Distinguishing Correlation and Causation - Statistics and Probability

1. Which feature is essential for a causal claim: random assignment or a large sample size?

   Random assignment. Random assignment eliminates confounding, unlike sample size alone.

2. Which feature of a randomized experiment helps eliminate confounding: random assignment or self-selection?

   Random assignment. Randomization eliminates systematic differences between groups.

3. Which option usually supports only correlation: observational study or randomized experiment?

   Observational study. Without random assignment, confounders may explain observed relationships.

4. What is the definition of correlation between two variables?

   A statistical association showing how two variables vary together. Measures how variables change together, not why they change.

5. What does it mean if two variables have a strong positive correlation?

   As one variable increases, the other tends to increase. They move in the same direction together.

6. Identify the term for correlation caused by a third variable rather than a direct link.

   Spurious correlation. The correlation exists but isn't meaningful or causal.

7. Identify whether this is correlation or causation: "Students who sleep more score higher." (observational)

   Correlation only. Observational data can't establish if sleep causes higher scores.

8. What is the lurking variable problem when interpreting a correlation?

   An unmeasured variable may be causing the observed association. Hidden third variables can create apparent relationships.

9. Identify the study type: researchers record exercise and blood pressure without assigning exercise levels.

   Observational study. No manipulation of variables means it's observational.

10. Identify the likely issue: ice cream sales and drowning deaths increase together in summer.

    Confounding by season/temperature. Both variables are influenced by hot weather, not each other.

11. Which design change best supports causation: random assignment to treatments or increasing sample size only?

    Random assignment to treatments. Random assignment eliminates confounding variables.

12. Which statement is valid: "$x$ causes $y$" or "$x$ is associated with $y$" for observational data?

    "$x$ is associated with $y$". Observational data supports association claims, not causal claims.

13. Which statement is always true: correlation implies causation, or correlation does not imply causation?

    Correlation does not imply causation. Variables can be related without one causing the other.

14. Choose the correct interpretation: In a randomized experiment, treated group mean $>$ control mean.

    The treatment likely caused an increase in the response. Randomization allows causal interpretation of group differences.

15. Identify the correct interpretation: a scatterplot shows a curved pattern but $r\,\approx\,0$.

    There may be a non-linear association despite near-zero $r$. $r$ only detects linear patterns, not curves.

16. What is the purpose of a control group in a randomized experiment?

    To provide a baseline for comparison against the treatment group. Shows what happens without the treatment.

17. What does a correlation of $r=0$ indicate about a linear relationship?

    No linear association is present. The correlation coefficient measures only linear relationships.

18. Find and correct the claim: "Because $r=-0.6$, increasing $x$ will cause $y$ to decrease."

    Correct: $r=-0.6$ shows association, not causation. Correlation coefficient describes association strength, not causal effect.

19. What is the definition of causation in a statistical study?

    A change in one variable directly produces a change in another variable. Causation requires one variable to be the reason for changes in another.

20. Which option indicates causation: observational study or randomized experiment?

    Randomized experiment. Random assignment controls confounders, enabling causal conclusions.

21. What does the phrase "correlation does not imply causation" mean?

    An observed association alone is insufficient to conclude cause and effect. Other factors may explain the relationship between correlated variables.

22. What does it mean if two variables have a strong negative correlation?

    As one variable increases, the other tends to decrease. They move in opposite directions.

23. Identify whether this is correlation or causation: "Randomly assigned tutoring increased scores."

    Causation supported. Random assignment allows causal inference about tutoring's effect.

24. Which option best indicates a causal claim: "is associated with" or "causes"?

    "Causes". Direct causal language indicates cause-effect claims.

25. What is reverse causation in an observed association?

    The supposed effect actually causes the supposed cause. The direction of causation is backwards from what's assumed.

26. Which design best supports causation: match subjects only, or randomize subjects to treatments?

    Randomize subjects to treatments. Randomization eliminates confounding; matching alone doesn't.

27. What is the direction problem when interpreting a correlation?

    It is unclear which variable influences the other, if either does. With correlation alone, we can't determine if A causes B or B causes A.

28. Identify the likely issue: people with higher stress report less sleep; a headline says stress is caused by low sleep.

    Reverse causation is possible. Sleep loss might cause stress, not vice versa.

29. Identify whether reverse causation is plausible: More firefighters at fires correlates with more damage.

    Yes; severity can cause both more firefighters and more damage. Fire severity could cause both more responders and more damage.

30. What is the definition of causation between two variables?

    A relationship where changes in one variable directly produce changes in the other. One variable must be the reason for changes in the other.

31. What is the definition of correlation in a statistical study?

    Association between variables; not necessarily a cause-and-effect link. Correlation measures relationship strength, not whether one causes the other.

32. Identify the term: a correlation caused by chance, bias, or confounding rather than a real link.

    Spurious correlation. The association exists but lacks a genuine causal mechanism.

33. Which option best reduces confounding in a study: random assignment or voluntary response?

    Random assignment. Randomly assigning treatments balances known and unknown confounders.

34. Identify the study type: researchers assign a new drug or placebo by chance and compare outcomes.

    Randomized experiment. Random assignment of treatments defines an experiment.

35. What is a confounding variable in interpreting an association?

    A third variable related to both variables that can create a misleading association. It affects both variables, creating false associations.

36. Identify the confounder type: Ice cream sales and drownings rise together due to hot weather.

    Confounding (lurking) variable: temperature. Temperature affects both variables, creating spurious correlation.

37. Which study type can directly support a cause-and-effect conclusion: observational study or randomized experiment?

    Randomized experiment. Only experiments with random assignment can prove causation.

38. Choose the correct conclusion: an observational study finds $r=0.80$ between $X$ and $Y$.

    Strong association, but causation is not established. Observational studies can't prove causation regardless of $r$.

39. Identify the best conclusion: A study finds $r=0.80$ between $x$ and $y$ in observational data.

    Strong correlation; causation cannot be concluded. High $r$ shows strong association, but observational data can't prove causation.

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