bProduct moment correlation coefficient/b
The product moment correlation coefficient (PMCC) ((r)) describes the linear correlation between two variables, and can take any value between (-1) and (1) inclusive.
When (r=1), there is a perfect positive linear corrleation. All the data points lie on a straight line with a positive gradient. When (r=-1), there is a perfect negative linear corrleation.All the data points lie on a straight line with a negative gradient. When (r=0), there is no linear correlation.
bHypothesis testing for zero correlation/b
(r) is the PMCC for a sample, and (\rho) is the PMCC for a population.
You can use a hypothesis test to test whether the PMCC for a sample ((r)) indicates that there is likely to be a linear relationship in the population.
For a bone-tailed test/b:
(H0:\rho=0) (H1:\rho > 0) or (H_1:\rho < 0)
For a btwo-tailed test/b:
(H0:\rho=0) (H1:\rho \neq 0)
To carry out the test, you will need to compare your (r) value with the table of critical values for a given significance level (usually (5%)) and sample size. The table of critical values are provided in the formula book.
mtaimg/images/topics/15/15-2-1.png/mtaimg
An observed value of (r) greater than the critical value would provide sufficient evidence to reject the null hypothesis and conclude that it is likely that (\rho>0).
An observed value of (r) less than the negative of the critical value would provide sufficient evidence to reject the null hypothesis and conclude that it is likely that (\rho<0).
[b]uProduct moment correlation coefficient[/u]/b
(-1 \leq r \leq 1)