Hypothesis testing is the capstone of the H2 statistics strand and appears in Paper 2 every year. It builds on sampling and the Normal distribution, using the distribution of the sample mean X̄ to decide, in a disciplined and quantitative way, whether a claim about a population mean μ deserves to be believed.
The calculation is almost entirely done on the GC, so marks are won and lost on language, setup and precision: choosing H₁ from the wording of the claim, justifying why a z-test is valid, and writing the conclusion in context with non-definitive language.
What these notes cover
- •The core logic of hypothesis testing
- •Null and alternative hypotheses
- •One-tail and two-tail tests
- •Test statistic, significance level and critical region
- •The p-value and the decision rule
- •Case 1: normal population, known variance
- •Case 2: large sample and the Central Limit Theorem
- •Unbiased estimates from raw and coded data
- •The six-step test procedure
- •Conclusion protocol and forbidden wording
- •Critical-value method and inverse problems
- •Common mistakes and exam technique
Key definitions
- Null hypothesis, H₀
- The status-quo assumption — the claim, the stated value or the previous situation — assumed true throughout the calculation. It is always an equality, H₀: μ = μ₀, never μ ≤ μ₀ or μ ≥ μ₀.
- Alternative hypothesis, H₁
- What is suspected to be true instead, framed from the wording of the claim in the question. It takes exactly one of three forms: μ ≠ μ₀ (two-tail), μ > μ₀ or μ < μ₀ (one-tail).
- Level of significance, α
- The probability of rejecting H₀ when H₀ is in fact true. Testing at the 5% level means accepting a 0.05 probability of wrongly rejecting a true null hypothesis.
- Critical region
- The set of values of the test statistic for which H₀ is rejected, also called the rejection region. Under H₀ its total probability is exactly α, and its boundary values are the critical values.
- p-value
- The probability, assuming μ = μ₀ is true, of obtaining a sample mean at least as extreme as the observed value, in the direction(s) specified by H₁. Reject H₀ if and only if p-value < α.
Key formulas
Test statistic — Case 1 (normal population, known variance)
Under H₀: μ = μ₀. X̄ is the sample mean, σ the known population standard deviation and n the sample size. Valid for any n when the population is stated to be normal; do not cite the CLT.
Test statistic — Case 2 (large sample)
Used when n is large: by the Central Limit Theorem X̄ is approximately normal, and the unbiased estimate s² replaces the unknown σ². If σ is given, use it in place of s.
Unbiased estimates of μ and σ²
s² divides by n − 1, not n. On the GC's 1-Var Stats, Sx divides by n − 1 (square it to get s²); σx divides by n and is not the value wanted.
Unbiased estimates from coded data
a is the coding constant. Subtracting a constant shifts the mean but leaves the spread untouched, so a is added back for the mean only.
Critical-value decision rule (two-tail)
Equivalent to rejecting H₀ when p-value < α. Critical values from invNorm: two-tail 5% ±1.9600, one-tail 5% 1.6449, two-tail 1% ±2.5758, one-tail 1% 2.3263.
Rejection region for x̄ (upper-tail test)
Found by unwinding the standardisation. Needed for inverse questions asking for the range of x̄, n, μ₀ or the significance level, which cannot be done by pressing Z-Test.
Worked example
A drinks bottler claims that the mean volume of cola in its bottles is 500 ml. Volumes are known to be normally distributed with standard deviation 4.2 ml. A quality auditor measures a random sample of 50 bottles and obtains a sample mean volume of 501.1 ml. Test, at the 5% level of significance, whether the mean volume differs from 500 ml. Interpret the p-value in the context of the question.
- 1Let X be the volume of cola, in ml, in a randomly chosen bottle and let μ be the population mean volume, with X ~ N(μ, 4.2²) and σ = 4.2 known.
- 2Test H₀: μ = 500 against H₁: μ ≠ 500; “differs from” questions the claim in both directions, so this is a two-tail test at the 5% level of significance.
- 3The population is normal, so X̄ is exactly normal and no Central Limit Theorem is needed: under H₀, X̄ ~ N(500, 4.2² ÷ 50), so Z = (X̄ − 500) ÷ (4.2 ÷ √50) ~ N(0, 1).
