Data analysis, probability and statistics

The definitive archive reference. Compiled 15 September 2026 from corpus/sm/questions.json (1,469 graded parts, 2006–2025), the VCAA papers and examination reports in corpus/sm/text/ and corpus/sm/raw/, and the page images in corpus/sm/pages/. Written for candidates targeting a study score of 45+.

Read 01-study-design.md first. This document does not restate the study design's structure; it assumes §1.9 and §5 of that file and extends them into a question-level catalogue.

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0. Sources, tags and what the numbers mean

What pct is. Every percentage quoted in this document as "pct" is VCAA's own published figure for the percentage of the state that earned full marks on that question part. It is not a mean score. A 2-mark part with pct = 44% means 44% of students scored 2/2; some of the remaining 56% scored 1. Where the report also published an average, it is quoted separately.

What a "separator" is. A part with pct ≤ 50. In this area of study there are 40 of them out of 108 graded parts.

Two corpus defects specific to this area, stated up front rather than papered over.

1. 2025 Exam 2 Section B Q6d is missing from [QJSON]. The 2025 Examination 2 report prints its full-mark percentage as 53,11 — a comma instead of a decimal point — and the parser dropped the row. The question exists, is statistics, is worth 1 mark, and its true pct is 53.11% (average 0.53). It is included in this document's catalogue and excluded from the separator list, because 53.11 > 50.
2. Six parts of 2023 Exam 2 Section B Q4 are tagged "Data analysis, probability and statistics" in [QJSON] but are a logistic differential equation question. 2023 Exam 2 Section B Q4 is the fish-pond logistic model (dP/dt = P(1 − P/1000)); parts b, c, d, e.i, e.ii and g are tagged statistics, which is a topic-tagger artefact — the words "population", "sample" and "distribution" do not appear, but the word "population" in the demographic sense does. They are Calculus. They are listed in §4 for completeness, clearly flagged, and they are not counted when this document describes what statistics tests.

Coverage gaps in [QJSON]. Northern Hemisphere Timetable (NHT) sittings appear in [QJSON] only for 2024 and 2025, and only Examination 2 Section A (20 rows each). NHT Examination 1 and NHT Examination 2 Section B are absent, as are all NHT sittings 2017–2023 and the 2026 NHT sitting. Those papers are in the corpus as text or PDF, and this document quotes them for wording; they simply carry no VCAA percentages, because VCAA does not publish mark distributions for NHT.

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1. What the study design puts in this area — and what the exams actually test

1.1 The content, verbatim

[SD] overview:

"In this area of study students cover the study of linear combinations of random variables and introductory statistical inference with respect to the mean of a single population, the determination of confidence intervals, and hypothesis testing for the mean using the distribution of sample means."

Topic 1 — Distribution of linear combinations of random variables. [SD]:

This topic includes:
- for n independent identically distributed random variables X₁, X₂ … Xₙ each with mean μ and variance σ²:
- E(X₁ + X₂ + … + Xₙ) = nμ
- Var(X₁ + X₂ + … + Xₙ) = nσ²
- for n independent random variables X₁, X₂ … Xₙ and real numbers a₁, a₂ … aₙ:
- E(a₁X₁ + a₂X₂ + … + aₙXₙ) = a₁E(X₁) + a₂E(X₂) + … + aₙE(Xₙ)
- Var(a₁X₁ + a₂X₂ + … + aₙXₙ) = a₁²Var(X₁) + a₂²Var(X₂) + … + aₙ²Var(Xₙ)
- for n normally distributed independent random variables X₁, X₂ … Xₙ and real numbers a₁, a₂ … aₙ the random variable a₁X₁ + a₂X₂ + … + aₙXₙ is also normally distributed.

Topic 2 — Distribution of the sample mean. [SD]:

This topic includes:
- the concept of the sample mean X̄ as a random variable whose value varies between samples where X is a random variable with mean μ and the standard deviation σ
- simulation of repeated random sampling, from a variety of distributions and a range of sample sizes, to illustrate properties of the distribution of X̄ across samples of a fixed size n including its mean μ, its standard deviation σ/√n (where μ and σ are the mean and standard deviation of X respectively) and its approximate normality if n is large.

Topic 3 — Confidence intervals for the population mean. [SD]:

This topic includes:
- determination of confidence intervals for means and the use of simulation to illustrate variations in confidence intervals between samples and to show that the likelihood of a confidence interval containing μ depends on the level of confidence chosen in the determination of the interval
- construction of an approximate confidence interval, (x̄ − z·σ/√n, x̄ + z·σ/√n) where σ is the population standard deviation and z is the appropriate quantile for the standard normal distribution, or construction of an approximate confidence interval (x̄ − z·s/√n, x̄ + z·s/√n) where s is the sample standard deviation and z is the appropriate quantile for the standard normal distribution, and n is large (n ≥ 30 in many practical contexts).

Topic 4 — Hypothesis testing for a population mean. The heading itself is load-bearing: "Hypothesis testing for a population mean with a sample drawn from a normal distribution of known variance, or for a large sample". [SD]:

This topic includes:
- concepts of null hypothesis, H₀, and alternative hypotheses, H₁, test statistic
- level of significance and p-value
- formulation of hypotheses and making a decision concerning a population mean based on:
- a random sample from a normal population of known variance
- a large random sample from any population
- 1-tail and 2-tail tests
- interpretation of the results of a hypothesis test in the context of the problem
- hypothesis test, relating the formulation, conduct, errors and results in terms of conditional probability.

1.2 The relevant Outcome 1 key knowledge and key skills, verbatim

The study design does not attach key knowledge/skills to areas of study; it attaches them to outcomes. The statistics-bearing items in Outcome 1 are:

Key knowledge: "the distribution of sample means"

Key skills: "set up and solve problems involving the distribution of sample means"; "construct confidence intervals, and approximate confidence intervals, for sample means"

Hypothesis testing appears in neither list. As 01-study-design.md §2.1 establishes, this is a genuine gap in VCAA's document, not an omission from this summary. It does not narrow scope — [SPEC] says "All of the content from the areas of study and the key knowledge and key skills that underpin the outcomes in Units 3 and 4 are examinable" — but it is worth knowing that the Outcome 1 lists under-describe what Examination 1 has actually asked. 2021 Exam 1 Question 3, 2022 Exam 1 Question 3 and the NHT Examination 1 papers have all set inference on the technology-free paper.

1.3 Which years this area did and did not appear in

This is the single most important practical fact in this document, because it determines which past papers are worth working.

Consequences for practice. The 2006–2015 papers contain nothing for this area. The 2020 papers are actively misleading — a student who works 2020 Exam 2 will find no Question 6. The usable archive for statistics is 2016, 2017, 2018, 2019, 2021, 2022, 2023, 2024, 2025 November, plus the NHT sittings 2017–2019 and 2021–2026, minus the redactions listed below.

Redacted questions. Three statistics parts were struck after the Independent Review into VCAA's examination-setting processes: 2022 Exam 1 Q3b, 2022 Exam 2 Section A Q19 and 2022 Exam 2 Section B Q6f. [QJSON] carries Section A Q19 with pct: null and the comment "This question has been redacted following the findings of the Independent Review into the VCAA's Examination-Setting Policies, Processes and Procedures for the VCE." One further part, 2023 Exam 2 Section B Q6h, shows pct = 100% because it was invalidated: "Following the identification of an error in the question stimuli, this question was invalidated" [RPT23 E2]. Treat that 100% as a full-marks-to-all award, not as evidence the question was easy.

1.4 What the exams actually test — the counts

Restricting to November sittings and excluding the six mis-tagged 2023 logistic parts:

The structural rule, and it has held without exception since statistics entered the course.

• Examination 1 carries exactly one statistics question, worth 3–5 marks. It has been Question 2 (2016), Question 4 (2017, 2018, 2025), Question 3 (2019, 2021, 2022) and Question 6 (2023, 2024). In NHT sittings: Question 6 (2024 NHT, 2026 NHT), Question 2 (2025 NHT).
• Examination 2 Section B's statistics question is always Question 6 — the last question on the paper. 2016, 2017, 2018, 2019, 2021, 2022, 2023, 2024, 2025 November; 2023 NHT, 2024 NHT, 2025 NHT, 2026 NHT. Thirteen sittings, thirteen Question 6s. It is worth 8–10 marks and is almost always built as a single escalating context with 6–9 parts.
• Examination 2 Section A's statistics items are always the last two to four. 2016: Q19–20. 2017: Q18–20. 2018: Q18–20. 2019: Q18–20. 2021: Q17–20. 2022: Q18, Q20 (Q19 redacted). 2023: Q19–20. 2024: Q19–20. 2025: Q20 only. The 2023 redesign cut Section A from five options to four and reduced the statistics allocation from three or four items to two.

How well the state does. Median full-mark rate across the 108 graded statistics parts is 58%. Split by section:

Read that table carefully. Statistics is, overall, the strongest-performing area in the subject — 01-study-design.md §1.9.4 records a 65% median across 2023–2025 written parts, highest of any area, and VCAA lists it among areas of strength in 2016, 2022, 2023 and 2024. But that is entirely an Examination 2 phenomenon. On Examination 1, statistics is brutal: 13 of 18 graded parts are separators and the median is 44%. [RPT21 E1] names it outright under "Areas of weakness": "poor understanding of hypothesis testing and confidence intervals." [RPT22 E1] lists "probability (Question 3a.)" as a weakness. Statistics on the technology-free paper is where 45+ candidates are separated from 40+ candidates.

