Mistake Master
What a study can honestly claim
These questions grade your judgment, not your arithmetic. A sample speaks for the population it came from; a margin of error brackets a population value, both directions; and an association observed is not a cause proven. The points slip away whenever a claim outruns the design that produced it. Audit the design, then size the claim to fit.
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What this topic is about
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Statistics questions on the SAT rarely ask you to compute; they ask what a study can honestly CLAIM. Three judgments carry the topic: how far a sample's conclusion reaches, what a margin of error does and does not say, and when an association can be called a cause.
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A sample speaks for its own population
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A random sample supports conclusions about exactly the population it was drawn from. Members of one gym speak for that gym's members; no wording can stretch them into "all adults."
- Name the sampling frame: who could have been selected? The conclusion reaches exactly that group.
- Self-selected samples (call-ins, online polls) do not even reach their own population; the flaw is WHO responds, and no sample size fixes it.
- Estimates scale: sample proportion times population size.
Worked example. A random sample of $50$ students from a school of $800$ finds $12$ bike to school. Estimate the school's bikers.
$\dfrac{12}{50}$ of the school: $\dfrac{12}{50} \cdot 800 = 192$.
The estimate is about THIS school. Neighboring schools were never in the frame.
§3
What a margin of error brackets
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An estimate of $46$ with margin of error $3$ says the POPULATION value is plausibly between $43$ and $49$. It does not make the estimate exact, does not bound individuals, and reaches both directions.
- Plausible range: estimate $\pm$ margin, both ways.
- The interval brackets the population MEAN or PROPORTION, never each individual.
- A value is consistent with the poll exactly when it lies inside the interval.
- Larger random samples shrink the margin; re-asking the same sample does not.
Worked example. A poll reports $58\%$ support, margin of error $4$ points. Is $53\%$ consistent with the poll?
$58 \pm 4$: from $54\%$ to $62\%$.
$53\%$ falls outside, so it is inconsistent; $56\%$ or $61\%$, inside, would be fully consistent.
§4
Association is not causation
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An observational study can show two things moving together. It cannot say one causes the other, in either direction, and it cannot predict what intervening would do. The upgrade to causation requires an experiment with random ASSIGNMENT.
- Observation supports "associated with" and nothing stronger.
- Random SAMPLING widens reach; random ASSIGNMENT unlocks cause. Different randomness, different powers.
- With random assignment and a significant difference, a causal claim is fair, for the population represented.
§5
Match the claim to the design
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Every study question reduces to an audit: who was sampled, how was treatment assigned, and does the stated claim stay inside what those two facts allow?
Let the sampling frame set the conclusion's reach, read the margin of error as a two-sided bracket on the population value, and reserve causal language for randomized experiments.
§6
Three patterns that cost real points
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Three patterns recur on sampling questions. They are the same ones the diagnostic routes on.
The conclusion outruns the sample.
One gym's members become "all adults," a magazine's subscribers become "all home cooks," or a sample count stands in for a population estimate unscaled.
Fix. Ask who could have been selected. The claim reaches that group and no further, and estimates scale by population over sample.
The margin of error over- or under-promises.
The estimate gets treated as exact, the interval gets applied to individuals, or consistency gets judged by closeness to the center instead of membership in the interval.
Fix. Estimate minus margin to estimate plus margin, applied to the population value only. Inside means consistent; outside means not.
Correlation gets promoted to cause.
Students with more absences have lower grades becomes "absences cause low grades," or a prediction that intervening will change outcomes.
Fix. Check the design first: observation earns "associated," random assignment earns "causes." No exceptions for plausible-sounding stories.
Ten quick checks across the patterns: the reach of a sample, scaling estimates, margin-of-error brackets and endpoints, self-selection, and association versus causation. Pick or type your answer, then check. Progress is saved.