Lesson 3: Free-Response Communication: State, Plan, Do, ConcludeAP Deep Dive

Statistics is the one AP subject where the writing is the mathematics. Two students can compute identical numbers and score 9 and 5, because the rubric grades statements: parameters defined in context, conditions verified with evidence, conclusions linked to α and phrased in the language of evidence. This lesson dissects a model response line by line, then runs an error clinic on the sentences that graders are trained to reject.

Anatomy of a model response (annotated)

Scenario: two study methods are compared. Method A (n = 25): mean score 78.4, s = 6.2. Method B (n = 25): mean 72.1, s = 7.8. Students were randomly assigned.

State. "Let μ_A and μ_B be the true mean scores of all students taught by methods A and B. H₀: μ_A = μ_B; Hₐ: μ_A ≠ μ_B." *Scoring note: parameters defined with population and variable in words — the phrase "true mean scores of all students taught by…" is what earns this; symbols alone do not.*

Plan. "A two-sample t test. Random: students were randomly assigned to methods, so the groups are independent. Normal: both samples are of size 25 and the score distributions show no strong skew or outliers in the data provided, so the t procedure is appropriate." *Scoring note: each condition is tied to evidence. "Random: met" scores zero — the grader must be shown the fact from the problem that satisfies the condition.*

Do. "t = (78.4 − 72.1)/√(6.2²/25 + 7.8²/25) = 6.3/1.99 = 3.16, with df ≈ 45.7 (Welch), p ≈ 0.0028. A 95% confidence interval for μ_A − μ_B is 6.3 ± 2.01(1.99) = (2.29, 10.31) points." *Scoring note: the statistic, degrees of freedom and p-value are reported as numbers with notation, not as a calculator command name ("I used T-Test") — calculator syntax is not statistical communication.*

Conclude. "Because p = 0.0028 < 0.05, we reject H₀. There is convincing evidence that the true mean score under method A differs from that under method B; the interval suggests A is higher by roughly 2.3 to 10.3 points. Because students were randomly assigned, this difference can be attributed to the methods themselves." *Scoring note: three separate earns — the α comparison, the conclusion in context with direction from the interval, and the causal scope justified by random assignment. Note also the duality on display: the 95% interval excludes 0 exactly as the α = 0.05 two-sided test rejects.*

The error clinic: five sentences that lose points

1. "There is a 95% probability that the true mean difference lies between 2.29 and 10.31." Wrong. The parameter is fixed; the *interval* is the random object. Correct: "We are 95% confident…" — confidence describes the success rate of the method over repeated sampling. 2. "p = 0.0028 means there is a 0.28% chance the null hypothesis is true." Wrong direction of conditioning. The p-value is P(data this extreme | H₀ true), not P(H₀ true | data). 3. "We accept H₀ and conclude the methods are equal." Never. Failing to reject means the evidence was insufficient — with low power, real differences hide routinely (Lesson 2's power of 0.64). 4. "The correlation is 0.9, proving exercise causes weight loss." Association in observational data cannot establish causation; lurking variables (diet, age, baseline fitness) are unexamined. Only random assignment licenses "causes." 5. "The sample mean is 78.4, so μ = 78.4." Statistics estimate parameters; equality claims erase the entire idea of sampling variability. Write "x̄ = 78.4 estimates μ."

Read those five again: they are, almost verbatim, the distractors in the multiple-choice section and the deduction triggers in FRQ rubrics. Recognising them on sight is a learnable, high-yield skill.

The investigative task: architecture

The final FRQ (the investigative task) chains parts: an early computation feeds a later design question, which feeds a critique. Strategy: (i) answer each part in context — later parts reuse earlier numbers, so a dropped context ("the mean *of what*?") compounds; (ii) when a part asks "explain whether the conclusion is justified," the rubric wants a *specific* flaw or strength named — sample type, randomisation, a condition, a bias direction ("voluntary response over-represents people with strong opinions, so the estimated support is biased upward" beats "the sample is biased"); (iii) budget time — roughly 25 minutes, and do not let one part consume the marks of three others. Direction-of-bias statements are a hallmark of top responses: not merely "biased," but biased *which way, and why that direction*.

