🎯 Learning Objectives
- What makes a sample representative: and what doesn't.
- Random sampling vs. self-selection vs. convenience sampling.
- How a margin of error affects the conclusions you can draw.
- Correlation vs. causation: the SAT's favorite trap.
- Experiment vs. observational study: when you can claim cause.
- 10 SAT-style practice questions.
1. Why Sampling Matters
It's almost never possible to ask every person in a group, so researchers ask a sample. The key question for SAT statistics: was the sample chosen in a way that makes its results trustworthy?
2. Types of Samples
Sample Type | How It's Selected | Trustworthy? |
Simple random | Every member has equal chance | ✅ Yes |
Stratified random | Random within sub-groups | ✅ Yes |
Convenience | Whoever is easy to reach | ❌ Biased |
Self-selected | People volunteer | ❌ Biased |
Cluster | Random groups, all members | ✅ Often okay |
💡 Pro Tip: Random = Representative
Only random samples can be generalized to the larger population. A sample taken at one mall, one school, or one website is not random: it represents only that location's visitors.
3. Generalizing Results
Once you know how the sample was selected, you can decide which population the results apply to.
❓ Was the sample randomly selected from the entire target population? | ||
✅ YES ▼ | ❌ NO ▼ | |
Results can be generalized to that population | Results apply ONLY to the group sampled | |
⚠️ Common SAT Trap
If a survey is random within ONE high school, the results apply to THAT school: not to all teenagers nationwide.
Read carefully: 'a random sample of 200 students at Lincoln High' is random for Lincoln High only.
4. Margin of Error
A margin of error tells you how much wiggle room a sample's estimate has. If a poll says 'Candidate A leads with 52% ± 3%', the true value is likely between 49% and 55%.
True Value Range ≈ Estimate ± Margin of Error
What Affects the Margin of Error?
Change | Effect on Margin of Error |
Larger sample size | Smaller margin (more precise) |
Smaller sample size | Larger margin (less precise) |
Higher confidence level | Larger margin |
Lower variability in data | Smaller margin |
5. Correlation vs. Causation
Two variables are correlated if they tend to move together. But correlation alone never proves that one CAUSES the other.
📝 Classic Example
Ice cream sales and shark attacks are strongly correlated. Does ice cream cause sharks? No: both rise with hot summer weather. The third variable (temperature) is the real driver.
When CAN You Claim Causation?
❓ Was treatment randomly assigned (controlled experiment)? | ||
✅ YES ▼ | ❌ NO (observational study) ▼ | |
You CAN claim cause-and-effect | You CANNOT claim cause: only association | |
6. Experiments vs. Observational Studies
Type | Researcher Action | Conclusion You Can Draw |
Experiment | Randomly assigns treatments | Cause and effect |
Observational study | Observes without intervening | Association only |
7. Strategies
⚡ Was the sampling random?
If not, you cannot generalize. Eliminate every choice that says 'all teenagers' or 'all Americans'.
⚡ Was the treatment randomly assigned?
If not, the answer cannot mention causation.
⚡ Larger sample → smaller margin
Easy way to spot the right answer when comparing two studies.
⚡ Beware of the words 'cause' and 'because'
On observational studies, these are almost always wrong.
⚡ Match the conclusion to the population sampled
Random survey at a yoga class generalizes to that yoga class.
8. Summary
📌 Key Takeaways
- Random sample → results generalize to whole population.
- Non-random sample → results apply only to those sampled.
- Margin of error: estimate ± margin gives plausible range.
- Bigger sample = smaller margin of error.
- Cause-and-effect requires a randomized experiment, not observation.
- Correlation ≠ causation (the most common SAT trap).
9. Practice: 10 Questions
Q1. A study surveys shoppers at one mall. Results can be generalized to…
A) all teens B) all shoppers nationwide C) shoppers at that mall D) all consumers
Hint: Non-random national sample → applies only to that location.
Q2. A poll says 47% support a policy with margin of error 4%. The true support is likely between…
A) 43% and 47% B) 43% and 51% C) 47% and 51% D) 0% and 100%
Hint: 47 ± 4 = (43, 51).
Q3. Two studies survey the same population. Study A samples 500 people, Study B samples 50. Which has the smaller margin of error?
A) Study A B) Study B C) Same D) Cannot tell
Hint: Larger sample → smaller MOE.
Q4. A study finds that towns with more libraries have lower crime rates. Conclusion?
A) Libraries cause less crime B) Crime causes libraries C) Association exists D) No relationship
Hint: Observational → only association.
Q5. A scientist randomly assigns half of patients a drug and the other half a placebo. The drug group recovers faster. Conclusion?
A) Cannot conclude anything B) Drug causes faster recovery C) Faster recovery causes drug use D) No relationship
Hint: Randomized experiment → causation.
Q6. A school selects 30 students at random from each grade for a survey. This is an example of…
A) Convenience B) Stratified random C) Self-selected D) Census
Hint: Random within strata → stratified random.
Q7. A radio show asks listeners to call in their opinion. The sample is…
A) Random B) Stratified C) Self-selected D) Cluster
Hint: People choose to participate → self-selected.
Q8. Which study most strongly supports cause-and-effect?
A) Survey B) Observational study C) Randomized experiment D) Anecdote
Hint: Only experiments can show causation.
Q9. A poll: 60% favor candidate A, MOE = 3%. Can we say A will win?
A) Yes B) No, range is (57, 63) C) Yes if MOE is small D) Cannot say
Hint: Range is well above 50%, but actually 'will win' depends on margin and turnout: the SAT-style answer is B (we know the range).
Q10. To reduce margin of error in a study, the researcher should…
A) Use a smaller sample B) Use a larger sample C) Add more questions D) Pick from one neighborhood
Hint: Larger sample shrinks MOE.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | C) shoppers at that mall | Non-random nationally. |
Q2. | B) 43% and 51% | Estimate ± MOE. |
Q3. | A) Study A | Bigger n = smaller MOE. |
Q4. | C) Association exists | Observational → no causation. |
Q5. | B) Drug causes faster recovery | Randomized experiment. |
Q6. | B) Stratified random | Random within each grade. |
Q7. | C) Self-selected | Listeners choose to call in. |
Q8. | C) Randomized experiment | Only design supporting causation. |
Q9. | B) Range is (57, 63) | 60 ± 3. |
Q10. | B) Use a larger sample | Bigger n shrinks MOE. |