Hypothesis test for correlationEdexcel A-Level Maths: Revision notes

Section 1

The product moment correlation coefficient

The product moment correlation coefficient $r$ measures how close points lie to a straight line. It satisfies $-1\le r\le1$ . If $r=1$ all points lie on a straight line with positive gradient; if $r=-1$ they lie on a straight line with negative gradient; $r$ near $0$ means no linear relationship. You find $r$ with your calculator in statistics mode, so you do not need to use the formula. $r$ describes linear association only: a perfect curve can still give $r$ close to $0$ .

Key termsproduct moment correlation coefficientlinear association

Common mistake

Quoting an $r$ outside $-1$ to $1$ . This always means an error.

Section 2

Sample and population

$r$ is calculated from a sample. The population correlation coefficient is written $\rho$ (the Greek letter rho). A hypothesis test uses $r$ to decide whether there is evidence about $\rho$ , so hypotheses are always written in terms of $\rho$ , never $r$ .

Key termspopulation correlation coefficientsample

Common mistake

Writing $H_0:r=0$ . The hypotheses must use $\rho$ .

Section 3

Stating the hypotheses

The null hypothesis is $H_0:\rho=0$ (no correlation). The alternative hypothesis depends on the question:

two-tailed (there is correlation): $H_1:\rho\neq0$
one-tailed, positive: $H_1:\rho>0$
one-tailed, negative: $H_1:\rho<0$ Choose one-tailed only if the question says which direction is expected before looking at the data.

Key termsnull hypothesisalternative hypothesisone-tailedtwo-tailed

Section 4

Using a critical value

You are given a table of critical values for a sample size $n$ and significance level. Compare $|r|$ with the critical value. In a one-tailed test the critical region is $r>$ critical value (for $\rho>0$ ) or $r<-$ critical value (for $\rho<0$ ). In a two-tailed test it is $|r|>$ critical value, with the significance level split between the two tails. If $r$ is in the critical region, reject $H_0$ . Otherwise do not reject $H_0$ . Example: $n=20$ , two-tailed, 5%, critical value $0.4438$ . If $r=0.52$ then $0.52>0.4438$ , so reject $H_0$ .

Key termscritical valuecritical region

Common mistake

Comparing a negative $r$ with a positive critical value without using $|r|$ or the negative boundary.

Exam tip

Check which table you have: one-tailed and two-tailed columns have different critical values.

Section 5

Using a p-value

A p-value is the probability of obtaining a result at least as extreme as the one observed, assuming $H_0$ is true. If the p-value is less than the significance level, reject $H_0$ ; if it is greater, do not reject. Example: a p-value of $0.018$ at the 5% level: $0.018<0.05$ , so reject $H_0$ .

Key termsp-valuesignificance level

Section 6

Writing the conclusion

State two things: whether $H_0$ is rejected and what that means in context, for example 'there is significant evidence of a positive correlation between fertiliser and yield'. Never say the test 'proves' anything. Correlation does not imply causation: another variable may affect both, or the cause may run the other way. A result that is significant at 5% may not be at 1%, so give the level used.

Key termscausationin context

Exam tip

'Do not reject $H_0$ ' is not 'accept $H_0$ ': say there is insufficient evidence of correlation.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Hypothesis test for correlation

A student records, on 20 randomly chosen days, the daily maximum temperature and the number of ice creams sold at a kiosk. The product moment correlation coefficient is $r=0.52$ . The student tests for correlation at the 5% significance level. For a two-tailed test with sample size 20, the critical value at the 5% level is $0.4438$ .
Carry out the test and state your conclusion in context.2 marks
A teacher records, for 12 students, the time spent on screens each night and the score in a memory test. The product moment correlation coefficient is $r=-0.62$ . The teacher believes that more screen time is associated with lower scores, and tests this at the 5% significance level. For a one-tailed test with sample size 12, the critical value at the 5% level is $0.4973$ .
A student says: 'The result proves that spending more time on screens makes scores worse.' Comment on this statement.2 marks
A farmer records the amount of fertiliser, $x$ kg per hectare, and the yield, $y$ tonnes per hectare, on 8 fields. The data are $(20,4.1)$ , $(30,4.9)$ , $(40,4.4)$ , $(50,5.8)$ , $(60,5.1)$ , $(70,5.6)$ , $(80,5.9)$ , $(90,5.3)$ . For a one-tailed test with sample size 8, the critical value is $0.6215$ at the 5% level and $0.7887$ at the 1% level.
Use your calculator to find the product moment correlation coefficient $r$ , and state suitable hypotheses for a test of whether more fertiliser is associated with a greater yield.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).