Common mistakes at the top band

• Concluding about the sample instead of the population ("the 25 students in method A scored higher") — inference is about parameters.
• Reporting a conclusion without the α comparison — the logical link between p and the decision must be visible on the page.
• Generalising beyond the sampled population, or claiming causation from random *sampling* (which licenses generalisation, not causation).
• Describing a distribution with fewer than shape, centre, spread and outliers — in context, with units — when the question asks for a description.
• Confusing r and r²: r² = 0.64 means 64% of the variation in y is *explained by the linear relationship* with x — not "64% correlation," and not "y causes 64% of x."

AP Exam Experience

1. A 95% confidence interval for a population mean is (12.4, 18.6). The correct interpretation is:

• (A) There is a 95% probability that μ lies between 12.4 and 18.6
• (B) 95% of the data values lie between 12.4 and 18.6
• (C) The interval was produced by a method that captures the true mean in about 95% of repeated samples
• (D) There is a 95% probability that a new observation lies between 12.4 and 18.6

Answer: (C). Confidence is a property of the procedure over repetitions. (A) is the single most common misconception in the course — the parameter is not random; the interval is.

2. A test yields p = 0.03. The p-value is

• (A) the probability that the null hypothesis is true
• (B) the probability of obtaining a statistic at least as extreme as the one observed, assuming the null hypothesis is true
• (C) the probability that the alternative hypothesis is true
• (D) the probability the result occurred by chance, with no assumptions

Answer: (B). Every word matters: "at least as extreme," and the conditioning on H₀. (D) is the colloquial paraphrase students give — and it smuggles in the error of (A), because "by chance" is meaningless without the null model that defines the chances.

3. Which conclusion is correctly worded for a test with p = 0.012 at α = 0.05?

• (A) "We have proved the treatment works."
• (B) "Because p < α, we reject H₀; there is convincing evidence that the treated seeds have a higher germination rate."
• (C) "We accept the alternative hypothesis with 98.8% confidence."
• (D) "The result is significant, so the effect must be large."

Answer: (B). It shows the comparison, the decision, and an evidence claim in context — and stops there. (A) overclaims proof; (C) invents a probability for a hypothesis; (D) confuses statistical significance with effect size, which small samples and large samples trade in opposite directions.

Free-response question. A school counsellor surveys 200 students on their grade band and preferred learning format, to test whether the two variables are associated.

(a) State the hypotheses for a chi-square test of independence, in context. (b) Calculate the expected count for the cell "Grades 9–10, In person," and verify the condition for the test. (c) The test statistic is χ² = 9.56 with p-value ≈ 0.008. State the degrees of freedom and the conclusion at α = 0.05, in context. (d) Identify the cells that contribute most to the test statistic, and describe what they show about the association.

Scoring guideline.

• (a) 1 point. H₀: grade band and preferred learning format are independent (no association) in the population of students at this school; Hₐ: the variables are associated. The population must be referenced ("students at this school"), and the hypotheses are about association, not about means or proportions — a test of independence has no directional alternative.
• (b) 2 points. 1 point for the expected count: (row total × column total)/grand total = (100 × 70)/200 = 35.0. 1 point for the condition with evidence: all expected counts (35, 32.5, 32.5 in each row) are ≥ 5, so the chi-square approximation is appropriate. Checking the *observed* counts instead earns nothing — the condition is on expected counts, a distinction this course has now made three times because the exam makes it every year.
• (c) 2 points. 1 point for df = (2 − 1)(3 − 1) = 2. 1 point for the conclusion: p = 0.008 < 0.05, so reject H₀ — there is convincing evidence of an association between grade band and preferred learning format among students at this school. (Scope note: the survey is of this school's students; a random sample supports generalising to the school, not beyond it.)
• (d) 2 points. 1 point for computing or identifying the largest contributions: (45 − 35)²/35 = 2.86 and (25 − 35)²/35 = 2.86 — the two "In person" cells together contribute 5.71 of the 9.56. 1 point for the interpretation in context: younger students (grades 9–10) chose in-person learning far more than expected under independence (45 vs 35), and older students far less (25 vs 35) — the association is driven mainly by the in-person column, with older students shifting toward hybrid. Naming the cells without the observed-versus-expected comparison earns the first point only.