IB Maths AI HLHypothesis Testing for Population ParametersPaper 1 & 2~9 min read
Hypothesis Testing for Mean (One Sample)
A one-sample test checks whether the mean of a normally distributed population has changed from a claimed value. The single decision you make first: is the population variance known? If yes, use a z-test; if no, use a t-test (with the unbiased variance estimate). After that the routine is the same every time — write hypotheses, let the GDC give a p-value, then compare it with the significance level.
📘 What you need to know
z-test: population variance σ2 is known. t-test: variance is unknown (use sn−12).
Hypotheses: H0: μ = μ0; H1 is <, > (one-tailed) or ≠ (two-tailed). Always say μ is the population mean.
Decision rule: if p-value < significance level → rejectH0.
z-test can also use a critical value / critical region via the inverse normal.
Conclusion in context: reject → the mean has changed/increased/decreased; accept → insufficient evidence it has.
For a t-test you won’t be asked for the critical value — just read off the p-value.
z-test or t-test?
Both test the mean of a normally distributed population. The only question that decides which one to run is whether the population variance is given.
Variance knownz-testσ2 is given. Sample mean follows N(μ, σ²n). Can also use a critical region.
Variance unknownt-testEstimate it with sn−12. Uses a t-distribution; just read the p-value off the GDC.
Distribution of the sample mean (z-test)X̄ ∼ N(μ, σ²n)
so standard deviation = σ√nUnbiased variance estimate is in the booklet ✓
🤔 Why does the t-test need an “unbiased estimate”?
When σ is unknown you estimate it from the sample, but the ordinary sample variance slightly underestimates the true spread. The unbiased estimate sn−12 = nn−1sn2 corrects this. The extra uncertainty is exactly why we switch from the normal to the wider t-distribution.
The four steps
🧭 Recipe — one-sample test on the GDC
Hypotheses: H0: μ = μ0, then H1 as <, > or ≠. State that μ is the population mean.
Choose the test: variance known → z-test (enter σ); unknown → t-test (enter sn−1).
Read the p-value from the GDC (raw data as a list, or summary stats).
Decide & conclude: p-value < significance level → reject; state the conclusion in context.
Choosing and deciding
One decision splits the path; from the p-value onwards both tests are identical.
Conclusion wording: rejecting H0 means “sufficient evidence the mean has increased / decreased / changed”. Accepting means “insufficient evidence — the mean has not increased / decreased / changed”. Always name the real variable and stay tentative.
Critical region (z-test only)
An exam may ask for the critical region instead of a p-value. The critical value c is found with the inverse normal, using the sample-mean distribution.
Critical region for a one-tailed z-test at α%H1: μ > μ0 → region X̄ > c, where P(X̄ > c) = α%
two-tailed → two regions, each with tail probability 12α%
🧠 Rounding the critical value
For X̄ < c keep the lower bound; for X̄ > c keep the upper bound. This stops the tail probability from creeping above the significance level.
Worked examples
WE 1–3 use the Burmese-cats data (z-test, variance known). WE 4–5 use the Calculus High IQ data (t-test, variance unknown).
WE 1
z-test — state the hypotheses
Burmese cats have mass C ∼ N(4.2, 1.3²) kg. Kamala claims cats in households with young children weigh more than average. State H0 and H1.
let μ = population mean mass of Burmese catsclaim is “weigh more” → one-tailed, testing for an increase.H₀: μ = 4.2 H₁: μ > 4.2
WE 2
z-test — critical region
A sample of n = 25 cats is taken, 5% significance. Find the critical region.
σ² known → z-test; sample mean ~ N(μ, σ²/n)C̄ ~ N(4.2, 1.3²/25), so σ = √(1.3²/25) = 0.26region C̄ > c with P(C̄ > c | μ=4.2) = 0.05inverse normal: P(C̄ < c) = 0.95 → c = 4.6276…critical region: C̄ > 4.63
WE 3
z-test — conclusion
The sample mean of the 25 cats is 4.65 kg. State the conclusion of the test.
compare sample mean with the critical region4.65 > 4.6276… → in the critical regionreject H₀sufficient evidence that Burmese cats in households with young children weigh more than average.
WE 4
t-test — hypotheses & unbiased variance
IQ at Calculus High ∼ N(126, …). The head plays classical music and suspects the mean IQ has changed. A sample of 15 students gives mean 127.1 and sample variance 14.7. State the hypotheses and find sn−12.
let μ = population mean IQ; “changed” → two-tailedH₀: μ = 126 H₁: μ ≠ 126unbiased estimate: s²ₙ₋₁ = (n/(n−1)) × s²ₙ= (15/14) × 14.7s²ₙ₋₁ = 15.75
WE 5
t-test — p-value & conclusion
Variance is unknown, so a t-test is used with x̄ = 127.1, sn−1 = √15.75, n = 15. The GDC gives p = 0.301. Is the head’s suspicion supported (5%)?
compare p with significance level0.301 > 0.05accept H₀p-value > significance level → insufficient evidence to support the headteacher’s suspicion that the mean IQ has changed.
💡 Top tips
Pick the test first: variance given → z; variance unknown → t. This is worth easy marks.
Define μ explicitly as the population mean of the real variable.
One-tailed vs two-tailed: “more/fewer/increased/decreased” → one-tailed; “changed/different” → two-tailed.
For a t-test, enter sn−1 (the standard deviation, not the variance) — convert if given the variance.
Decision is always p < α → reject. Write the conclusion in context, tentatively.
If asked for a critical region, it’s a z-test — use the inverse normal on N(μ, σ²/n).
⚠ Common mistakes
Wrong test — running a z-test when the variance is only estimated from the sample.
Forgetting the n correction — using σ instead of σ/√n in the sample-mean distribution.
Variance vs SD mix-up — entering sn−12 where the GDC wants sn−1.
Tail error — using a one-tailed setup for a “changed” question (or vice versa).
Comparing the wrong way — reject when p < α, not when p > α.
Definitive conclusions — “proves the mean increased” instead of “sufficient evidence to suggest…”.
Next up — Hypothesis Testing for Mean (Two Sample), where you compare the means of two populations. The same z-vs-t decision returns, plus a new idea: for unknown variances you assume they’re equal and use a pooled sample — and you’ll learn when a paired t-test is the smarter choice.
Need help with Hypothesis Testing for Population Parameters?
Get 1-on-1 help from an IB examiner who knows exactly what Paper 1 & 2 are looking for.