- 4With x̄ = 501.1, z = (501.1 − 500) ÷ (4.2 ÷ √50) = 1.1 ÷ 0.59397 = 1.8519 (5 s.f.).
- 5The test is two-tail, so p-value = 2P(Z ≥ 1.8519) = 0.0640 (3 s.f.).
- 6Since 0.0640 > 0.05, we do not reject H₀: there is insufficient evidence, at the 5% level of significance, that the mean volume of cola in the bottles differs from 500 ml.
- 7Check by rejection region: with critical values ±1.9600, H₀ is rejected when x̄ ≤ 500 − 1.9600 × 4.2 ÷ √50 = 498.84 or x̄ ≥ 500 + 1.9600 × 4.2 ÷ √50 = 501.16, and 501.1 falls just short of 501.16, giving the same conclusion.
- 8Interpretation: if the true mean volume were 500 ml, there would be a probability of 0.0640 of obtaining a sample mean at least as far from 500 ml as the observed 501.1 ml, in either direction.
Watch out: Never round the boundary values early: the observed 501.1 ml falls only just short of the upper boundary 501.16 ml.
Common mistakes
Saying “accept H₀”
Wrong: Since p-value = 0.251 > 0.05, we accept H₀. The mean mass is 375 g.
Right: Since p-value = 0.251 > 0.05, we do not reject H₀. There is insufficient evidence, at the 5% level of significance, that the mean mass of a packet differs from 375 g.
Choosing H₁ by looking at the sample data
Wrong: The sample mean is x̄ = 61.3 > 60, so I will test H₁: μ > 60.
Right: H₁ comes from the wording of the claim, decided before the data is examined. If a customer suspects underfilling, H₁: μ < μ₀ even if the sample mean happens to be above μ₀.
Wrong critical value for a two-tail test
Wrong: Two-tail test at the 5% level, rejecting H₀ when |z| > 1.6449.
Right: A two-tail test at the 5% level puts 2.5% in each tail, so the critical values are ±1.9600. The value 1.6449 is the one-tail 5% critical value.
Questions students ask
What is the difference between a one-tail and a two-tail hypothesis test?
If H₁ is μ ≠ μ₀ the test is two-tail, so a sample mean that is unusually large or unusually small counts as evidence against H₀. If H₁ is μ > μ₀ or μ < μ₀ the test is one-tail and only one direction counts. The direction is decided entirely by the wording of the question, such as “has changed” (two-tail) or “has increased” (one-tail, upper).
Do I need to assume the population is normal for a z-test in H2 Math?
If n is large, no assumption is needed: by the Central Limit Theorem the sample mean X̄ is approximately normal regardless of the distribution of the population. If n is small and normality is not stated, you must assume the population is normally distributed so that X̄ is normally distributed. The mark is for connecting sample size → CLT → normality of X̄.
Why can't you write “accept H₀” in a hypothesis test conclusion?
A hypothesis test never proves anything: failing to reject H₀ only means there is insufficient evidence against it, not that H₀ is true. The only permitted phrase is “do not reject H₀”, followed by a statement about the evidence at the stated level of significance, in context. Writing “accept H₀”, “H₀ is true” or “the claim is proved” is penalised every year.
How do you find the smallest level of significance at which H₀ is rejected?
H₀ is rejected when p-value < α ÷ 100, so the boundary level of significance equals the p-value expressed as a percentage. For example, a p-value of 0.028614 gives a smallest level of significance of 2.86% (3 s.f.). Do not answer 100 − 2.86 = 97.1%; a plausible significance level is typically under 10%.
In the full PDF
- •Diagrams of the one-tail and two-tail rejection regions under the standard normal curve
- •The wording-to-H₁ table, including how to handle “overstated” and “understated” claims
- •The decision table for choosing Case 1 or Case 2, and when no valid test exists in 9758
- •Further worked examples: a one-tail test on coded data at the 2% level, and inverse problems for the range of x̄ and of μ₀
- •The remaining pitfalls: reversed inequalities in inverse problems, misusing the CLT and misinterpreting the level of significance
- •Model answers for the assumption question, GC (TI-84) keystrokes, the exam technique checklist and the five-takeaway summary