AP Statistics Unit 4 Essentials: Key Terms & Core Concepts

Key Concept 1

When σ is unknown, means inference uses the t-distribution

Inference for a mean starts from the sampling distribution of x̄ — center μ, spread σ/√n, approximately normal under the conditions. But σ is almost never known, so you estimate it with the sample standard deviation s and switch from z to the t-distribution, which has heavier tails and degrees of freedom (n − 1 for one sample) to account for that extra uncertainty. As always, you first check the Random, 10%, and Normal/Large Sample conditions.

Sampling Distribution t-distribution Conditions

Key Concept 2

One-mean inference: intervals estimate, tests decide — and paired data is a one-sample problem

A one-sample t-interval, x̄ ± t*·(s/√n), gives a plausible range for μ, while a one-sample t-test, t = (x̄ − μ₀)/(s/√n), weighs evidence against a claim using a p-value. Paired (matched-pairs) data — before/after or otherwise linked observations — reduces to exactly the same one-sample procedure applied to the differences, with the parameter being the mean difference μ_d.

Confidence Intervals t-tests Paired Data

Key Concept 3

Two means extend the framework — and design decides the procedure

To compare two independent groups, use two-sample t-procedures for μ₁ − μ₂, whose standard error combines both samples' variability. The most-tested judgment is paired vs. two-sample: matched or linked subjects call for a paired (one-sample) analysis, while separate independent samples call for two-sample procedures. Because t-procedures are robust, they remain reliable under mild non-normality — but you still interpret every interval and p-value in context.

Two Means Design Choice Robustness

Sampling distribution of x̄

The distribution of the sample mean: center μ, standard deviation σ/√n, approximately normal under the conditions.

Foundations

Central Limit Theorem

For large samples, the sampling distribution of x̄ is approximately normal regardless of population shape.

Foundations

t-distribution

The distribution used for mean inference when σ is unknown; heavier tails than the normal, set by degrees of freedom.

Foundations

Degrees of freedom

n − 1 for a one-sample or paired t-procedure; it selects the correct t-distribution.

Foundations

Standard error of the mean

s/√n, the estimated standard deviation of the sample mean.

Foundations

Random / 10% / Normal-Large Sample

The three conditions for inference about a mean.

Foundations

Confidence interval for a mean

x̄ ± t*·(s/√n), a plausible range for the population mean.

One Mean

Critical value (t*)

The multiplier from the t-distribution, determined by the confidence level and degrees of freedom.

One Mean

Margin of error

t* times the standard error; smaller with larger n, larger with higher confidence.

One Mean

One-sample t-test statistic

t = (x̄ − μ₀)/(s/√n), measuring how many standard errors x̄ is from μ₀.

One Mean

Null / alternative hypotheses

H₀: μ = μ₀ versus Hₐ; always statements about the population mean.

One Mean

p-value

The probability, assuming H₀ is true, of a result at least as extreme as observed.

One Mean

Paired data

Matched or linked observations (such as before and after on the same subjects).

Paired Data

Matched-pairs t-procedure

A one-sample t-procedure applied to the differences of paired data.

Paired Data

Mean difference (μ_d)

The population mean of the paired differences; the parameter in a paired analysis.

Paired Data

Two-sample t-interval

(x̄₁ − x̄₂) ± t*·√(s₁²/n₁ + s₂²/n₂), for the difference of two means.

Two Means

Two-sample t-test

A test of H₀: μ₁ = μ₂ using t = (x̄₁ − x̄₂)/√(s₁²/n₁ + s₂²/n₂).

Two Means

Independent samples

Two separate groups whose observations are not linked, requiring two-sample procedures.

Two Means

Robustness

The property that t-procedures stay fairly accurate under mild violations of normality.

Two Means

Standard deviation vs. standard error

s measures spread of individuals; s/√n measures spread of the sample mean.

Foundations

Significance level (α)

The pre-set threshold for rejecting H₀.

One Mean

Confidence level

The long-run percentage of such intervals that capture the true mean.

One Mean