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Correlation and causation

Learning goals

  • Distinguishes association from causation.
  • Identifies plausible confounding or alternative explanations.
  • Explains what study design would be needed for a causal claim.
  • Evaluates causal claims against the available evidence.

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Correlation and causation 1
A report says, “Students who use a weekly planner tend to have higher course grades.” A headline rewrites this as, “Using a weekly planner raises students' course grades.” Which sentence states an association, and which sentence makes a causal claim?

Hints

- Look for wording that only says two variables occur together. - Look for wording that says one variable produces a change in the other. - Focus on the difference between “tend to have” and “raises.”

Solution

1. “Students who use a weekly planner tend to have higher course grades” describes two variables occurring together, so it states an association. 2. “Using a weekly planner raises students' course grades” says that one variable produces a change in the other, so it makes a causal claim.

Answer

The report states an association. The headline makes a causal claim.
Correlation and causation 2
Two studies examine whether a new review routine improves quiz scores. Study A asks students whether they already use the routine and compares their quiz scores with scores of students who do not use it. Study B randomly assigns students to use either the new routine or the usual routine for the same amount of study time, then gives everyone the same quiz. Which study provides stronger evidence that the review routine causes a change in quiz scores? Give the key reason.

Hints

- Compare how students enter the groups in the two studies. - Identify which study assigns the review routine instead of merely observing it. - Causal evidence is stronger when the treatment is deliberately randomized.

Solution

1. Study A observes choices students have already made, so other differences between the groups could help explain a score difference. 2. Study B randomly assigns the review routine, so the treatment is deliberately varied rather than chosen by the students. 3. Therefore, Study B provides stronger evidence for a causal effect.

Answer

Study B. Random assignment provides stronger evidence that the review routine, rather than preexisting differences between the groups, caused a change in quiz scores.
Correlation and causation 3
Students who open their school's optional tutorial site more often also tend to finish more class projects. A principal says, “Using the site makes students finish more projects.” Give one other explanation for this pattern. What can the school conclude from the observation alone?

Hints

- Separate what the school measured from what someone might claim caused it. - Could the same students differ from other students in another way that matters here? - Think about whether the observation rules out other causes of completing projects.

Solution

1. Students who are already motivated may both open the tutorial site often and finish more projects. 2. The observation shows that site use and finished projects are associated. 3. Without better evidence, the school cannot tell whether visiting the site caused the difference.

Answer

Motivated students might be more likely to use the site and finish projects. The school can report an association, not proof that using the site caused more completed projects.
Correlation and causation 4
A store sells more sunglasses on days when it also sells more bottled water. Give one possible third factor that could increase both sales. Explain why these sales numbers do not prove that buying sunglasses causes someone to buy water.

Hints

- Think about conditions that might make both products more useful on the same day. - A third factor can change two quantities at once. - Check whether a direct cause between the two purchases is actually needed to explain the pattern.

Solution

1. Hot, sunny weather can increase demand for sunglasses and bottled water. 2. Because weather could affect both sales totals, the two totals can rise together without either purchase causing the other.

Answer

Hot, sunny weather is one possible common cause. It can increase demand for both products, so their association does not establish that one purchase causes the other.
Correlation and causation 5
Students who volunteered for an after-school basketball club made more free throws on the final skill test than students who did not join. A coach says the club caused the entire difference. Name one difference that might already have existed between the groups before the club started. Explain why it makes the coach's claim uncertain.

Hints

- How did students end up in the club group? - Might the volunteers have differed from the others before the first meeting? - How could that difference affect the final test?

Solution

1. Students who volunteered may already have practiced more often or been more interested in improving. 2. Such differences could affect their final free-throw scores too. 3. Because students chose their groups, the comparison alone cannot show how much of the difference the club caused.

Answer

Students who volunteered might already have been better players or practiced more. The final results do not show how much improvement the club caused.
Correlation and causation 6
Cities with more bookstores often have more people who read regularly. Someone claims that having more bookstores must be what makes people read more. Explain one way the cause could run in the opposite direction.

Hints

- Ask what bookstores need to stay open. - Consider whether reader demand might affect business decisions. - An association does not tell us which variable influences the other.

Solution

1. People who already read often may buy more books. 2. In cities with many readers, there may be enough customers to support more bookstores. 3. The number of readers could therefore help explain the number of bookstores, rather than only the other way around.

