1.3StatisticsStretch

Hypothesis Testing for Zero Correlation

A sample PMCC may suggest correlation, but is it significant in the population? A hypothesis test compares the sample r against a critical value to decide whether the population correlation ρ is zero.

26 min Video by Zeeshan Zamurred Regression, Correlation and Hypothesis Testing

Edexcel A Level Maths: 1.3 Hypothesis Testing for Zero CorrelationWatch the full walkthrough before the notes below.

Open on YouTube

What you'll be able to do

State hypotheses about ρ
Choose one- or two-tailed tests
Compare r with the critical value
State a conclusion in context

Setting up the test

The population correlation coefficient is $ρ$ . The null hypothesis is $H_{0} : ρ = 0$ (no correlation). The alternative is $ρ \neq = 0$ (two-tailed) or $ρ > 0$ / $ρ < 0$ (one-tailed).

H_{0} : ρ = 0

No correlation in the population.

Making the decision

Compare the sample $∣ r ∣$ with the critical value (from the table for the given significance level and sample size). If $∣ r ∣$ exceeds the critical value, reject $H_{0}$ — there is evidence of correlation.

0.78 > 0.6319

2Reject

H_{0}

: evidence of positive correlation.

AnswerSignificant — reject H₀.

Tip — For a two-tailed test at 5%, use the 2.5% critical value.

Formula recap

H_{0} : ρ = 0, H_{1} : ρ \neq = 0

Two-tailed.

∣ r ∣ > critical value \Rightarrow reject H_{0}

Decision rule.

Common mistakes to avoid

Writing hypotheses about r instead of ρ.

Hypotheses are about the population parameter ρ.

Using the full significance level for a two-tailed test.

Halve it (e.g. 5% two-tailed uses the 2.5% value).

Key takeaways

Test H₀: ρ = 0 against a one- or two-tailed alternative.
Reject H₀ if |r| exceeds the critical value.
State the conclusion in context.

Test yourself

Ready to lock in Hypothesis Testing for Zero Correlation? Pick a mode and earn XP & Dobloons.