What the reports say the area is good and bad at, verbatim:

1.5 What is on the formula sheet, and what is not

The current sheet's "Data analysis, probability and statistics" block gives, in full:

• E(aX₁ + b) = a·E(X₁) + b and Var(aX₁ + b) = a²·Var(X₁)
• E(a₁X₁ + … + aₙXₙ) = a₁E(X₁) + … + aₙE(Xₙ) and Var(a₁X₁ + … + aₙXₙ) = a₁²Var(X₁) + … + aₙ²Var(Xₙ)
• For i.i.d. variables: E(X₁ + … + Xₙ) = nμ and Var(X₁ + … + Xₙ) = nσ²
• Approximate confidence interval for μ: (x̄ − z·s/√n, x̄ + z·s/√n)
• Distribution of the sample mean: E(X̄) = μ, Var(X̄) = σ²/n

Not on the sheet, and therefore memorised or derived: the test statistic z = (x̄ − μ)/(σ/√n); the definition of a p-value; the decision rule; Type I and Type II error definitions; any statement of the Central Limit Theorem; the normal probability density function; E(X) = ∫ x·f(x) dx for a continuous random variable; the normal quantiles 1.645, 1.96, 2.5758.

Two observations follow. First, the sheet gives Var in terms of the coefficients squared — the single most-missed rule in this area is printed in front of the candidate, which means every mark lost to it is lost to not reading. Second, the confidence-interval formula on the sheet is the s version; the σ version is identical in structure and VCAA uses whichever the stem provides.

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2. The complete catalogue of question types

Twenty-six types. For each: a name, VCAA's literal wording quoted from a real paper, what is really being tested, the method, three or more archive instances with ref and pct, typical marks, and the traps the reports name.

Where an instance has no percentage, it is an NHT paper (VCAA publishes no mark distributions for NHT) or a sample paper.

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Type 1 — Mean and variance of a sum of independent, non-identical random variables

VCAA wording (2024 Exam 1 Q6a, [PAPERS]): "The production of a brand of weed trimmer involves three stages, Stage 1, Stage 2 and Stage 3, which take W₁ hours, W₂ hours and W₃ hours, respectively. Here W₁, W₂ and W₃ are independent random variables, which may be assumed to be normally distributed. … Find the mean and the variance of the total time to produce one weed trimmer."

Really testing: that variances add and standard deviations do not.

Method: E(ΣXᵢ) = ΣE(Xᵢ); Var(ΣXᵢ) = ΣVar(Xᵢ) = Σ(sd)². Take a square root only if the question says "standard deviation".

Instances:

Traps. [RPT23 E1] on Q6a: "While many students correctly found the mean, a large number of students gave the standard deviation as (the sum of the standard deviations of the random variables)." [RPT25 E2] on Q6a.i: "Some students wrote the variance rather than the standard deviation." Both errors are the same error in opposite directions — not tracking whether the question asked for variance or standard deviation. [RPT23 E1] also lists this type among areas of strength, which is the point: at 75–86% it is a mark a 45+ candidate never drops.

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Type 2 — Weighted linear combination aX + bY: forward

VCAA wording (2017 Exam 2 Section A Q18, [PAPERS]): "U and V are independent normally distributed random variables, where U has a mean of 5 and a variance of 1, and V has a mean of 8 and a variance of 1. The random variable W is defined by W = 4U − 3V. In terms of the standard normal variable Z, Pr(W > 5) is equivalent to …"

Really testing: Var(aX + bY) = a²Var(X) + b²Var(Y), with the a² surviving a negative a or b.

Method: mean = aμ_X + bμ_Y; variance = a²σ_X² + b²σ_Y²; sd = √variance; standardise.

Instances:

Traps. In 2017 Exam 2 Q18, 25% chose option A, Pr(Z > 9/7) — that is sd = 4(1) + 3(1) = 7, adding the standard deviations weighted by the coefficients instead of adding the squared-weighted variances. The correct sd is √(16 + 9) = 5. In 2025 Exam 2 Q20 the correct sd is √(9·4 + 4·9 + 1·36) = √108 = 6√3; option D, Pr(Z > 30/7), corresponds to √(2² + 3² + 6²) = 7, i.e. combining the standard deviations as if every coefficient were 1, and drew 10%. [RPT24 E1] on Q6b: "Some students did not apply the formula for the variance of a sum of independent and identically distributed random variables, frequently forgetting to square either the cost values or the standard deviation at each stage."

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Type 3 — Weighted linear combination: inverse (solve for the coefficients)

VCAA wording (2018 Exam 1 Q4, [PAPERS]): "X and Y are independent random variables. The mean and the variance of X are both 2, while the mean and the variance of Y are 2 and 4 respectively. Given that a and b are integers, find the values of a and b if the mean and the variance of aX + bY are 10 and 44 respectively."

Really testing: setting up and solving a simultaneous linear/quadratic system, by hand, on Examination 1.

Method: write aμ_X + bμ_Y = given mean (linear) and a²σ_X² + b²σ_Y² = given variance (quadratic); substitute; solve; discard solutions that violate the stated domain (here, integers).

Instances:

Traps. [RPT18 E1]: "Common problems included failing to reject the non-integer solution and only stating the solution with minimal or no working. Students are reminded that in a question worth more than one mark, appropriate working must be shown. Algebraic errors were common, with some students having difficulty solving a quadratic equation. Quite a few students 'squared' both sides of the first equation to obtain 4a² + 4b² = 100" — that is, squaring 2a + 2b = 10 termwise and losing the cross term.

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Type 4 — Difference of two independent random variables ("differ by less than / by more than")

VCAA wording (2022 Exam 2 Section A Q18, [PAPERS]): "The time taken, T minutes, for a student to travel to school is normally distributed with a mean of 30 minutes and a standard deviation of 2.5 minutes. Assuming that individual travel times are independent of each other, the probability, correct to four decimal places, that two consecutive travel times differ by more than 6 minutes is …"

Really testing: (i) Var(X₁ − X₂) = σ² + σ²= 2σ², not zero and not σ² − σ²; (ii) that "differ by" is a two-sided event.

Method: let D = X₁ − X₂. Then E(D) = 0 and Var(D) = 2σ². "Differ by less than k" is Pr(−k < D < k); "differ by more than k" is Pr(|D| > k) = 2·Pr(D > k) by symmetry.

Instances:

Traps. In 2022 Exam 2 Q18, option A = 0.0448 is exactly half the correct answer — the one-tailed value — and took 16%. This is the single cleanest illustration of the two-sided reading. [RPT24 E1] on Q6c: "This required finding the expected value and the variance (or going directly to the standard deviation) of the difference of the two random variables. … some students gave the correct final answer with little or no evidence of appropriate working."

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Type 5 — Difference of two sample means

VCAA wording (2019 Exam 2 Section B Q6b, [PAPERS]): "Assume that the machine is working properly. Find the probability that the means of the two random samples differ by less than 2 grams. Give your answer correct to three decimal places."

Really testing: composing Type 4 with the sample-mean standard deviation — Var(X̄₁ − X̄₂) = σ²/n₁ + σ²/n₂.

Method: Var(X̄ᵢ) = σ²/nᵢ; add the two variances; centre at zero; take the two-sided probability.

Instances:

Traps. This is the hardest single thing in the area. [RPT19 E2] on Q6b: "Many students did not make a reasonable start to this question, or they were unable to correctly find the variance for the combined distributions. Very few students indicated an understanding that the difference between the samples could be negative and found Pr(X̄₁ − X̄₂ < 2) rather than correctly finding Pr(−2 < X̄₁ − X̄₂ < 2) or Pr(|X̄₁ − X̄₂| < 2)." The 2019 report named "working with linear combinations of sample means" as an area of weakness for the whole paper. Note the bimodal mark distribution: 67% scored zero, 15% scored full — almost nobody scored 2 of 3. Either you see the structure or you do not.

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Type 6 — Scaling a random variable, then summing

VCAA wording (2024 Exam 2 Section A Q20, [PAPERS]): "The masses of avocados in a crop may be assumed to be normally distributed, with a mean of 200 grams and a standard deviation of 7.5 grams. After an avocado of mass M grams is peeled and the stone is removed, the mass of edible flesh F grams is given by F = 0.70M. Four avocados are randomly selected from the crop. What is the probability, correct to four decimal places, that a total of more than 570 grams of edible flesh is obtained?"

Really testing: applying Var(aX) = a²Var(X) first, then the i.i.d. sum rule — two rules stacked.

Method: F ~ N(0.7μ, (0.7σ)²). Then ΣFᵢ ~ N(4·0.7μ, 4·(0.7σ)²). Here: mean 560, sd 2 × 5.25 = 10.5, so Pr(Z > 10/10.5) = 0.1705.

Instances:

Traps. [RPT22 E2] names this as an area of weakness for the whole paper: "working with random variables that are functions of other variables." The 2019 Exam 1 chocolate question is the purest example: V = πr²L = (π/4)L, so Var(V) = (π/4)²Var(L) = π²/1600 — but S = 2πr² + 2πrL = π/2 + πL is not a pure scaling, and E(S) = π/2 + 3π = 7π/2. [RPT19 E1] on Q3b: "Students could use fractions to find [the variance]. Students who used this approach tended to score more highly than those using decimals, who sometimes were not able to evaluate correctly. A number of students omitted the π from their answer." On Q3c: "Some students were unable to evaluate correctly."