Answer

Cities with more regular readers may create greater demand for books, leading more bookstores to open. The association could partly run from readers to bookstores.
Correlation and causation 7
A school compared project completion with attendance at an optional review session. <table><tr><th>Review session</th><th>Completed project</th><th>Did not complete</th><th>Total</th></tr><tr><td>Attended</td><td>\(42\)</td><td>\(18\)</td><td>\(60\)</td></tr><tr><td>Did not attend</td><td>\(30\)</td><td>\(30\)</td><td>\(60\)</td></tr><tr><td>Total</td><td>\(72\)</td><td>\(48\)</td><td>\(120\)</td></tr></table> a) Find the conditional relative frequency of completing the project for each attendance group. b) Describe the association shown by the table. c) Does the table prove that attending the review session caused students to complete the project? Explain.

Hints

- Use each row total as the denominator for a within-group completion rate. - Compare the two percentages using percentage points. - Separate evidence of association from evidence produced by a randomized experiment.

Solution

1. Among students who attended, the completion rate is \(\frac{42}{60}=70\%\). 2. Among students who did not attend, the completion rate is \(\frac{30}{60}=50\%\). 3. The completion rate is \(20\) percentage points higher for students who attended, so the variables show a positive association in this group. 4. The table is observational. Other differences between the groups, such as prior preparation or motivation, could help explain the result, so causation is not established.

Answer

a) Attended: \(70\%\). Did not attend: \(50\%\). b) Project completion is associated with attending the review session; the completion rate is \(20\) percentage points higher among attendees. c) No. The observational table does not rule out confounding variables or self-selection.
Correlation and causation 8
Neighborhood fires attended by more firefighters tend to have greater property damage. Explain the likely causal structure behind this association.

Hints

- Ask what determines how many firefighters are sent. - Identify a variable connected to both response size and damage. - Consider the sequence of events.

Solution

1. More severe fires generally cause more property damage. 2. Those severe fires also lead emergency services to send more firefighters. 3. Fire severity is a common cause, and the likely causal direction is from severity to both staffing and damage, not from firefighters to damage.

Answer

Larger, more severe fires both require more firefighters and cause more damage. The correlation does not mean firefighters cause the damage.
Correlation and causation 9
Two regions report the following annual data. <table><tr><th>Region</th><th>Population</th><th>Hospitals</th><th>Reported illness cases</th></tr><tr><td>Lake</td><td>\(50{,}000\)</td><td>\(2\)</td><td>\(2500\)</td></tr><tr><td>Metro</td><td>\(300{,}000\)</td><td>\(8\)</td><td>\(12{,}000\)</td></tr></table> a) Which region has more hospitals and more reported illness cases? b) Find the reported illness rate in each region. c) Explain why the totals do not support the conclusion that hospitals cause illness.

Hints

- Compare the raw totals first. - Divide reported cases by population to compare per-person rates. - Distinguish an association between totals from evidence that one variable causes the other.

Solution

1. Metro has \(8\) hospitals and \(12{,}000\) reported cases, both larger totals than Lake’s \(2\) hospitals and \(2500\) cases. 2. Lake’s illness rate is \(\frac{2500}{50{,}000}=0.05=5\%\). 3. Metro’s illness rate is \(\frac{12{,}000}{300{,}000}=0.04=4\%\). 4. Metro’s larger population can produce both more hospitals and more total cases. The larger totals therefore do not show that hospitals cause illness; the per-person illness rate is actually lower in Metro.

Answer

a) Metro b) Lake: \(5\%\). Metro: \(4\%\). c) Population size affects both totals. More hospitals and more total cases do not establish causation, and Metro has the lower illness rate.
Correlation and causation 10
A teacher randomly assigns students to use either retrieval practice or rereading for the same amount of study time. Everyone takes the same quiz under the same conditions. The retrieval-practice group has a higher average score. What causal conclusion is supported, and to whom does it apply most directly?

Hints

- Identify how students entered the two groups. - Check which conditions were held constant. - Separate causal validity for participants from broader generalization.

Solution

1. Random assignment tends to balance prior ability, motivation, and other student characteristics between the groups. 2. The study holds study time and quiz conditions constant, isolating the review method. 3. The score difference supports a causal effect of retrieval practice for students similar to those who participated.

Answer

The experiment supports that retrieval practice caused a higher average quiz score than rereading for students like those in the study.
Correlation and causation 11
Identical products are randomly assigned to two packing methods before being shipped on the same route. Method A has a lower damage rate. What conclusion is supported, and why is shipping route control important?

