Lesson 20: Full AP Exam ReviewAP

Everything in this course converges on one skill: given a scenario, choose the right tool, execute it cleanly, and say what it means. This final lesson is the synthesis — a procedure-selection framework, three integrated worked examples, and the traps that cost the most points when topics mix. It is also your revision map: every row of the framework points back to the lesson that taught it.

The selection framework

Ask four questions in order. (1) What is the data type? One categorical variable → proportion methods or chi-square goodness of fit. Two categorical variables → chi-square table tests. One quantitative variable → mean methods. Two quantitative variables → regression/slope. (2) What is the goal? Estimate (confidence interval) or test a claim (significance test) — or describe/design (Units 1–3: then no inference at all). (3) What is the design? One sample, paired data, or separate populations? Random sample (generalise) and/or random assignment (causation)? (4) Which parameter? p, μ, μ_d (paired mean difference), β (slope), or a distribution of counts. Name the parameter, and the procedure usually names itself.

Worked synthesis 1 (selection drill). (i) "Do students at School A and School B differ in their distribution of transport modes?" Two separate populations, one categorical variable → chi-square homogeneity. (ii) "A runner times herself on the same 20 routes with two shoe models, order randomised; is one model faster on average?" Paired quantitative differences → paired t-test on the 20 differences. (iii) "Is there a linear relationship between a car's engine size and its fuel consumption, and how strong is the evidence?" Two quantitative variables, inference about association → t-test for the slope. Three scenarios, three branches of the framework, zero formulas needed — selection *is* the skill being graded in part (a) of half the FRQs ever written.

Worked synthesis 2 (the interval–test duality). A club claims 65% of members support a new meeting time. A sample of 60 members finds 48 in favour (p̂ = 0.80). The 95% confidence interval: SE = √(0.80 × 0.20/60) ≈ 0.0516, giving 0.80 ± 1.96 × 0.0516 ≈ (0.699, 0.901). Since the claimed 0.65 lies *outside* the interval, the two-sided test of H0: p = 0.65 at α = 0.05 rejects — the interval and the test are the same evidence wearing two coats. But note the asymmetry of purposes: the interval also tells you the support is plausibly as high as 0.90, information no p-value carries. When a part asks "is the claim plausible?", either tool works; when it asks "estimate", only the interval does.

Worked synthesis 3 (scope of inference). A university randomly selects 400 students for a survey and finds exercisers report better sleep. A second study recruits 60 volunteers, randomly assigns half to an exercise programme, and finds the same. The survey (random sample, no assignment) generalises to the student body but cannot claim exercise *causes* better sleep (confounding: health-conscious students do both). The experiment (volunteers, random assignment) supports causation but generalises only to people like the volunteers. Every FRQ conclusion should end by checking its claim against this 2×2 — scope errors are the most common "last part" deductions.

The traps that span units

• Paired vs independent (Lessons 14, 16): same subjects twice = differences, always.
• Random sample vs random assignment (Lessons 9, 10): generalisation vs causation — never interchangeable.
• Which standard error (Lessons 13, 15): p̂ inside intervals, p0 inside tests.
• Counts vs percents for chi-square (Lesson 17): raw counts, expected ≥ 5.
• df bookkeeping: proportions z — none; one mean — n − 1; slope — n − 2; chi-square GOF — k − 1; tables — (r−1)(c−1).
• Language audit: no "accept H0", no "prove", no probability attached to a parameter, no causation without assignment, every conclusion in context with linkage.

Your final-week plan

Rework one FRQ per unit under time (13 minutes), writing full four-step responses and grading yourself against the checklists in Lessons 13–19 — self-grading with a rubric teaches faster than re-reading notes. For multiple choice, practise the selection framework on scenario stems before touching calculations: most MCQ errors are procedure-selection errors, not arithmetic. Sleep the night before is a statistically significant variable. Use it.

AP Exam Experience

Question 1. A researcher records, for a random sample of adults, each person's employment status (employed/unemployed) and whether they rent or own their home, and asks whether the two variables are associated in the population. The correct procedure is:

• (A) Two-proportion z-test
• (B) Chi-square test for independence
• (C) Chi-square test for homogeneity
• (D) Paired t-test
• (E) One-proportion z-test

Answer: (B). One sample, two categorical variables, association question — independence. (C) would require separate samples from distinct populations; AP Statistics does not include a two-proportion z-test, eliminating (A) on CED grounds alone.

Question 2. A study measures blood pressure in 24 patients before and after a month of treatment and will test whether the mean change differs from 0. The correct analysis is:

• (A) One-sample t-test on the after measurements
• (B) Chi-square goodness of fit
• (C) Paired t-test on the 24 differences, df = 23
• (D) One-sample t-test on all 48 measurements, df = 47
• (E) Test for the slope of a regression line

Answer: (C). Before/after on the same patients creates 24 paired differences; the paired t-test uses df = 23. (D) pretends the measurements are independent — they are not — and (A) discards the pairing that is the design's whole strength.