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Type 7 — n·X versus X₁ + X₂ + … + Xₙ

VCAA wording (2022 Exam 1 Q3a, [PAPERS]): "The time taken by a coffee machine to dispense a cup of coffee varies normally with a mean of 10 seconds and a standard deviation of 1.5 seconds. Find the probability that more than 34 seconds is needed to dispense a total of four cups of coffee. Give your answer correct to two decimal places."

Really testing: that four independent cups give variance 4σ² (sd 2σ = 3), not 16σ² (sd 4σ = 6).

Method: sum of n i.i.d.: mean nμ, variance nσ², sd σ√n. Contrast Var(4X) = 16σ².

Instances:

Traps. [RPT21 E2] on Q6b: "There was evidence of confusion between the correct sum of four random variables and incorrectly scaling a random variable by a factor of four." [RPT22 E1] on Q3a: "A large number of students did not find the correct standard deviation and so were unable to move towards evaluating [the probability] … Students who successfully evaluated [it] often drew diagrams of the probability density function and were aware of the approximate probabilities for a normal distribution." In 2023 Exam 2 Q19, 17% chose option C (0.413), which is what you get from sd = 16 × 200 = 3200 — multiplying by n instead of √n.

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Type 8 — Inverse problem on a sum: the largest n

VCAA wording (2021 Exam 2 Section B Q6a, [PAPERS]): "The maximum load of a lift in a chocolate company's office building is 1000 kg. The masses of the employees who use the lift are normally distributed with a mean of 75 kg and a standard deviation of 8 kg. On a particular morning there are n employees about to use the lift. What is the maximum possible value of n for there to be less than a 1% chance of the lift exceeding the maximum load?"

Really testing: recognising that the total, not the mean, is the random variable, and then solving an inequality in n where n appears in both the mean (75n) and the sd (8√n).

Method: T ~ N(75n, 64n). Require Pr(T > 1000) < 0.01, i.e. (1000 − 75n)/(8√n) > 2.3263. Solve numerically or by trial; n = 12 works, n = 13 does not.

Instances:

Traps. [RPT21 E2] on Q6a: "Successful students used a trial-and-error approach or used a standardised value to solve for n. A common error was to approach this as a sampling problem" — that is, using σ/√n for the sample mean instead of σ√n for the total. Note the mark distribution: 73% scored zero.

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Type 9 — Write down the mean and standard deviation of X̄

VCAA wording (2025 Exam 2 Section B Q6a.i, [PAPERS]): "The volume of water, V mL, consumed by a student during a school day may be assumed to be normally distributed with a mean of 1000 mL and a standard deviation of 80 mL. Write down the mean and standard deviation of the sampling distribution for the average volume of water consumed by randomly selected samples of 25 students. Give your answers in millilitres."

Really testing: E(X̄) = μ, sd(X̄) = σ/√n. Nothing else.

Method: divide the population standard deviation by √n. The mean is unchanged.

Instances:

Traps. [RPT25 E2]: "Some students wrote the variance rather than the standard deviation." [RPT18 E2] on Q6b: "A variety of correct, exact forms were accepted" — on Examination 2, 15/√50 = 3√2/2 and 1.5√2 and 2.12 are all acceptable unless a form is specified. This is a free mark and the highest-scoring type in the whole area.

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Type 10 — Standardising a sample mean: Pr(X̄ > k) or Pr(a < X̄ < b)

VCAA wording (2017 Exam 1 Q4, [PAPERS]): "The volume of soft drink dispensed by a machine into bottles varies normally with a mean of 298 mL and a standard deviation of 3 mL. The soft drink is sold in packs of four bottles. Find the approximate probability that the mean volume of soft drink per bottle in a randomly selected four-bottle pack is less than 295 mL. Give your answer correct to three decimal places."

Really testing: on Examination 1, whether the candidate divides by √n and can then evaluate the tail from the 68–95–99.7 rule or a supplied value.

Method: Z = (x̄ − μ)/(σ/√n). On Examination 1 the stem supplies whatever normal value is needed (Pr(Z < 1) = 0.84, Pr(−1 < Z < 1) = 0.68, Pr(−1.96 < Z < 1.96) = 0.95); on Examination 2 use the CAS normal CDF.

Instances:

Traps. [RPT17 E1] is the most detailed diagnosis in the whole archive and worth quoting nearly in full: "This question was answered well by students who found the standard deviation of the sample, but many used the standard deviation of the population. Students' notation was often not clear and did not distinguish between the standard deviation of X and the standard deviation of X̄. Some arithmetic errors were made when dividing by √4. Other typical errors included: not working with the mean, leading to finding Pr(X < 295) using the total volume and taking the standard deviation to be 12 rather than 6 … finding z = +2 by incorrect standardisation … Some used the 68% or 99.7% approximation instead of 95%, while others made attempts to find a confidence interval."

In 2021 Exam 2 Q17, option B (0.8413) drew 20% — that is Pr(Z > −1), the answer for a single bottle, i.e. forgetting √6 entirely. That one distractor is the whole question.

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Type 11 — The Central Limit Theorem clause

VCAA wording (2025 Exam 1 Q4b, [PAPERS]): "For random samples of 25 waiting times, it may be assumed that the sample means are approximately normally distributed. Find the probability that the average waiting time for a random sample of 25 patients is between 0.44 hours and 0.5 hours. Use s = 0.3 and Pr(Z < 1) = 0.84."

Really testing: that a non-normal parent distribution is legitimate provided n is large enough, and that VCAA will hand you the licence in the stem rather than ask you to state the theorem.

How VCAA uses the CLT. The study design's only reference is "its approximate normality if n is large", and the confidence-interval dot point puts a number on "large": n ≥ 30 "in many practical contexts". The archive contains no question asking a candidate to state or name the Central Limit Theorem. Instead:

• when the parent is normal, the stem says so ("may be assumed to be normally distributed") and no appeal to the CLT is needed at any sample size;
• when the parent is not normal, VCAA states the normality of the sample mean as a given, as in 2025 Exam 1 Q4b above (the parent there is a continuous distribution on [0, 1] with pdf f(t) = 3/(2·logₑ2)·1/((t+1)(2−t)), which is not normal), or in 2019 Exam 2 Section A Q20, where the parent is the uniform distribution on [0, 1] (μ = 0.5, σ = 0.2887) and n = 100.

Instances: 2025 Exam 1 Q4b (49%), 2019 Exam 2 Section A Q20 (70%), 2025 Exam 1 Q4a (37% — the pdf work that sets it up).

Trap. Do not volunteer a CLT justification when the parent is already normal; it costs time and earns nothing. Do check n when the parent is not normal and the stem has not granted normality — and note that no archive question has yet withheld that grant.

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Type 12 — Constructing a confidence interval with σ known

VCAA wording (2025 Exam 2 Section B Q6b, [PAPERS]): "The manufacturer of Wasser bottled water knows that the volume of water dispensed into bottles may be assumed to be normally distributed with a standard deviation of 5 mL. Engineers at the company take a random sample of 30 bottles and measure the volume of water in each bottle. The sample mean is found to be 750 mL. Find a 95% confidence interval for the mean volume of water dispensed into each Wasser bottle. Give your values in millilitres, correct to one decimal place."

Really testing: mechanical accuracy and the reporting convention.

Method: (x̄ − z·σ/√n, x̄ + z·σ/√n) with z = 1.6449 (90%), 1.96 (95%), 2.5758 (99%). On Examination 2, use the CAS zInterval with Data: Stats. Report as an ordered pair in brackets to the stated accuracy.

Instances:

Traps. [RPT23 E2] on Q6a: "This routine question was handled well. Many students recognised that the confidence interval should be expressed with brackets in the form (a, b)." [RPT18 E2] on Q6g: "Some students appeared to use a 95% confidence interval rather than the required 99% confidence interval. It is important for students to read questions carefully." [RPT21 E1] on Q3c: "Students frequently used σ rather than σ/√n. Arithmetic errors were also observed, as was the use of [a different z] rather than [the given one] as instructed."

The Examination 1 versions are the separators, because there is no CAS and the z value must come from the stem.

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Type 13 — Constructing an approximate confidence interval from a sample standard deviation

VCAA wording (2016 Exam 2 Section A Q19, [PAPERS]): "A random sample of 100 bananas from a given area has a mean mass of 210 grams and a standard deviation of 16 grams. Assuming the standard deviation obtained from the sample is a sufficiently accurate estimate of the population standard deviation, an approximate 95% confidence interval for the mean mass of bananas produced in this locality is given by …"

Really testing: that the s-version of the interval is used only when n is large, and that the structure is otherwise identical.

Method: identical to Type 12 with s in place of σ. There is no t-distribution anywhere in VCE Specialist Mathematics; the quantile is always normal.

Instances: 2016 Exam 2 Section A Q19 (78%, answer B), 2019 Exam 2 Section A Q18 (76%, answer D, a 98% interval so z = 2.326), 2023 Exam 2 Section A Q19 (NHT, answer — a 95% interval from n = 50, s = 8), 2018 Exam 2 Section B Q6g (52%).

Trap. In 2016 Exam 2 Q19 the distractors are built from the wrong divisor: option A (178.7, 241.3) uses s undivided; option C (209.2, 210.8) uses s/n. Only 7% and 9% respectively fell for them, which is why this is one of the few statistics MCQs above 75%.

