IB Maths AI HLHypothesis Testing for Population ParametersPaper 1 & 2~9 min read
Hypothesis Testing for Mean (Two Sample)
A two-sample test compares the means of two populations. Same z-vs-t decision as before (variances known → z, unknown → pooled t), but there’s a sharper trap here: when the two samples are actually linked — same people measured twice, or matched pairs — you must run a paired t-test on the differences instead. Spotting “paired vs two-sample” is the marks-deciding skill on this page.
📘 What you need to know
Two-sample test: compares μ1 and μ2 from two separate normally distributed populations.
Variances known → z-test; unknown → t-test with a pooled sample (assume equal variances).
Hypotheses: H0: μ1 = μ2; H1 is <, > or ≠. Always say which mean is which.
Paired t-test: when each value in one sample pairs with one in the other (same subject twice). Work with the differences as a single sample.
Both compare the means of two independent normally distributed populations. As with one sample, the only choice is whether the population variances are given.
Variances knownz-testEnter σ1 and σ2. Can use summary stats directly.
Variances unknownpooled t-testAssume equal variances, choose the pooled option, enter the two data lists.
🤔 What does “pooled” mean?
When the variances are unknown but assumed equal, both samples are combined to make one shared estimate of the population variance — a “pooled” estimate. More data behind the estimate gives a more reliable t-test. On the GDC you just tick the Pooled: Yes option.
The steps
🧭 Recipe — two-sample test on the GDC
Hypotheses: H0: μ1 = μ2, then H1. Define each μ clearly (which population is 1, which is 2).
z or t? Variances known → two-sample z; unknown → two-sample t with Pooled: Yes.
Enter data as two lists (or summary stats for a z-test); read the p-value.
Decide & conclude: p < significance level → reject; state in context.
Conclusion wording: with H1: μ1 < μ2, rejecting means “the mean of population 1 is smaller”. With ≠, “the two means are different”. Accepting flips each to “not smaller / not different”. Always name the real populations.
Paired or two-sample?
This is the examiner’s favourite trap. Two columns of data with the same length can look like a two-sample setup, but if each row is the same subject (measured twice, or before/after), it’s paired — and you test the differences.
Telling them apart
Same number of values in each group does NOT make it paired — the rows must be the same subjects.
🧠 Paired = “differences become one sample”
For a paired test, compute d = (value A − value B) for each subject, then run a one-sample t-test on the list of d‘s against 0. Keep the subtraction order consistent. You only need the differences to be normally distributed, not the populations.
Worked examples
WE 1–3 use the puzzle-times data (two-sample pooled t). WE 4–5 use the French/Spanish scores (paired t).
WE 1
Two-sample — hypotheses
Children’s and adults’ puzzle times (minutes) are recorded. The creator claims children are faster. A t-test runs at 1%. State the hypotheses.
let μ_C, μ_A = mean times for children, adults“faster” = shorter time → testing μ_C < μ_A (one-tailed).H₀: μ_C = μ_A H₁: μ_C < μ_A
WE 2
Two-sample — p-value
Variances are unknown. Find the p-value for the test.
variances unknown → two-sample pooled t-testenter the two lists in the GDC, choose Pooled: Yes.p = 0.007259…p = 0.00726 (3sf)
WE 3
Two-sample — conclusion
Is the creator’s claim supported at the 1% level? Give a reason.
compare p with significance level0.00726 < 0.01reject H₀sufficient evidence that children are generally faster than adults — this supports the creator’s claim.
WE 4
Paired — spot it & state hypotheses
Nine students each sit a French and a Spanish test. The head wants to know if there’s a difference in scores between subjects. 10% level. State the hypotheses.
same student does both → paired testlet d = French − Spanish; μ_D = mean difference for the population.“a difference” → two-tailedH₀: μ_D = 0 H₁: μ_D ≠ 0
WE 5
Paired — p-value & conclusion
The differences (French − Spanish) are −13, 3, −6, 14, 4, −10, −7, −10, −4. Run a one-sample t-test on these. The GDC gives p = 0.296. Conclude at 10%.
enter the 9 differences as a list → one-sample tp = 0.2958… → 0.296 (3sf)compare with significance level0.296 > 0.10accept H₀insufficient evidence of a difference in scores between French and Spanish.
💡 Top tips
Same subject twice → paired. Two unrelated groups → two-sample. Decide this before anything else.
Label the populations (1 and 2, or by name) so your H1 direction is unambiguous.
“Faster / fewer / smaller” means a one-tailed <; “different / changed” means two-tailed ≠.
Pooled option ON for the two-sample t-test (this course assumes equal variances).
For paired: compute differences, keep the order consistent, then it’s literally a one-sample t-test against 0.
Same decision rule: p < α → reject. Write the conclusion in context.
⚠ Common mistakes
Running two-sample on paired data — the classic trap when both columns have equal length.
Forgetting to pool — leaving Pooled: No on the two-sample t-test.
Direction confusion — “children faster” is a smaller time, so μ_C < μ_A, not >.
Inconsistent subtraction — mixing French − Spanish with Spanish − French across rows.
Comparing the wrong way — reject when p < α, not when p > α.
Vague conclusions — not naming which population’s mean is bigger/smaller.
Next up — Binomial Hypothesis Testing, where the parameter being tested is a proportion rather than a mean. These tests are always one-tailed, and you’ll meet a new way to decide: the critical region, found with the inverse binomial function.
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