Scope of Inference

Unit 3 · Collecting Data

What AP Stats asks here

What can a study conclude? Two design choices set the ceiling. Random selection from the population unlocks generalization to that population. Random assignment to treatments unlocks causal claims within the assigned group. Both together are the gold standard; either alone gives a weaker conclusion; neither gives only a descriptive observation in the study group.

Scope matrix

Random select + random assign

\text{causal AND generalizable}

Random select only

\text{generalizable association}

Random assign only

\text{causal, but only within the study group}

Neither

\text{descriptive only}

AP Tip: Two questions, two answers. Was selection random? → generalization. Was assignment random? → causation. Treat them independently.

Caution: Headlines that say 'shows', 'causes', or 'leads to' for an observational study are misreading the scope. The honest verb for observational data is 'is associated with'.

Type 1

Random selection × random assignment matrix

Identify what each axis of the matrix delivers. Selection ⇒ external validity (generalization). Assignment ⇒ internal validity (causation).

Example 1

A study takes a random sample of 500 employees from one company, then randomly assigns them to two ergonomic chair designs. What scope of inference is valid? (A) Association between chair design and outcomes among these 500 employees only. (B) Causation generalizable to the company's employees: chair design caused the difference, and the result extends to all employees of this company. (C) Causation generalizable to all employees in all companies. (D) Only a descriptive comparison.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 2

Causation-claim violation

An observational study showing an association cannot conclude causation; lurking variables and reverse causation remain plausible.

Example 2

A study surveyed 5000 randomly-selected Americans and reports that regular church-goers had 30% lower depression rates. The author concludes 'attending church reduces depression.' Which critique is correct? (A) The conclusion is fine — the sample was random. (B) The causal claim is unjustified because participants chose whether to attend church; reverse causation and lurking variables (social support, lifestyle) can explain the association. (C) The conclusion is fine — survey is the strongest possible design. (D) The 30% gap must be due to sampling error.

Practice more of this type— AI-generated · always-new problems

Generate Problems →

Type 3

Generalization-claim violation

Random assignment licenses causation in the study group only. Without random selection, do not generalize to a broader population.

Example 3

A study at one elite university randomly assigns 100 students to two study strategies. Strategy A produces higher scores. The paper concludes 'Strategy A is better for college students in general.' Which critique is correct? (A) The conclusion is fine — random assignment justifies both causation and generalization. (B) Strategy A caused higher scores at this university (random assignment); generalizing to all college students is unjustified because the sample is not a random selection from that broader population. (C) The conclusion is fine — students are students. (D) The result is purely observational.

Practice more of this type— AI-generated · always-new problems

Generate Problems →