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Type 14 — Reverse confidence interval: recover x̄, σ or n from the interval

VCAA wording (2023 Exam 2 Section A Q20, [PAPERS]): "The lifespan of a certain electronic component is normally distributed with a mean of μ hours and a standard deviation of σ hours. Given that a 99% confidence interval, based on a random sample of 100 such components, is (10500, 15500), the value of σ is closest to …"

Really testing: that the interval is symmetric about x̄ and that its half-width is z·σ/√n.

Method: x̄ = (L + U)/2; half-width h = (U − L)/2; then σ = h√n / z or n = (zσ/h)².

Instances:

Traps. The distractor family is always "forgot to multiply back by √n". Note that VCAA has moved this type onto Examination 1 in the NHT papers, where the z value is supplied in the stem and the arithmetic must be exact — 2026 NHT Exam 1 Q6 says "Use Pr(Z < 1.645) = 0.95 in your calculations", which is the giveaway that a 90% interval is intended (one-sided 0.95 → two-sided 0.90).

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Type 15 — The effect of sample size on interval width

VCAA wording (2017 Exam 2 Section A Q19, [PAPERS]): "A confidence interval is to be used to estimate the population mean μ based on a sample mean x̄. To decrease the width of a confidence interval by 75%, the sample size must be multiplied by a factor of …"

Second wording (2023 Exam 2 Section B Q6c, [PAPERS]): "The ranger wants to decrease the width of the 95% confidence interval by 60% to get a better estimate of the population mean. How many adult male koalas should be sampled to achieve this?"

Really testing: width ∝ 1/√n, and the difference between "decrease the width by 60%" (new width is 40% of old) and "decrease the width to 60%".

Method: if the width is to be multiplied by k, then √n must be multiplied by 1/k, so n_new = n_old / k². Round up to an integer.

Instances:

Traps. [RPT23 E2] names this type as an area of weakness for the whole 2023 paper: "determining sample size to achieve a given change in the width of a confidence interval." The report's own comment on Q6c is terse: "This question was challenging for students." In 2017 Exam 2 Q19, the distractors are 2, 4, 9, 25 — every one of them a plausible mis-square. This type is, on the evidence, the most reliable single separator in the whole area of study: 44%, 28%, and its close cousin Type 16 at 46% and 53%.

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Type 16 — Minimum sample size for a stated margin of error

VCAA wording (2024 Exam 2 Section B Q6g, [PAPERS]): "What minimum size sample should be used so that, with 95% confidence, the sample mean is within 1 mL of the population mean volume dispensed by the new machine? Assume a population standard deviation of 4 mL."

Second wording (2025 Exam 2 Section B Q6d, [PAPERS]): "What is the minimum size of the sample required to ensure that the difference between the sample mean and the mean volume dispensed is no more than 1 mL at the 95% confidence level?"

Really testing: solving z·σ/√n ≤ E for n, and rounding in the right direction.

Method: n ≥ (zσ/E)², then round up.

Instances:

Traps. [RPT25 E2] on Q6d is exact about the rounding: "97. Found solving the inequality … Some responses rounded down to quote 96, but this would have resulted in more than 1 mL." [RPT24 E2] gives the model answer as "The confidence interval extends [1 mL] each side of the mean, so solve [the inequality]. This yields [a non-integer]. As n is an integer, then n = 62."

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Type 17 — The simulation-count question ("in how many of these intervals…")

VCAA wording (2024 Exam 2 Section B Q6f, [PAPERS]): "Forty samples, each consisting of 50 randomly chosen bottles, are taken, and a 95% confidence interval is calculated for each sample. In how many of these confidence intervals would the population mean volume dispensed by the machine be expected to lie?"

Really testing: the frequentist meaning of a confidence level — the study design's dot point "the use of simulation to illustrate variations in confidence intervals between samples and to show that the likelihood of a confidence interval containing μ depends on the level of confidence chosen".

Method: multiply the confidence level by the number of samples (not the sample size), then round sensibly to an integer.

Instances:

Traps. [RPT24 E2] on Q6f: "Some students incorrectly used 50 rather than 40 as the number of samples." That is the entire question — every stem deliberately puts two numbers in play (the number of samples and the size of each sample) and the marker is checking which one you multiplied. [RPT25 E2] on Q6c: "285. Found by seeking 95% of 300."

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Type 18 — State H₀ and H₁

VCAA wording (2023 Exam 2 Section B Q6d, [PAPERS]): "It is thought that the mean mass of adult male koalas in the forest is 12 kg. The ranger thinks that the true mean mass is less than this and decides to apply a one-tailed statistical test. … Write down the null hypothesis, H₀, and the alternative hypothesis, H₁, for the test."

Really testing: that H₀ is always an equality about the population mean μ, that H₁ carries the direction named in the stem, and that the notation is the notation.

Method: H₀: μ = μ₀. Then H₁: μ > μ₀, H₁: μ < μ₀ or H₁: μ ≠ μ₀ according to the stem's words. Never write H₀: x̄ = …; never write H₀: μ > ….

Instances:

Traps. [RPT16 E2]: "Some students failed to use appropriate notation or state the hypotheses clearly. The alternate hypothesis was occasionally written for a two-tail test, that is H₁: μ ≠ 1.1." [RPT18 E2]: "Common errors included: poor notation such as H₀ = 150 or similar, and not understanding the nature of a one-tailed test, evidenced by answers such as H₁: μ ≠ 150." [RPT19 E2] (where the test was two-tailed): "While most students correctly stated the null and alternative hypotheses for a two-tailed test, answers indicating a one-tailed test were relatively frequent." [RPT21 E1]: "The alternative hypothesis was sometimes written with the incorrect inequality (< or >) and some idiosyncratic notation was observed." [RPT22 E2]: "Some incorrect responses involved hypotheses for a two-tailed test."

The trend is the headline. 70–75% in 2016–2021, 85–91% in 2022–2025. This is now a near-free mark, and dropping it in 2026 would be a significant self-inflicted wound.

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Type 19 — Identify the correct hypothesis pair (multiple choice)

VCAA wording (2021 Exam 2 Section A Q18 (NHT), [PAPERS]): "It is believed that the average life span of people in Okinawa is 80 years. A random sample of the life spans of 200 of these people had an average of 84 years. A one-tailed statistical test, using a 5% level of significance, is to be carried out to investigate if this result provides evidence that the average life span of people in Okinawa is greater than 80 years. Which one of the following statements would form suitable null and alternative hypotheses for this test?" Options include H₀: μ > 80, H₁: μ > 84; H₀: μ = 84, H₁: μ > 80; and the correct H₀: μ = 80, H₁: μ > 80.

Really testing: that H₀ uses the claimed value and not the observed sample mean.

Method: identify μ₀ from the claim, not from the data. The sample mean never appears in a hypothesis.

Instances: 2021 Exam 2 Section A Q18 (NHT); 2022 Exam 2 Section A Q19 (NHT) — "which one of the following statements is necessarily correct?", a conceptual variant mixing sampling-distribution and confidence-interval misconceptions in one item.

Trap. Every distractor in the 2021 NHT item plants 84 — the sample mean — in one of the two hypotheses.

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Type 20 — Compute the p-value

VCAA wording (2018 Exam 2 Section B Q6c, [PAPERS]): "Write down an expression for the p value of the statistical test and evaluate your answer correct to four decimal places."

Really testing: (i) the correct tail; (ii) the sample-mean standard deviation σ/√n; (iii) the conditional notation Pr(X̄ < k | μ = μ₀).

Method: one-tailed p = Pr(X̄ < x̄ | μ = μ₀) (lower) or Pr(X̄ > x̄ | μ = μ₀) (upper); two-tailed p = 2 × the smaller tail. On Examination 2, CAS normCdf with σ/√n, or zTest.

Instances:

Traps. [RPT18 E2] is the notation report: "Most students obtained the correct value of p. A small number of these did not write p to the required four decimal places. Some students inappropriately used calculator syntax in place of correct working or notation. Students must take care with notation as some responses incorrectly stated that p = Pr(X < 145 | μ = 150)" — i.e. X where X̄ was meant. [RPT16 E2]: "Transcription errors caused some students to miss out on marks, with answers such as 0.009 occurring [for 0.0009]. High-scoring answers using the z-distribution were prevalent." [RPT24 E2] on Q6b.i: "Some students did not divide the standard deviation by 3 to account for the sample size." In 2019 Exam 2 Section A Q20, option E (0.9525) is the z-value 0.9525 misread as a probability.

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Type 21 — State the conclusion, with a reason, in context

VCAA wording (2025 Exam 2 Section B Q6f.ii, [PAPERS]): "Is the company's claim correct? Explain your conclusion in terms of the p value."

Really testing: whether the candidate writes all three of (a) the numerical comparison, (b) the decision about H₀, (c) the meaning in the words of the stem.

Method: see §3.3 for the complete template. If p < α, reject H₀; if p ≥ α, do not reject H₀. State the comparison with both numbers, then translate.

