Mathfolis

Inference from Sample Statistics & Margin of Error

Unit 3 · Problem-Solving & Data Analysis

What the SAT asks here

Inference questions check whether you can interpret a confidence interval, predict how the margin of error reacts to a larger sample, and recognise when a sample-based conclusion can validly be generalized. The SAT does not ask you to compute MoE — it asks you to reason about it. About one problem per test.

Confidence interval (CI)

Sample estimate ± margin of error
Interpretation
P(μ∈CI)≈confidence levelP(\mu \in \text{CI}) \approx \text{confidence level}

What changes the margin of error

Sample size ↑ → MoE ↓
Confidence level ↑ (95% → 99%) → MoE ↑
Variability in data ↑ → MoE ↑
SAT Tip: A 95% CI refers to the population MEAN, not individual values. 'We are 95% confident the population mean lies in (a, b).'
Caution: A sample's conclusions only generalize when the sample was RANDOMLY drawn from the target population. Self-selected groups (volunteers, club members) do not qualify.
Type 1

Interpret a confidence interval

Translate the CI into a sentence about the population mean (not individual values).

Example 1
A poll estimates the mean weekly study time for high school students is 12.4 hours, with a 95% confidence interval of (11.8, 13.0) hours. What does this interval indicate?

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Type 2

Effect of sample size

Larger sample → smaller margin of error, holding other factors equal.

Example 2
A researcher plans to double the sample size in a survey. What happens to the margin of error, holding other factors equal?

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Type 3

Validity of generalization

Check whether the sample was randomly drawn from the target population. If not, the conclusion cannot be generalized beyond the studied group.

Example 3
A researcher surveys students at a single high school's chess club and concludes 'most teenagers enjoy strategy games.' Why is this conclusion not valid?

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