Practice Causation in Math

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

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Quick Recap

Causation exists when one variable directly produces or influences a change in another variable — distinct from mere correlation or association.

XX causes YY means changing XX will change YY. Not just 'they move together.'

Showing a random 20 of 50 problems.

Example 1

easy
A study shows people who exercise are happier. Could the causal arrow go the other way?

Example 2

easy
Causation means changing X will change Y. If forcing X higher leaves Y unchanged, is X a cause of Y?

Example 3

medium
Cite-and-explain Bradford Hill criterion: dose-response. Why does it strengthen causal claims?

Example 4

easy
Shoe size and reading ability are strongly correlated in elementary students. What is the obvious confounder?

Example 5

medium
Sunscreen use correlates with higher skin-cancer rates in some data. Give a confounder that explains this without sunscreen causing cancer.

Example 6

easy
Cities with more firefighters have more fire damage. Does adding firefighters cause more damage?

Example 7

easy
Ice cream sales and drownings both rise in summer. What is the likely confounder making them correlate?

Example 8

medium
Why is 'X precedes Y in time' necessary but not sufficient for 'X causes Y'?

Example 9

medium
An observational study shows coffee drinkers have higher heart-disease rates, but coffee drinkers also smoke more. How does smoking threaten the causal claim?

Example 10

hard
A pharmaceutical company's observational study finds Drug X correlates with recovery. Explain three criteria needed to claim Drug X causes recovery, and why a randomized trial is required.

Example 11

easy
Countries with more TV sets per capita have higher life expectancy. Does this mean buying TVs causes longer lives? Identify the likely confounding variable.

Example 12

easy
Children who attend preschool tend to do better in college. Why is this not direct evidence of causation?

Example 13

challenge
Critique the headline: 'Eating chocolate makes you smarter — Nobel laureates eat more chocolate per capita.'

Example 14

easy
Roosters crow before sunrise every day. Do roosters cause the sun to rise?

Example 15

medium
A city's parks correlate with high property values. Explain confounding and reverse-causation candidates.

Example 16

easy
Name three alternative explanations to 'X causes Y' for an observed correlation.

Example 17

easy
To prove causation in a study, which design rules out confounders best?

Example 18

challenge
A headline claims 'students who own more books score higher, so buy your child books.' Critique the causal leap and propose how to test causation properly.

Example 19

challenge
Berkson's paradox: a hospital-based study finds two diseases negatively correlated, despite being unrelated in the population. Explain.

Example 20

hard
In a DAG X→YX \to Y, Z→XZ\to X, Z→YZ\to Y, can we estimate XX's causal effect on YY from observational data?
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