Hints

- Identify the treatment and the outcome. - Ask what the common shipping route keeps from varying. - Limit the causal statement to the tested conditions.

Solution

1. Random assignment balances product differences between the packing groups. 2. Using the same route prevents route roughness or distance from being mixed with packing method. 3. The experiment supports that Method A caused a lower damage rate under the tested shipping conditions.

Answer

Method A caused a lower damage rate under the tested conditions. Keeping the route the same prevents transportation differences from confounding the result.
Correlation and causation 12
Patients treated by specialists sometimes have lower recovery rates than patients treated by general practitioners. Explain the selection effect that makes a causal conclusion unreliable.

Hints

- Ask which patients are referred to specialists. - Compare the groups before treatment begins. - Look for selection based on the seriousness of the case.

Solution

1. Specialists often receive the most severe or complicated cases. 2. Those cases have lower recovery chances even before the specialist provides treatment. 3. Patient severity influences both the type of doctor and the outcome, so the observed association does not show that specialists cause worse recovery.

Answer

Specialists are more likely to treat severe cases. Case severity can explain the lower recovery rate, so the comparison does not establish harmful specialist care.
Correlation and causation 13
A dog-training center offered an optional class. Owners chose whether to attend, and afterward their dogs took the same skills test. <table><tr><th>Group</th><th>Dogs passing</th><th>Total dogs</th></tr><tr><td>Owners took class</td><td>\(36\)</td><td>\(40\)</td></tr><tr><td>Owners did not</td><td>\(24\)</td><td>\(40\)</td></tr></table> A headline claims, “The class caused the higher pass rate.” Find each group's pass rate and explain why the numbers do not prove this claim.

Hints

- What portion of the dogs passed in each group? - How were the owners sorted into the two groups? - What might be different about the dogs before either group took the skills test?

Solution

1. The dogs whose owners took the class had a pass rate of \(36\div40=90\%\). 2. The other dogs had a pass rate of \(24\div40=60\%\). 3. The rates differ by \(30\) percentage points, showing an association in these groups. 4. Owners chose whether to attend. The dogs in the two groups may already have differed in training or behavior, so the rates alone do not show what caused the difference.

Answer

Owners took class: \(90\%\). Owners did not: \(60\%\). The difference is \(30\) percentage points, but the owners chose whether to attend. The dogs may have differed beforehand, so the class's effect is not proved.
Correlation and causation 14
A garden center wants to test whether a new plant food helps tomato plants grow taller. It has \(60\) similar plants. Describe how to divide the plants fairly into a plant-food group and a comparison group. What conditions should stay the same, and what result should be compared?

Hints

- How could the garden center avoid putting only the biggest plants in one group? - What else besides the plant food can affect tomato growth? - What measurement would show whether the plants grew taller?

Solution

1. Randomly assign the \(60\) plants to two groups of \(30\). 2. Give one group the new plant food and the other group no plant food. 3. Keep sunlight, watering, soil, and growing time the same for both groups. 4. Compare the plants' growth over the same period, such as the mean increase in height. Random assignment helps balance differences between the plants before treatment.

Answer

Randomly assign \(30\) plants to receive the food and \(30\) to receive none. Keep sunlight, water, soil, and time alike. Compare how much the plants grew, such as their average height increase.
Correlation and causation 15
A teacher wants to learn whether background music affects how quickly students solve puzzles. One group works with music first thing in the morning. Another group works without music late in the afternoon. Name one important change the teacher should make to test the effect of music more fairly. Explain what problem the change fixes.

Hints

- List everything besides music that differs between the groups. - Try to make the conditions alike except for the factor being tested. - Ask what else could change how quickly students solve puzzles.

Solution

1. The groups differ in two ways: music and time of day. 2. Test both groups at the same time of day, or randomly assign students to music and silence within the same session. 3. This prevents time of day from being a competing explanation for differences in puzzle speed.

Answer

Test both music and no-music groups at the same time of day. Then a difference in speed is less likely to be explained by morning versus afternoon conditions.
Correlation and causation 16
Two community pools test a new swimming lesson. Each compares \(30\) swimmers who took the lesson with \(30\) who did not. <table><tr><th>Pool</th><th>How groups were formed</th><th>Lesson group passing</th><th>Other group passing</th></tr><tr><td>A</td><td>Swimmers chose whether to join</td><td>\(25\)</td><td>\(10\)</td></tr><tr><td>B</td><td>Swimmers were randomly assigned</td><td>\(18\)</td><td>\(12\)</td></tr></table> Everyone took the same swimming skills test afterward. Which pool provides stronger evidence that the lesson helped swimmers pass, despite the difference in pass counts? What important limit remains?

