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 Regression, Correlation and Hypothesis Testing

Edexcel A Level Maths: 1.3 Hypothesis Testing for Zero CorrelationCurated by Kepler, not produced by us. Watch it before the notes below.

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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 $\rho$ . The null hypothesis is $H_0: \rho = 0$ (no correlation). The alternative is $\rho \ne 0$ (two-tailed) or $\rho > 0$ / $\rho < 0$ (one-tailed).

H_0: \rho = 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.

Example 1r = 0.78, critical value 0.6319 (5%, one-tailed). Conclusion?

0.78 > 0.6319

2Reject

H_0

: evidence of positive correlation.

Answer

Significant — reject H₀.

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

Formula recap

H_0: \rho = 0, \ H_1: \rho \ne 0

Two-tailed.

|r| > \text{critical value} \Rightarrow \text{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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

The null hypothesis is usually:
- $\rho = 0$
- $\rho = 1$
- $r = 0$
- $\rho > 0$
Show answer and explanation
Answer: $\rho = 0$
No correlation.
Hypotheses are stated in terms of:
- $\rho$
- $r$
- $n$
- $x$
Show answer and explanation
Answer: $\rho$
Population parameter.
We reject H₀ when:
- $|r| >$ critical value
- $|r| <$ critical value
- $r = 0$
- always
Show answer and explanation
Answer: $|r| >$ critical value
r in the critical region.
A two-tailed test at 5% uses the critical value at:
- 2.5%
- 5%
- 10%
- 1%
Show answer and explanation
Answer: 2.5%
Split between tails.
With r = 0.78 > 0.6319, we:
- reject H₀
- accept H₀
- cannot decide
- increase n
Show answer and explanation
Answer: reject H₀
r in critical region.

Next step

Check your understanding

Work through questions on Hypothesis Testing for Zero Correlation. Use the feedback to decide what needs another look.

2 past-paper questions · 6 marks

Practise this topic

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