Inference, now for averages. When the population standard deviation is unknown, you use the t-distribution to build confidence intervals and run significance tests for a single mean, for paired (matched-pairs) data, and for the difference between two means.
All 10 topics from the College Board CED, in order.
Topic 4.1
Sampling Distributions for Sample Means
The mean, standard deviation, and shape of the distribution of x̄, and the role of the CLT.
Topic 4.2
Constructing a Confidence Interval for a Mean or Mean Difference
The one-sample and paired t-interval: x̄ ± t*·(s/√n).
Topic 4.3
Justifying a Claim Based on a Confidence Interval for a Mean
Using a t-interval to support or reject a claim about a mean.
Topic 4.4
Setting Up a Test for a Mean or Mean Difference
Stating hypotheses and checking conditions for a t-test.
Topic 4.5
Carrying Out a Test for a Mean or Mean Difference
The one-sample and paired t-test statistic, p-value, and conclusion.
Topic 4.6
Sampling Distributions for the Difference Between Two Means
The distribution of x̄₁ − x̄₂.
Topic 4.7
Constructing a Confidence Interval for the Difference Between Two Means
The two-sample t-interval for μ₁ − μ₂.
Topic 4.8
Justifying a Claim Based on a Two-Mean Interval
Deciding whether two means plausibly differ.
Topic 4.9
Setting Up a Test for the Difference Between Two Means
Hypotheses and conditions for a two-sample t-test.
Topic 4.10
Carrying Out a Test for the Difference Between Two Means
The two-sample t-test statistic, p-value, and conclusion.
About Unit 4
Unit 4 mirrors Unit 3, but the parameter is now a mean instead of a proportion. The key change: in real problems the population standard deviation σ is unknown, so you estimate it with the sample standard deviation s and use the t-distribution — a bell curve with heavier tails and degrees of freedom (n − 1 for one sample) — instead of the normal z. Everything starts from the sampling distribution of x̄, which has mean μ and standard deviation σ/√n and is approximately normal when the population is normal or the sample is large (the Central Limit Theorem).
From there you build a one-sample t-interval and t-test for a single mean, apply the same procedures to paired (matched-pairs) data by analyzing the differences, and extend to the difference between two independent means with two-sample t-procedures. Throughout, you check the Random, 10%, and Normal/Large Sample conditions, interpret intervals and p-values in context, and recognize when a paired design is more appropriate than two independent samples.
This unit is 10–20% of the AP Statistics exam and takes about 18 class periods. The four statistical practices below run through every topic:
Practice 1
Formulate Questions — pose a valid investigative question
Practice 2
Collect Data — identify the appropriate t-procedure
Practice 3
Analyze Data — compute intervals and t-statistics
Practice 4
Interpret Results — justify conclusions in context