Instances:

Traps. This is the most-reported failure mode in the entire area, and the wording of the complaint has barely changed in ten years:

• [RPT16 E2]: "Some students did not explicitly test at the 5% level of significance. Two-tail approaches appeared occasionally."
• [RPT17 E2]: "Some students [did] not continue to explicitly answer the question or state a correct conclusion."
• [RPT18 E2]: "Some students did not supply a reason for their conclusion as required by the question. Occasional errors caused some students to miss out on the mark. For example, some responses incorrectly stated that 0.0092 > 0.05."
• [RPT21 E2]: "Some students stated a correct conclusion but did not give a reason by referencing the p-value."
• [RPT22 E2]: "Generally well done but some students did not justify their response with reference to the p value."
• [RPT23 E2]: "Some students stated a correct conclusion but did not give a reason by referencing the p value."
• [RPT24 E2]: "This question was generally well done; however, some students did not fully answer the question regarding whether or not the machine should be paused."
• [RPT25 E2]: "Responses needed to comment on the company's claim and also quote the significance level."
• [RPT21 E1]: "A number of students drew an incorrect conclusion from the p value. This was sometimes due to students confusing the p value with the significance level."

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Type 22 — The p-value decision rule (multiple choice)

VCAA wording (2017 Exam 2 Section A Q20, [PAPERS]): "In a one-sided statistical test at the 5% level of significance, it would be concluded that A. H₀ should not be rejected if p = 0.04 B. H₀ should be rejected if p = 0.06 C. H₀ should be rejected if p = 0.03 D. H₀ should not be rejected if p ≥ 0.05 E. H₀ should not be rejected if p = 0.01"

Really testing: the direction of the inequality, in the abstract, with no numbers to compute.

Instances: 2017 Exam 2 Section A Q20 (77%, answer C; A 5 / B 9 / C 77 / D 5 / E 3).

Trap. Option D is true but is not the "concluded that" the question wants — it is the general rule rather than an instance, and VCAA's key is C. This is the only archive instance of the type; its 77% makes it the highest-scoring hypothesis-testing MCQ in the corpus.

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Type 23 — The critical value of the sample mean (one-tailed)

VCAA wording (2025 Exam 2 Section B Q6g, [PAPERS]): "At the 1% level of significance for a sample size of 50 bottles, find the critical value of the sample mean, below which a sample mean value would support the conclusion that the mean volume of water dispensed is now less than 750 mL. Give your answer correct to three decimal places."

Second wording (2016 Exam 2 Section B Q6d, [PAPERS]): "For this test, what is the smallest value of the sample mean that would provide evidence that the mean level of pollutant has increased? That is, find x_c such that Pr(X̄ > x_c | μ = 1.1) = 0.05."

Really testing: inverse-normal work at the sample-mean scale, plus tail identification.

Method: solve Pr(X̄ < x_c | μ = μ₀) = α (lower-tail test) or Pr(X̄ > x_c | μ = μ₀) = α (upper-tail test) with sd = σ/√n. On CAS: invNorm(α, μ₀, σ/√n) for a lower tail, invNorm(1 − α, μ₀, σ/√n) for an upper tail.

Instances:

Traps. [RPT19 E2] on Q6f: "This question was often not attempted. Most students who did attempt it answered correctly. The most frequent incorrect response was 372.5, resulting from Pr(X̄ < x_c) = 0.05" — the test was two-tailed at 5%, so the correct tail probability is 0.025. [RPT25 E2] on Q6g: "Most responses identified the critical value for this significance level. Some responses used the wrong tail of the distribution." [RPT22 E2] on Q6d: "Some transcription errors were apparent, giving 14.59 as the answer [for 14.95]." [RPT16 E2] on Q6d: "Rounding errors caused some students to miss out on the mark."

Note the wording variety and read it slowly: "smallest value … for H₀ to be not rejected", "smallest sample mean above which the inspectors would not reject", "largest mean concentration … for the null hypothesis not to be rejected", "critical value … below which a sample mean would support the conclusion". They are all the same computation with the tail and the inequality flipped by the context.

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Type 24 — The critical region for a two-tailed test, or the rejection range

VCAA wording (2024 Exam 2 Section B Q6d, [PAPERS]): "Let X̄ denote the sample mean of a random sample of nine bottles. As a quality-control measure, the machine will be paused if X̄ < a or if X̄ > b, where Pr(X̄ < a) = 0.01 and Pr(X̄ > b) = 0.01. Assume μ = 1000 mL and σ = 4.2 mL. Find the values of a and b correct to one decimal place."

Second wording (2021 Exam 2 Section B Q6d, [PAPERS]): "Find the range of values for the mean daily sales of another 14 randomly selected days that would lead to the null hypothesis being rejected when tested at the 1% level of significance. Give your answer correct to the nearest integer."

Really testing: producing a region, not a number — and in the 2021 case, realising the answer must be stated as an inequality or interval.

Instances:

Trap. [RPT21 E2] on Q6d is the definitive one: "Some students calculated 63,108.7 but did not proceed to answer the question correctly as a range of values." They found the critical value and stopped. A 1-mark question at 16% is almost always a question where the state computed the right number and answered the wrong question.

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Type 25 — Type II error probability

VCAA wording (2024 Exam 2 Section B Q6c, [PAPERS]): "Assuming that the mean volume dispensed by the machine each time is in fact 997 mL and not 1000 mL, find the probability of a type II error for the test using nine bottles at the 5% level of significance. Assume that the population standard deviation is 4.2 mL, and give your answer correct to two decimal places."

Second wording, unlabelled (2021 Exam 2 Section B Q6e, [PAPERS]): "The advertising campaign has been successful to the extent that the mean daily sales is now 63000. A statistical test is applied at the 5% level of significance. Find the probability that the null hypothesis would be incorrectly accepted, based on the sales of another 14 randomly selected days and assuming a standard deviation of 5000."

Third wording, fully contextual (2025 Exam 2 Section B Q6h, [PAPERS]): "Assume that, after the service, the true mean volume of water in the Apa bottles was found to be 747.5 mL and that the population standard deviation, σ, is 5 mL. At the 1% level of significance, for a sample size of 50, find the probability that the company will conclude that the service has not reduced the mean volume of water in an Apa bottle."

Really testing: the two-step structure — a critical value computed under H₀, then a probability computed under the true mean.

Method: 1. Find the critical sample mean x_c from μ₀, σ/√n and α (this is Type 23). 2. Compute β = Pr(X̄ falls in the non-rejection region | μ = μ_true) using the same σ/√n but the new mean.

For a lower-tail test: β = Pr(X̄ > x_c | μ = μ_true). For an upper-tail test: β = Pr(X̄ < x_c | μ = μ_true).

Instances:

Traps. [RPT24 E2] on Q6c lists them exactly: "Common errors were: students sometimes did not find the critical value for X̄ when H₀ is true; students used the wrong tail." [RPT25 E2] on Q6h: "Most students were able to find this Type II error if they were successful in part g" — the two parts are chained, and a wrong critical value propagates. [RPT18 E2] on Q6f (11%): "Only a small number of students attempted this question."

This is the hardest reliably-recurring type in the area, and since 2023 VCAA has used the label "type II error" as well as the circumlocutions, so a candidate who knows the term is now rewarded directly.

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Type 26 — Type I / Type II error definition, and the diagram

VCAA wording, definition (2024 Exam 2 Section A Q19, [PAPERS]): "When conducting a hypothesis test, a type II error occurs when A. a null hypothesis is not rejected when the alternative hypothesis is true. B. a null hypothesis is rejected when it is true. C. a null hypothesis is rejected when the alternative hypothesis is true. D. a null hypothesis is not rejected when it is doubtful."

VCAA wording, diagram (2023 Exam 2 Section B Q6h, [PAPERS]): "The frequency curves for the sampling distributions associated with H₀ and H₁ are shown below. Label the critical sample mean on the diagram and shade the region that represents the type II error."

Really testing: the study design's deepest dot point — "hypothesis test, relating the formulation, conduct, errors and results in terms of conditional probability". Type I is Pr(reject H₀ | H₀ true) = α; Type II is Pr(do not reject H₀ | H₀ false) = β.

Instances:

Trap. Option B in the 2024 item is the Type I definition and took 8%; option D ("doubtful") is not a statistical statement at all and took 15% — more than the genuine confusion. Read all four.

The diagram version is worth practising even though the live instance was invalidated, because the NHT paper set it in 2023 and it is the natural graphical expression of the conditional-probability dot point: two normal curves with the same standard deviation σ/√n, centred at μ₀ and μ_true, a vertical line at x_c, and β the area under the H₁ curve on the non-rejection side.

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Type 27 — Expected value of a continuous random variable by integration

VCAA wording (2025 Exam 1 Q4a, [PAPERS]): "The waiting time, T hours, to see a particular doctor at a clinic has a distribution with a probability density function f defined by f(t) = 3/(2·logₑ2) · 1/((t + 1)(2 − t)) for 0 ≤ t ≤ 1, and 0 elsewhere. Use integration to show that E(T) = ½."

Really testing: that Mathematical Methods Units 3 and 4 is assumed knowledge, so a continuous-random-variable integral can open a Specialist statistics question without warning — and that it can be made hard by requiring partial fractions.

Method: E(T) = ∫ t·f(t) dt over the support. Here t/((t+1)(2−t)) must be split into partial fractions before integrating.

Instances: 2025 Exam 1 Q4a (37%, 3 marks, dist 32/23/9/37, average 1.5); it then feeds 2025 Exam 1 Q4b (49%).

Traps. [RPT25 E1]: "Students should know how to find the expected value (mean) of a continuous random variable. This question required partial fractions to be applied. A small number of students did not realise this and were unable to progress with the problem. A significant number of responses were not awarded full marks, either because: the initial expression missed the t term on the numerator, or the coefficients for the partial fractions were incorrect, or the working towards the given answer was unclear." The last clause matters: this is a "show that", so the answer is given and only the working earns marks.