Question 3. A 99% confidence interval for a true slope β is (−0.4, 2.1). For the two-sided test of H0: β = 0 at α = 0.01, the correct decision and reason are:

• (A) Reject H0, because the interval is wide
• (B) Reject H0, because 2.1 > 0
• (C) Fail to reject H0, because 0 lies inside the 99% interval
• (D) Reject H0, because the interval contains mostly positive values
• (E) The test cannot be decided from an interval

Answer: (C). The duality: a two-sided test at level α rejects exactly when the (1 − α) confidence interval excludes the null value. 0 ∈ (−0.4, 2.1), so at the 1% level we fail to reject. Width and the interval's "lean" are irrelevant — only exclusion matters.

Free-response question (mini investigative task). A school district will survey students about a proposed later start time and also pilot the change in one school. (a) The district takes a random sample of 300 students; 171 favour the change. Construct a 95% confidence interval for the proportion of all district students who favour the change, and state what the district may conclude about majority support. (b) In the pilot school, the mean self-reported sleep of 40 randomly sampled students rises from 6.9 to 7.4 hours (mean difference d̄ = 0.5 hours, s_d = 1.1 hours) after the change. Perform a paired t-test of H0: μ_d = 0 vs Ha: μ_d > 0 at the 5% level (t ≈ 2.87, p ≈ 0.0033) and conclude in context. (c) A board member says: "Both studies prove the later start causes better sleep for all district students." Identify the scope error in each study's case.

Scoring guideline.

• Part (a): Essentially correct if p̂ = 171/300 = 0.57, SE = √(0.57 × 0.43/300) ≈ 0.0286, interval 0.57 ± 1.96 × 0.0286 ≈ (0.514, 0.626), with conditions noted (random sample; counts 171 and 129 ≥ 10; 10% condition) and the conclusion that the entire interval lies above 0.5, so the district can be 95% confident a majority of all its students favour the change. Partially correct if the interval is right but the majority conclusion is not drawn from it.
• Part (b): Essentially correct if: the procedure is identified as a paired one-sample t-test on the differences with df = 39; the given t ≈ 2.87 and p ≈ 0.0033 are used; and the conclusion reads: since 0.0033 < 0.05, reject H0 — there is convincing evidence that the true mean sleep duration increased after the later start *for students at the pilot school*. Partially correct if the decision is right but scope or context is missing.
• Part (c): Essentially correct if BOTH scope errors are named: (i) the survey is observational — it measured opinion (and even if it measured sleep, association in a survey cannot establish causation due to confounding); (ii) the pilot involved ONE school (and a sample within it), so results do not automatically generalise to all district students (no random sample from the district), and within the pilot, students were not randomly assigned to start times, so even the local causal claim is shakier than a designed experiment's would be. Partially correct if only one study's error is identified. Incorrect if it accepts "prove" — no statistical study proves.

Deeper dive: one last synthesis — design decides everything downstream

Trace a single question through the whole course. "Does tutoring improve exam scores?" Unit 3 decides the ceiling: randomly assign students to tutoring and the study *can* support causation; survey volunteers who chose tutoring and it cannot, whatever the p-value says. Unit 4 supplies the probability model for the analysis; Unit 5 the sampling distributions; Units 6–9 the mechanics of the paired or two-group comparison, the interval and the test; Lesson 19 the communication. Notice where marks are lost at each stage in published chief-reader reports: not in the arithmetic of the middle, but in the design judgements at the start and the conclusion language at the end. That distribution of errors is your revision priority list in disguise. The strongest final-week exercise in this course is therefore not another computation drill: take any five past FRQs, and for each, write only parts (a) and the conclusion sentence — the two places where statistics, as opposed to calculation, is actually examined.

Error clinic: six verdicts in one line each

(1) "The p-value is 0.03, so there is a 3% chance the null is true." — Wrong: the p-value conditions on the null; it never prices it. (2) "We surveyed both classes completely, then ran a z-test to generalise to the school." — Wrong tool for the job: a census needs no inference; the parameter is already known. (3) "r = 0.92, so studying causes 92% of the grade." — Two errors in nine words: causation from observation, and r confused with a percent of anything. (4) "The chi-square test on the percentages (62%, 48%, 41%, 49%) gives..." — Stop: chi-square eats counts, not percents; without n there is no test. (5) "The 95% CI for the mean difference was (−0.89, 0.05), so the paired test at 5% must reject." — Only the *two-sided* duality applies: 0 in the interval means the two-sided test fails to reject, whatever a one-sided test said. (6) "Our SRS of 25 from the class register tells us about all teenagers." — Scope error: inference travels to the population *sampled from* — this class's register — not to teenagers everywhere. Six errors, six different lessons, one shared moral: on this exam, the set-up and the sentence earn the marks; the arithmetic is the easy middle.