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This lesson extends correlation and regression to include Spearman's rank correlation coefficient, hypothesis testing for correlation, and deeper analysis of the product-moment correlation coefficient (PMCC).
The PMCC r measures the strength and direction of a linear relationship between two variables:
r=SxxSyySxy
where Sxy=∑(xi−xˉ)(yi−yˉ), Sxx=∑(xi−xˉ)2, Syy=∑(yi−yˉ)2.
| Value of r | Interpretation |
|---|---|
| r=1 | Perfect positive linear correlation |
| r=−1 | Perfect negative linear correlation |
| r=0 | No linear correlation |
To test whether there is a significant linear correlation in the population:
H0:ρ=0 (no linear correlation in the population) H1:ρ=0 (two-tailed) or ρ>0 / ρ<0 (one-tailed)
Compare the sample r with the critical value from the PMCC table for n data points at the chosen significance level.
If ∣r∣>critical value, reject H0.
For n=10 pairs, r=0.65. Test at 5% (two-tailed).
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