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3. The standard wordings

VCAA reuses a small number of sentences almost verbatim. Learn them as triggers; each one names a specific deliverable.

3.1 The hypothesis-statement sentence

What it demands. Two lines, in the notation H₀: and H₁:, each about μ, with H₀ an equality. Nothing else — no p-value, no conclusion. The word "suitable" is not an invitation to editorialise.

The direction is always in the stem's preceding sentence. "to test whether the mean level of pollutant has increased" → H₁: μ > 1.1. "The ranger thinks that the true mean mass is less than this" → H₁: μ < 12. "to test whether the machine is working properly and is still producing packets with a mean mass of 375 grams. … A two-tailed test at the 5% level of significance is to be carried out" → H₁: μ ≠ 375. When VCAA wants two tails it says "two-tailed"; when it wants one it says "one-tailed" or "one-sided" and names a direction.

3.2 The critical-value sentence

What it demands. A number on the scale of the sample mean, to the stated accuracy. "Smallest … for H₀ not to be rejected" is an upper-tail critical value on a lower-tail test; "largest … for H₀ not to be rejected" is a lower-tail critical value on an upper-tail test. Draw the picture before touching the calculator.

3.3 The conclusion sentence, and what a complete conclusion must contain

What the reports say a complete conclusion must contain. Assembling the eight report comments quoted under Type 21, a full-mark conclusion has three components, and VCAA's published sample answers show all three:

1. The numerical comparison, with both numbers present. [RPT25 E2]'s model answer: "as 0.0023 < 0.01 (the significance level)". [RPT18 E2]'s: "As p = 0.0092 < 0.05, reject H₀." Writing only "p is small" or only "p < α" without the figures is where marks go. [RPT18 E2] also records responses that wrote 0.0092 > 0.05 — a comparison stated backwards.
2. The decision about H₀. "Reject H₀" or "do not reject H₀". VCAA's own older sample answers use "accept" ([RPT17 E2]: "p > 0.05, accept the dairy's claim") but the 2023–2025 answers use "reject / not reject", and "fail to reject" is safest.
3. The translation into the context, answering the literal question asked. [RPT24 E2]: "some students did not fully answer the question regarding whether or not the machine should be paused." [RPT25 E2]: "Responses needed to comment on the company's claim and also quote the significance level." [RPT19 E2]'s model answer: "As p < 0.05 reject H₀. The sample suggests the machine is not working properly."

A conclusion missing any one of the three has lost the mark in at least one archive year. The safe sentence is:

"As p = 0.0023 < 0.01, reject H₀ at the 1% level of significance. The sample provides evidence that the mean volume dispensed is less than 750 mL, so the company's claim is supported."

3.4 The p-value sentence

• "Find the p value for this test correct to three decimal places." (2024 E2 Q6b.i)
• "Determine the p value for this test. Give your answer correct to four decimal places." (2025 E2 Q6f.i; 2026 NHT E2 Q6d.i)
• "Write down an expression for the p value of the statistical test and evaluate your answer correct to four decimal places." (2018 E2 Q6c; 2019 NHT E2 Q6c; 2017 NHT E2 Q6b)

"Write down an expression … and evaluate" is a two-part instruction worth two marks: the conditional-probability statement p = Pr(X̄ < 145 | μ = 150) earns one and the value 0.0092 earns the other. Omitting the expression halves the mark.

3.5 The Type II error sentence

Four registers, all meaning the same thing:

1. Named: "find the probability of a type II error for the test" (2024 E2 Q6c; 2024 NHT E2 Q6f; [SAMPLE] E2 Q7).
2. Semi-named: "what is the probability that H₀ will be accepted at the 5% level of significance?" (2018 E2 Q6f); "find the probability that the null hypothesis would be incorrectly accepted" (2021 E2 Q6e).
3. Conditional-probability notation: "If the mean level of pollutant in the river, μ, is in fact 1.2 mg/L after the spill, find Pr(X̄ < 1.163 | μ = 1.2)." (2016 E2 Q6e).
4. Fully contextual, no statistical vocabulary at all: "find the probability that the company will conclude that the service has not reduced the mean volume of water in an Apa bottle" (2025 E2 Q6h); "find the probability that the snowboarder … will conclude that the wax is ineffective" (2026 NHT E2 Q6e); "what is the probability that the inspectors will not query the manufacturer's claim if the mean volume … is in fact 749.5 mL and not the claimed 750 mL?" (2023 NHT E2 Q6b.ii).

Register 4 is the dangerous one. Nothing in the sentence says "error", "hypothesis" or "Type II". The tell is always the phrase "is in fact" or "the true mean … is" followed by a value different from μ₀.

3.6 The simulation-count sentence

• "In how many of these confidence intervals would the actual mean mass … be expected to lie?" (2023 E2 Q6b)
• "In how many of these confidence intervals would the population mean volume dispensed by the machine be expected to lie?" (2024 E2 Q6f)
• "In how many of these confidence intervals would the engineers expect the value of the true mean volume dispensed to be included?" (2025 E2 Q6c)
• "If the manufacturer took 200 such samples, how many of the 99% confidence intervals would be expected to contain the value of the population mean?" (2025 NHT E2 Q6d)

3.7 Accuracy instructions, which are marks

Every statistics part in 2023–2025 carries an explicit accuracy instruction: "correct to four decimal places", "correct to three decimal places", "correct to one decimal place", "correct to the nearest integer", "give your answers in millilitres". [RPT24 E2] states the general rule: "Answers must be left in exact form unless a specific number of decimal places is required." [RPT17 E2] on Q6a: "many students either did not express their answer as a percentage or did not give the required level of accuracy"; on Q6c: "A significant proportion of otherwise correct answers were not given in the required form."

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4. The separators — every statistics part with pct ≤ 50

All 40, grouped by type, as ref — pct% — description.

4.1 Linear combinations and sums (12)

• 2021 Exam 2 Section B Q6a — 14% — maximum n so that Pr(total mass of n employees > 1000 kg) < 1%.
• 2017 Exam 2 Section B Q6d — 16% — maximum allowable machine standard deviation so that 99.9% of crates of 10 exceed 20 L.
• 2021 Exam 2 Section B Q6b — 24% — probability of getting a hot drink in time, fourth in a queue: sum of four i.i.d. dispensing times.
• 2022 Exam 1 Q3a — 31% — by hand: Pr(total time for four cups > 34 s).
• 2019 Exam 1 Q3b — 30% — variance of the volume of a chocolate cylinder, V = (π/4)L.
• 2024 Exam 1 Q6b — 37% — variance of 10W₁ + 20W₂ + 15W₃.
• 2022 Exam 2 Section B Q6e — 40% — masses of two randomly selected cans differ by no more than 3 g.
• 2022 Exam 2 Section A Q20 — 41% — pulley system with three randomly-massed blocks; Pr(the 4 kg block moves up).
• 2017 Exam 2 Section A Q18 — 42% — W = 4U − 3V; express Pr(W > 5) in terms of Z.
• 2022 Exam 2 Section A Q18 — 42% — two consecutive travel times differ by more than 6 minutes.
• 2018 Exam 1 Q4 — 43% — solve for integer a, b given E(aX + bY) = 10, Var(aX + bY) = 44.
• 2021 Exam 2 Section A Q20 — 43% — two coffee machines' times differ by less than 3 s.

4.2 Differences of independent variables and of sample means (3)

• 2019 Exam 2 Section B Q6b — 15% — two sample means of 50 differ by less than 2 g.
• 2024 Exam 1 Q6c — 45% — Pr(W₂ < W₁) by hand with Pr(−1 < Z < 1) = 0.68.
• 2024 Exam 2 Section A Q20 — 48% — total edible flesh of four avocados, F = 0.70M, exceeds 570 g.

4.3 The sample mean and standardising (6)

• 2019 Exam 2 Section B Q6a — 29% — Pr(at least one of two sample means of 50 lies between 370 g and 375 g) — a binomial layered on top of a sample-mean probability.
• 2017 Exam 1 Q4 — 37% — by hand: Pr(X̄ < 295) for a four-bottle pack.
• 2023 Exam 1 Q6b — 40% — by hand: express Pr(7 min 45 s < X̄ < 8 min 30 s) for n = 12 as Pr(a < Z < b) and find a and b.
• 2025 Exam 1 Q4b — 49% — by hand: Pr(0.44 < X̄ < 0.5) for n = 25 using Pr(Z < 1) = 0.84.
• 2019 Exam 1 Q3c — 50% — expected surface area of the chocolate cylinder (a non-scaling function of L).
• 2021 Exam 2 Section A Q17 — 50% — Pr(mean volume of six bottles ≥ 1.25 L).

4.4 Confidence intervals (4)

• 2016 Exam 1 Q2 — 17% — by hand: approximate 95% CI from a bag of 25 peaches with total mass 2625 g, "use an integer multiple of the standard deviation".
• 2023 Exam 2 Section B Q6c — 28% — sample size needed to decrease the interval's width by 60%.
• 2017 Exam 2 Section A Q19 — 44% — factor by which n must be multiplied to decrease the width by 75%.
• 2024 Exam 2 Section B Q6g — 46% — minimum n for the sample mean to be within 1 mL with 95% confidence.