Hints

- Is a bigger difference always better evidence about a cause? - How could two groups be made more alike before a lesson begins? - Would testing swimmers at one pool settle what happens at every other pool?

Solution

1. At pool A, the pass-count difference is \(25-10=15\), but swimmers chose their own groups. 2. At pool B, the difference is \(18-12=6\). Random assignment makes the groups more comparable before the lesson. 3. Pool B therefore gives stronger evidence that the lesson caused a difference, despite the smaller observed gap. 4. The result is based on swimmers at one pool and does not guarantee the lesson will work equally well at other pools.

Answer

Pool B gives stronger causal evidence because it used random assignment. Pool A's larger difference could reflect who chose the lesson. Results from pool B may not apply to all swimmers or pools.
Correlation and causation 17
A study across several high schools reports that students who play a musical instrument have higher average mathematics grades. A music teacher then argues, “Every student should be required to learn an instrument so that our school's overall mathematics performance will improve.” Evaluate the teacher's argument. Distinguish association from causation, and identify one possible confounding variable that could explain the observed association.

Hints

- Does observing a difference between two groups prove that the group characteristic caused the difference? - Identify a factor that could affect both access to music lessons and school performance. - Consider what kind of study would provide stronger causal evidence.

Solution

1. The study reports a positive association: students who play an instrument have higher average mathematics grades than students who do not. 2. The teacher assumes a causal relationship, but an association does not show that learning an instrument produces higher mathematics grades. 3. A possible confounding variable is family educational support or socioeconomic resources. Such factors could make music lessons more available and also support academic achievement. 4. Without a well-designed experiment or stronger causal evidence, the proposed requirement is not justified by the reported association alone.

Answer

The argument confuses association with causation. Playing an instrument may be associated with higher mathematics grades without causing them. Family educational support or socioeconomic resources could influence both, so the study alone does not justify requiring every student to learn an instrument.
Correlation and causation 18
An observational study finds that students from homes with more books tend to earn higher standardized test scores. Would placing \(100\) more books in every home necessarily raise scores? Explain why or why not, and give one possible confounder.

Hints

- Identify whether households were assigned different numbers of books. - Think about family characteristics related to both variables. - Separate access to objects from the behaviors and resources surrounding them.

Solution

1. The number of books was observed rather than randomly assigned. 2. Parent education, household income, or time spent reading with children could influence both the home library and test performance. 3. These factors can create the observed correlation without the physical presence of additional books being the direct cause.

Answer

The causal claim is not justified. Parent education is one plausible confounder because it may be related to both the number of books at home and student test scores.
Correlation and causation 19
An observational study finds that adults who own pets report lower average stress. Give one self-selection explanation and one possible confounder. Explain why the association does not prove that adopting a pet reduces stress for everyone.

Hints

- Ask who is able and willing to own a pet. - Look for a resource related to both ownership and stress. - Notice that the claim applies to every person, while the data show an average association.

Solution

1. People who enjoy animals and have stable routines may be more likely to choose pet ownership, creating self-selection. 2. Housing security or household income may make pet ownership easier and may also reduce stress. 3. These alternative explanations mean the observational association does not prove a universal causal benefit.

Answer

People with lower-stress lifestyles may be more likely to adopt pets, and housing security could affect both pet ownership and stress. The study does not prove adopting a pet reduces stress for everyone.
Correlation and causation 20
People who exercise more often also report having more daily energy. Give one reverse-causality explanation and one confounding explanation. Explain why the association does not identify a single cause.

Hints

- Reverse the proposed direction between the two measured variables. - Look for a third lifestyle factor related to both. - Notice the unsupported word “only.”

Solution

1. Reverse causality is possible because people who naturally feel more energetic may choose to exercise more often. 2. Sleep quality, overall health, or work schedule could influence both energy and exercise frequency. 3. The association therefore does not show that exercise is the only cause of the reported energy difference.

Answer

More energetic people may exercise more, and sleep quality could affect both exercise and energy. The correlation does not establish a single causal explanation.

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