When is the test two-tailed?

When asking whether there is any correlation: H1: \rho\neq0.

When is the test one-tailed?

When the question states a direction in advance: positive or negative correlation.

What does the product moment test assume?

The data are random pairs from a bivariate Normal distribution.

When is the Spearman test used?

When the data are ranks or the relationship is monotonic but not linear, or the data are not bivariate Normal.

Where does the critical value come from?

The tables, for the sample size n, the significance level and the number of tails.

Decision rule for a two-tailed test?

Reject H0 if |r| is greater than or equal to the critical value.

Decision rule for a one-tailed negative test?

Reject H0 if r\le- critical value.

Conclusion when H0 is not rejected?

There is insufficient evidence of correlation. It does not prove there is none.

Does a significant correlation show causation?

No. Another factor may cause both variables to change.

Testing for zero correlationEdexcel International A Level Further Maths: Flashcards

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What do $\rho$ and $\rho_s$ represent?

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What do $\rho$ and $\rho_s$ represent?: The population product moment and Spearman rank correlation coefficients.
What is the null hypothesis for a test of zero correlation?: $H_0$ : $\rho=0$ (or $\rho_s=0$ ).
When is the test two-tailed?: When asking whether there is any correlation: $H_1$ : $\rho\neq0$ .
When is the test one-tailed?: When the question states a direction in advance: positive or negative correlation.
Hypotheses for 'positive correlation'?: $H_0$ : $\rho=0$ , $H_1$ : $\rho>0$
Hypotheses for 'negative correlation'?: $H_0$ : $\rho=0$ , $H_1$ : $\rho<0$
What does the product moment test assume?: The data are random pairs from a bivariate Normal distribution.
When is the Spearman test used?: When the data are ranks or the relationship is monotonic but not linear, or the data are not bivariate Normal.
Where does the critical value come from?: The tables, for the sample size $n$ , the significance level and the number of tails.
Decision rule for a two-tailed test?: Reject $H_0$ if $|r|$ is greater than or equal to the critical value.
Decision rule for a one-tailed negative test?: Reject $H_0$ if $r\le-$ critical value.
Conclusion when $H_0$ is not rejected?: There is insufficient evidence of correlation. It does not prove there is none.
Does a significant correlation show causation?: No. Another factor may cause both variables to change.

Exam questions on Testing for zero correlation

A teacher records the hours of revision and the test score of each of 10 randomly chosen students. The data may be assumed to come from a bivariate Normal distribution. The product moment correlation coefficient for the sample is $r=0.62$ . The teacher tests at the 5% significance level whether there is any correlation between revision hours and score. For $n=10$ the two-tailed 5% critical value is 0.6319.
Suppose instead that the relationship between revision hours and score were known to be increasing but clearly curved. Name a more suitable measure of correlation and give a reason.2 marks
A tutor believes that students who complete more practice papers achieve higher mock exam ranks. For a random sample of 8 students, the number of practice papers and the mock mark are each ranked with rank 1 for the highest, and $\sum d^2=26$ . For $n=8$ the 5% critical value for Spearman's coefficient is 0.6429 for a one-tailed test and 0.7381 for a two-tailed test.
Carry out the test at the 5% significance level and state the conclusion in context.2 marks
A cafe owner believes that daily sales of hot drinks fall as the outside temperature rises. Over 12 randomly chosen days the owner records the temperature and the number of hot drinks sold. The data may be assumed to come from a bivariate Normal distribution, and the sample product moment correlation coefficient is $r=-0.632$ . For $n=12$ the one-tailed critical values for the product moment correlation coefficient are 0.4973 at the 5% level and 0.6581 at the 1% level.
State the hypotheses for the owner's test and the critical region at the 5% level.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).