4.5 p-values and conclusions (3)

• 2021 Exam 1 Q3b.i — 41% — by hand: the p-value for a one-tailed test with n = 36, using Pr(−3 < Z < 3) = 0.9973.
• 2017 Exam 2 Section B Q6e — 45% — carry out the whole one-sided test and state a reason (2 marks, conclusion included).
• 2021 Exam 1 Q3b.ii — 49% — "What should the company be told if the test was carried out at the 1% level of significance?"

4.6 Critical values and rejection regions (4)

• 2021 Exam 2 Section B Q6d — 16% — the range of sample means leading to rejection at 1%.
• 2019 Exam 2 Section B Q6f — 36% — smallest sample mean for H₀ not to be rejected, two-tailed at 5%.
• 2016 Exam 2 Section B Q6d — 43% — x_c such that Pr(X̄ > x_c | μ = 1.1) = 0.05.
• 2018 Exam 2 Section B Q6e — 48% — smallest sample mean height for H₀ to be not rejected.

4.7 Type II errors (5)

• 2021 Exam 2 Section B Q6e — 9% — Pr(H₀ incorrectly accepted) when the true mean is 63000 and α changes to 5%.
• 2018 Exam 2 Section B Q6f — 11% — Pr(H₀ accepted at 5%) when the true mean height is in fact 145 cm.
• 2023 Exam 2 Section B Q6g — 39% — Pr(type II error) when the true mean mass is 11.4 kg.
• 2024 Exam 2 Section B Q6c — 44% — Pr(type II error) when the true mean is 997 mL, n = 9, α = 5%.
• 2016 Exam 2 Section B Q6e — 46% — Pr(X̄ < 1.163 | μ = 1.2).

4.8 Continuous random variables (1)

• 2025 Exam 1 Q4a — 37% — "Use integration to show that E(T) = ½" for a pdf requiring partial fractions.

4.9 Parts tagged statistics in [QJSON] that are really the logistic differential equation (2)

These two are separators in the data and are listed for completeness, but they belong to Calculus, not to this area of study.

• 2023 Exam 2 Section B Q4e.i — 21% — express d²Q/dt² in terms of Q for the logistic model. [RPT23 E2]: "A common error involved not recognising the need to use the chain rule when differentiating with respect to t."
• 2023 Exam 2 Section B Q4g — 40% — maximum sustainable fish population when 5.5% is harvested annually.

4.10 Which types separate most

Ranking the types by the median pct of their graded instances:

What the reports say went wrong, in one paragraph. Across all 40 separators the reports name five failure modes and only five. (1) Not squaring the coefficients in a variance — [RPT24 E1], [RPT23 E1], and the 25% who took sd = 7 in 2017 Exam 2 Q18. (2) Using σ where σ/√n belongs, or vice versa — [RPT17 E1] ("many used the standard deviation of the population"), [RPT21 E1] ("Students frequently used σ rather than σ/√n"), [RPT21 E2] ("A common error was to approach this as a sampling problem"), [RPT24 E2] ("Some students did not divide the standard deviation by 3"). (3) The wrong tail, or one tail where two are needed — [RPT19 E2] ("Very few students indicated an understanding that the difference between the samples could be negative"; "The most frequent incorrect response was 372.5, resulting from Pr(X̄ < x_c) = 0.05"), [RPT25 E2] ("Some responses used the wrong tail of the distribution"), [RPT24 E2] ("students used the wrong tail"). (4) Computing the right number and then not answering the question — [RPT21 E2] ("did not proceed to answer the question correctly as a range of values"), [RPT24 E2] ("did not fully answer the question regarding whether or not the machine should be paused"), [RPT23 E2]/[RPT22 E2]/[RPT21 E2] on conclusions without a reason. (5) Rounding and transcription — [RPT16 E2] ("Rounding errors caused some students to miss out on the mark"; "answers such as 0.009 occurring"), [RPT22 E2] ("giving 14.59 as the answer"), [RPT25 E2] ("Some responses rounded down to quote 96").

None of these is a conceptual gap. All five are executional.

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5. What makes a hard one hard

5.1 The variance of a difference

Var(X − Y) = Var(X) + Var(Y). The plus sign is the whole difficulty. Nothing in the algebra of X − Y looks like addition, and the study design's own formula, Var(a₁X₁ + a₂X₂) = a₁²Var(X₁) + a₂²Var(X₂), hides the point behind the squares: a₂ = −1 gives a₂² = +1.

Three compounding layers appear in the archive:

1. Two independent variables — 2022 Exam 2 Q18 (42%), 2021 Exam 2 Q20 (43%).
2. Two independent variables with coefficients — 2017 Exam 2 Q18 (42%), where 4U − 3V has variance 16 + 9 = 25, and 25% of the state wrote sd = 4 + 3 = 7.
3. Two sample means — 2019 Exam 2 Q6b (15%), where Var(X̄₁ − X̄₂) = σ²/50 + σ²/50 = 2σ²/50.

The habit that survives pressure is: write the variance line before the standard-deviation line, always, and never write a standard-deviation sum.

5.2 One observation versus a sample mean

Every stem in this area contains a number that is either "the standard deviation of the population" or "the standard deviation of the sample mean", and the question is which one the event is about. The signals:

2021 Exam 2 Section A Q17 is the cleanest test of this reading: 50% got the sample-mean answer and 20% chose 0.8413, the single-bottle answer. [RPT17 E1] writes the diagnosis explicitly: "Students' notation was often not clear and did not distinguish between the standard deviation of X and the standard deviation of X̄."

The countermeasure is notational discipline. Write X ~ N(298, 3²) and X̄ ~ N(298, (3/2)²) as separate lines before computing anything. [RPT17 E2] named "careful use of notation in the statistics area of study" as an area of weakness for the whole 2017 paper.

5.3 Tail choice

Four separate tail decisions occur in this area and they are independent of each other:

1. The alternative hypothesis's direction. From the stem's verb: "has increased", "is less than", "differs significantly from".
2. One tail or two. VCAA always states it ("a one-tailed statistical test", "a one-sided statistical test", "A two-tailed test at the 5% level of significance is to be carried out"). If two, the p-value doubles and each critical value uses α/2.
3. Which side the critical value sits on. "Smallest value for H₀ not to be rejected" on a lower-tail test means the critical value is below μ₀ and the non-rejection region is above it.
4. Which side β is measured on. The Type II region is always the non-rejection region, measured under the true mean, so for a lower-tail test β = Pr(X̄ > x_c | μ = μ_true).

2019 Exam 2 Q6f (36%) is the archive's demonstration of how decisions 2 and 3 interact: the test was two-tailed at 5%, so the critical value uses 0.025, and "the most frequent incorrect response was 372.5, resulting from Pr(X̄ < x_c) = 0.05" [RPT19 E2].

Draw the curve. [RPT22 E1] observes that "Students who successfully evaluated [the probability] often drew diagrams of the probability density function"; [RPT25 E1] says of Q4b that "Some students drew a diagram to aid in identifying the required area under the standard normal curve." Two reports, five years apart, recommending the same thirty seconds of pencil work.

5.4 Writing a conclusion in context

This is the only part of the area of study where the mark is awarded for prose, and it is the most frequently reported loss. §3.3 gives the three required components. The specific failure patterns, with their archive evidence:

• Stating the decision without the comparison. "Reject H₀" alone. [RPT18 E2], [RPT21 E2], [RPT22 E2], [RPT23 E2] all record it.
• Stating the comparison without the context. "p < 0.05, so reject H₀" when the question asked whether the machine should be paused. [RPT24 E2].
• Omitting the significance level. [RPT16 E2]: "Some students did not explicitly test at the 5% level of significance." [RPT25 E2]: "Responses needed to comment on the company's claim and also quote the significance level."
• Confusing p with α. [RPT21 E1]: "A number of students drew an incorrect conclusion from the p value. This was sometimes due to students confusing the p value with the significance level."
• Reversing the inequality. [RPT18 E2]: "some responses incorrectly stated that 0.0092 > 0.05."

One 1-mark part, five distinct ways to lose it. Write the same three-clause sentence every time.

5.5 Rounding

The accuracy instruction is part of the question. Three distinct rounding hazards:

1. Stated decimal places. "correct to four decimal places" means 0.0023, not 0.002. [RPT16 E2] on Q6c.i: "Transcription errors caused some students to miss out on marks, with answers such as 0.009 occurring."
2. Direction of rounding for a sample size. n must be rounded up, always, because the inequality is n ≥ something. [RPT25 E2] on Q6d: "Some responses rounded down to quote 96, but this would have resulted in more than 1 mL." [RPT24 E2] on Q6g: "As n is an integer, then n = 62."
3. Intermediate rounding propagating. [RPT16 E2] on both Q6d and Q6e: "Rounding errors caused some students to miss out on the mark." Keep the unrounded value on the CAS stack and round only the reported answer. [RPT25 E2] on Q6h: "Most students were able to find this Type II error if they were successful in part g" — the chain means a rounded x_c becomes a wrong β.

And one presentational hazard: [RPT23 E2] notes that students who earned the confidence-interval mark "recognised that the confidence interval should be expressed with brackets in the form (a, b)".

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6. A worked method sheet

The nine highest-yield types, with CAS steps where the CAS matters. Notation follows the papers: μ population mean, σ population standard deviation, s sample standard deviation, x̄ observed sample mean, X̄ the sample-mean random variable, n sample size, α level of significance.

