo unbiased models predict the same target: A has prediction variance 9, B has variance 16. If you average them (equal weight, independent), what is the variance of the averaged prediction?
Example 22
challengeA linear model fit on years 1-5 (sales 10, 20, 30, 40, 50) is . The real year-10 sales were 70, not 100. What does the gap reveal about extrapolation?
Example 23
easyA model is trained on . Predict when .
Example 24
easyA weather app says 'High 72°F, give or take 3°F.' What is the predicted high, and what is the uncertainty?
Example 25
easyThe model predicts test score from hours studied. Predict the score for 10 hours.
Example 26
easyFor the linear model , predict at .
Example 27
mediumA model predicts daily revenue: , where is ads shown (thousands). Predict revenue when .
Example 28
mediumA regression on plant height vs days gives cm for days. Predict the height at and at (one year). Which prediction would you trust?
Example 29
mediumTwo unbiased predictions: A reports , B reports . Which is more useful for planning, and why?
Example 30
mediumA model trained on weekday traffic predicts weekend traffic and is wildly wrong. What kind of failure is this?
Example 31
mediumA model predicts with 80% interval . Customer requires 90% certainty. Will the 90% interval be narrower or wider, and why?
Example 32
mediumA model predicts test scores with a residual standard deviation of . About 95% of actual scores should fall within how many points of the predicted value?
Example 33
mediumIn a weather model: today, but it does not rain. Does this prove the model wrong?
Example 34
hardA regression on years 2000-2020 gives global average temperature . Predict the temperature in year 2100 and discuss two reasons to distrust the result.
Example 35
hardA doctor's model predicts a patient has an 85% probability of recovery. The patient does not recover. Does this single case mean the model is broken?
Example 36
hardYou have two unbiased models for the same quantity, with variances and . Combining them with weights inversely proportional to variance gives optimal weights and . Find the combined variance.
Example 37
hardA model says with 95% interval . The next observation is . Is this evidence the model is wrong?
Example 38
challengeA nonlinear model fit on gives . The relationship beyond is actually (linear). At the model predicts , but reality is . Compute the relative error.
Related Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.
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