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Inferences & Claims
From Statistics

Master how to evaluate statistical claims, assess sampling methods, and determine what conclusions data can actually support on the SAT.

SAT Math

What Are Statistical Inferences?

A statistical inference is a conclusion drawn about a population based on data from a sample. On the SAT, you must decide whether a given claim is supported by the data: what was measured, who was studied, and how the study was designed.

Population vs. Sample

Population Sample Inference generalizes from sample to population
Population
The entire group you want to learn about (e.g., all high school students in the U.S.).
Sample
The subset actually studied (e.g., 500 students selected from 10 schools).

Why the Sampling Method Matters

The SAT tests whether you can identify which conclusions are valid based on how the sample was collected. Two critical concepts:

Random Selection
Every member of the population has an equal chance of being chosen. This allows you to generalize results to the whole population.
Random Assignment
Participants are randomly placed into treatment or control groups. This allows cause-and-effect conclusions.

The Inference Validity Framework

Study Design Random Selection? Random Assignment? Yes No Yes No Can generalize Cannot generalize Can claim causation Only association Both needed for: generalizable causal claims

Margin of Error & Confidence Intervals

When a survey reports a result like "42% with a margin of error of \(\pm 3\%\)", it means the true population value is likely between 39% and 45%.

39% 42% 45% Sample Estimate Margin of Error: ±3%

Causation vs. Association

Causation Association
Meaning X directly causes Y X and Y occur together
Requires Random assignment Observational data only
Study type Controlled experiment Survey / observational study
SAT language "causes," "leads to," "results in" "is associated with," "is related to"
Example "The drug reduced symptoms" "Students who slept more tended to score higher"

Types of Bias in Studies

Selection Bias
The sample is not representative of the population. Example: surveying only gym members about exercise habits.
Response Bias
Participants give inaccurate answers, often due to leading questions or social pressure.
Nonresponse Bias
A significant portion of selected participants do not respond, and they may differ systematically from those who do.

How to Evaluate a Statistical Claim

1
Identify the population. Who is the study trying to learn about?
2
Check the sample. Was it randomly selected from that population? If not, you cannot generalize.
3
Check the design. Were participants randomly assigned to groups? If not, you cannot claim causation.
4
Evaluate the claim. Does the conclusion match what the study design can support?
5
Watch for bias. Look for volunteers, self-selection, leading questions, or small sample sizes.

Example 1: Survey Study

Question: Which of the following is a valid conclusion?

✓
Random selection from the voter registry means we can generalize to all registered voters in that city.
✕
We cannot generalize to all adults in the city: only registered voters were sampled.
✕
No random assignment was used, so we cannot claim that any factor causes exercise habits.

Example 2: Controlled Experiment

Analysis:

Generalization?
Yes: Students were randomly selected from the school, so results apply to all students at that school.
Causation?
Yes: Random assignment to groups means we can say the new program caused the improvement.

Example 3: Margin of Error

Can we conclude Candidate A is ahead?

41% 44% 45% 47% 48% 51% B: 44% A: 48% Overlap

How Sample Size Affects Precision

Sample Size (\(n\))
 Margin of Error 100 400 900 1600 2500 10% 8% 6% 4% 2%
\(n=100\)
\(n=400\)
\(n=2500\)

Scope of Inference Summary

Study Design Generalize? Causation?
Random selection + Random assignment Yes Yes
Random selection + No assignment Yes No
No selection + Random assignment No Yes (for sample only)
No selection + No assignment No No

Common Mistakes on the SAT

✕ Overgeneralizing
Concluding results apply to a broader group than was actually sampled. If only college students were studied, you cannot generalize to all adults.
✕ Claiming Causation
Saying one variable caused another from an observational study. Without random assignment, you can only show association.
✕ Ignoring Bias
Accepting results from a volunteer or convenience sample as if it were randomly selected. Self-selected groups are inherently biased.

Question 1 of 5

A researcher randomly selects 500 employees from a large company and surveys them about job satisfaction. She finds that 68% are satisfied. Which conclusion is valid?

Question 2 of 5

A study finds a positive association between hours spent reading and test scores among high school students who volunteered for the study. Which statement is best supported?

Question 3 of 5

A poll of 2,000 randomly selected voters finds that 52% support a ballot measure with a margin of error of \(\pm 2.5\%\). Which conclusion is most appropriate?

Question 4 of 5

To reduce the margin of error of a survey from \(\pm 6\%\) to \(\pm 3\%\), a researcher should approximately:

Question 5 of 5

Researchers randomly selected 300 patients at a hospital and randomly assigned them to receive either a new treatment or a placebo. Patients receiving the new treatment recovered 4 days faster on average. Which conclusion is valid?