M1 — Mean and variance of a linear combination

Given independent X₁ … Xₙ and constants a₁ … aₙ:

E(a₁X₁ + … + aₙXₙ) = a₁E(X₁) + … + aₙE(Xₙ)
Var(a₁X₁ + … + aₙXₙ) = a₁²Var(X₁) + … + aₙ²Var(Xₙ)

Three lines, every time, in this order:

1. Var = a₁²σ₁² + a₂²σ₂² + … (numbers substituted)
2. Var = … (evaluated)
3. sd = √Var only if the question says standard deviation

If the variables are normal, the combination is normal — say so if the question asks you to justify using the normal distribution. [RPT24 E1]'s model answer for Q6c opens: "Let [D = W₂ − W₁]. As a linear combination of normally distributed variables, [D] is also normally distributed."

Worked: 2024 Exam 1 Q6b. C = 10W₁ + 20W₂ + 15W₃; Var(C) = 100(0.3²) + 400(0.4²) + 225(0.5²) = 9 + 64 + 56.25 = 129.25.

M2 — Sum of n i.i.d. variables versus n times one

ΣXᵢ  ~  N(nμ, nσ²)      sd = σ√n
nX   ~  N(nμ, n²σ²)     sd = nσ

Same mean, different spread. "Four cups of coffee are dispensed" is ΣXᵢ. "The order is quadrupled" would be 4X. Every archive instance has been ΣXᵢ.

Worked: 2022 Exam 1 Q3a. Four cups: mean 40, sd = 1.5√4 = 3. Pr(T > 34) = Pr(Z > (34 − 40)/3) = Pr(Z > −2) = 0.98.

M3 — Difference of two independent variables

D = X − Y  ⇒  E(D) = μ_X − μ_Y,  Var(D) = σ_X² + σ_Y²

"Differ by less than k" → Pr(−k < D < k). "Differ by more than k" → 1 − Pr(−k < D < k), or 2Pr(D > k) when E(D) = 0.

CAS: normCdf(−k, k, E(D), sd(D)).

Worked: 2026 NHT Exam 2 Q6a. Two descents, each N(200, 25). D ~ N(0, 50), sd = √50. Pr(|D| < 5) = normCdf(−5, 5, 0, √50).

M4 — The sampling distribution of X̄

X̄ ~ N(μ, σ²/n)   ⇒   sd(X̄) = σ/√n

If the parent is normal, this is exact at any n. If the parent is not normal, it is approximate and needs n large — but the archive always grants it in the stem.

CAS: normCdf(lower, upper, μ, σ/√n). Define σ/√n as a stored variable first. [RPT25 E2]: "Students must make sure variables are defined if they are being used in formulas."

Worked: 2025 Exam 2 Q6a.i–ii. μ = 1000, σ = 80, n = 25 → X̄ ~ N(1000, 16²); Pr(X̄ > 970) = normCdf(970, ∞, 1000, 16) = 0.9696.

M5 — Confidence interval for μ

(x̄ − z·σ/√n,  x̄ + z·σ/√n)     z = 1.6449 (90%), 1.96 (95%), 2.5758 (99%)

CAS (TI-Nspire): Statistics → Confidence Intervals → z Interval, set Data Input Method: Stats, enter σ, x̄, n, C Level. CAS (ClassPad): Interactive → Distribution/Inv. Dist → One-Sample Z Int, mode Variable.

On Examination 1 there is no CAS and the z value will be given in the stem — 2021 Exam 1 Q3c supplies Pr(−1.96 < Z < 1.96) = 0.95; 2026 NHT Exam 1 Q6 supplies Pr(Z < 1.645) = 0.95; 2016 Exam 1 Q2 instructs "use an integer multiple of the standard deviation", meaning z = 2.

Report as an ordered pair in brackets, to the stated accuracy.

M6 — Reverse confidence interval

Given (L, U):

x̄ = (L + U)/2
half-width h = (U − L)/2 = z·σ/√n
⇒  σ = h√n/z       or       n = (zσ/h)²

Worked: 2026 NHT Exam 1 Q6. (11.51, 18.09), σ = 6, z = 1.645. x̄ = 14.8; h = 3.29 = 1.645 × 6/√n → 6/√n = 2 → n = 9.

M7 — Sample size and interval width

width ∝ 1/√n

• To multiply the width by k: n_new = n_old / k².
• "Decrease the width by q%" means k = 1 − q/100.
• For a stated margin of error E: n ≥ (zσ/E)².
• Always round n up.

CAS: solve the inequality with solve(z·σ/√n ≤ E, n), then take the ceiling.

Worked: 2023 Exam 2 Q6c. n_old = 20, decrease by 60% → k = 0.4 → n_new = 20/0.16 = 125. Worked: 2025 Exam 2 Q6d. 1.96 × 5/√n ≤ 1 → √n ≥ 9.8 → n ≥ 96.04 → n = 97.

M8 — The hypothesis test, end to end

1. Hypotheses. H₀: μ = μ₀; H₁: μ >, < or ≠ μ₀ per the stem.
2. Sampling distribution under H₀. X̄ ~ N(μ₀, (σ/√n)²).
3. p-value. - upper-tail: p = Pr(X̄ > x̄ | μ = μ₀) = normCdf(x̄, ∞, μ₀, σ/√n) - lower-tail: p = Pr(X̄ < x̄ | μ = μ₀) = normCdf(−∞, x̄, μ₀, σ/√n) - two-tailed: p = 2 × the smaller of the two
4. Decision. p < α → reject H₀. p ≥ α → do not reject H₀.
5. Conclusion, three clauses. Comparison with both numbers; decision about H₀; translation into the stem's own words.

CAS shortcut: TI-Nspire Statistics → Stat Tests → z Test, Stats input, enter μ₀, σ, x̄, n, and the alternative. ClassPad: Interactive → Distribution/Inv. Dist → One-Sample Z-Test. Both return p directly. Write the conditional-probability expression by hand anyway — 2018 Exam 2 Q6c, 2019 NHT Exam 2 Q6c and 2017 NHT Exam 2 Q6b all ask for "an expression for the p value" as a separately-marked step, and [RPT18 E2] warns that "Some students inappropriately used calculator syntax in place of correct working or notation."

M9 — Critical value of the sample mean

lower-tail test:  x_c = invNorm(α,     μ₀, σ/√n)
upper-tail test:  x_c = invNorm(1 − α, μ₀, σ/√n)
two-tailed:       a  = invNorm(α/2,     μ₀, σ/√n),  b = invNorm(1 − α/2, μ₀, σ/√n)

Read the question's wording to decide which of x_c and its complement is wanted, and whether the answer is a number or a region.

Worked: 2025 Exam 2 Q6g. Lower-tail test, α = 0.01, μ₀ = 750, σ/√n = 5/√50. x_c = invNorm(0.01, 750, 5/√50) = 748.355. Worked: 2024 Exam 2 Q6d. Two-sided quality control, Pr(X̄ < a) = Pr(X̄ > b) = 0.01, μ₀ = 1000, σ/√n = 4.2/3. a = invNorm(0.01, 1000, 1.4), b = invNorm(0.99, 1000, 1.4).

M10 — Type II error probability

Two steps, and the second uses a different mean but the same standard deviation.

1. Critical value under H₀: x_c = invNorm(α, μ₀, σ/√n) (lower-tail) or invNorm(1 − α, μ₀, σ/√n) (upper-tail).
2. β under the true mean: - lower-tail test: β = Pr(X̄ > x_c | μ = μ_true) = normCdf(x_c, ∞, μ_true, σ/√n) - upper-tail test: β = Pr(X̄ < x_c | μ = μ_true) = normCdf(−∞, x_c, μ_true, σ/√n)

Sanity check. β should be large when μ_true is close to μ₀ and small when it is far away. If you get β = 0.9 for a true mean a long way from μ₀, you have used the wrong tail.

Worked: 2025 Exam 2 Q6g → Q6h. x_c = 748.355; μ_true = 747.5; σ/√n = 5/√50. β = Pr(X̄ > 748.355 | μ = 747.5) = 0.113. Worked: 2024 Exam 2 Q6c. n = 9, σ = 4.2, μ₀ = 1000, α = 0.05, lower-tail → x_c = invNorm(0.05, 1000, 1.4) = 997.697; μ_true = 997; β = Pr(X̄ > 997.697 | μ = 997) = 0.31.

M11 — The one-minute checklist before you leave Question 6

• [ ] Did I use σ/√n where the event is about a mean, σ√n where it is about a total, and σ where it is about one observation?
• [ ] Did I square every coefficient in every variance?
• [ ] Is the alternative hypothesis's inequality the one the stem's verb names?
• [ ] Is the test one-tailed or two-tailed, and did I double the p-value if two?
• [ ] Does my conclusion contain both numbers, the decision on H₀, and the stem's own noun?
• [ ] Did I round to the stated accuracy — and up, if the answer is a sample size?
• [ ] Is the confidence interval written as (a, b)?
• [ ] For a Type II error: did I compute the critical value under μ₀ and the probability under μ_true?

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7. Fast reference — every graded statistics part, 2016–2025

Separators in bold.

Plus the six parts of 2023 Exam 2 Section B Q4 mis-tagged into this area (Q4b 57%, Q4c 79%, Q4d 85%, Q4e.i 21%, Q4e.ii 57%, Q4g 40%), and the four NHT Section A items carried in [QJSON] without percentages (2024 Exam 2 Section A Q19 (NHT), Q20 (NHT); 2025 Exam 2 Section A Q19 (NHT), Q20 (NHT)).