IB Maths AI HLHypothesis Testing for Population ParametersPaper 1 & 2~8 min read
Hypothesis Testing for Correlation
Now the parameter is the population correlation coefficientρ between two variables, and the null is always ρ = 0 (no linear correlation). You run a t-test for linear regression on the GDC. After the discrete binomial and Poisson tests, this one feels familiar — but unlike them it can be two-tailed: “any correlation” → ≠, while “positive” or “negative” → one-tailed.
📘 What you need to know
Tests for linear correlation between two normally distributed variables, using a t-test.
Hypotheses: H0: ρ = 0 (no correlation); H1: ρ < 0, ρ > 0, or ρ ≠ 0. State ρ is the population correlation coefficient.
One vs two-tailed: “positive / negative” → one-tailed; “any” linear correlation → two-tailed.
On the GDC: enter the two lists, choose t-test for linear regression, read the p-value.
Decision (p-value): p < significance level → reject H0.
Decision (PMCC): if |r| > |critical value| → reject. The critical value is given in the exam.
One-tailed or two-tailed?
This is the only setup decision. It’s driven entirely by the wording: are they testing for a specific direction of correlation, or just whether any correlation exists?
“any” correlationtwo-tailedH1: ρ ≠ 0. Testing whether a linear relationship exists at all.
“positive” / “negative”one-tailedH1: ρ > 0 or ρ < 0. Testing a specified direction.
🤔 What is ρ versus r?
ρ (rho) is the true correlation coefficient for the whole population — unknown. r is the PMCC you calculate from the sample. The test asks: is the sample r far enough from 0 to conclude the population ρ isn’t 0? The hypotheses are always written in terms of ρ, not r.
The steps
🧭 Recipe — correlation t-test on the GDC
Hypotheses: H0: ρ = 0, then H1 as <, > or ≠. Define ρ as the population correlation coefficient between the two named variables.
Run the test: enter the data as two lists, choose t-test for linear regression; read the p-value (and PMCC if asked).
Conclude in context, tentatively — there is / is not (positive / negative) linear correlation.
Two routes to the same decision
p-value route: reject H0 if p < α
PMCC route: reject H0 if |r| > |rcrit|
Critical value is given in the exam if the PMCC method is required ✓
Conclusion wording: reject with H1: ρ > 0 → “positive linear correlation”; ρ < 0 → “negative”; ρ ≠ 0 → “a linear correlation”. Accepting flips each to “not …”. Always name the two real variables.
Where this fits among the tests
Every test in this unit follows write hypotheses → get a p-value → compare with α. What changes is the parameter and the distribution behind it.
Same skeleton, different parameter
Correlation is the only one tested with a regression t-test — and the only discrete-vs-continuous “outsider” that can be two-tailed alongside the mean tests.
🧠 PMCC sign tells you the direction
If a one-tailed test rejects, the sign of r confirms the direction: positive r → positive correlation, negative r → negative. For a two-tailed test you just report “there is a linear correlation” — but you can still note the direction from the sign of r.
Worked examples
All five use Jessica’s data: distance run d km vs hours of sleep t the night after, over 9 days, at a 5% significance level.
Distance d (km)
1.2
2.3
1.5
1.3
2.5
1.8
1.9
2.0
1.1
Sleep t (hours)
7.9
8.1
7.6
7.3
8.1
8.4
7.8
7.9
6.8
WE 1
State the hypotheses
Jessica wants to know if there is any linear correlation between distance and sleep. State H0 and H1.
let ρ = correlation coefficient between Jessica’s distances and hours of sleep“any” linear correlation → two-tailed.H₀: ρ = 0 H₁: ρ ≠ 0
WE 2
Run the test on the GDC
Describe how to find the p-value and find the PMCC for the data.
enter d and t as two lists → t-test for linear regressionPMCC: r = 0.693…a moderately strong positive sample correlation.
WE 3
Find the p-value
State the p-value the GDC returns for this test.
read the p-value from the regression t-testp = 0.03833…p = 0.0383 (3sf)
WE 4
Decide & conclude in context
Perform the test at 5% and state the conclusion clearly.
compare p with the significance level0.0383 < 0.05reject H₀sufficient evidence of a linear correlation between the distance Jessica runs and the hours she sleeps.
WE 5
The PMCC method (cross-check)
Suppose the exam gives the critical value 0.666 for n = 9 at 5% (two-tailed). Reach the same conclusion using the PMCC.
compare |r| with the critical value|0.693| > 0.666reject H₀same conclusion — a linear correlation exists; the positive sign of r shows it is positive.
💡 Top tips
Define ρ as the population correlation coefficient between the two named variables — easy mark.
“Any” → two-tailed ≠; “positive / negative” → one-tailed. Decide this from the wording first.
Use the regression t-test on the GDC, not the normal/binomial menus.
PMCC method: compare absolute values, |r| vs |rcrit|; the critical value is given.
Sign of r tells you the direction of any correlation you find.
Same decision rule: p < α → reject. Write the conclusion in context, tentatively.
⚠ Common mistakes
Writing hypotheses in r — they must be in ρ, the population coefficient.
Wrong tail — using ≠ for a “positive correlation” question (should be ρ > 0).
Forgetting absolute values in the PMCC comparison (a strong negative r still rejects).
Correlation ⇒ causation — concluding running causes more sleep; only state correlation.
Comparing the wrong way — reject when p < α, not when p > α.
Vague conclusions — not naming the two variables, or being definitive instead of tentative.
Next up — Type I & Type II Errors, the last topic in this unit. You’ll learn what a “false positive” and “false negative” mean for a test, how the critical region fixes their probabilities, and why shrinking one error grows the other (only a bigger sample shrinks both).
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