The Pearson correlation coefficient significance

The test of significance for Pearson product-moment correlation coefficient is used to verify the hypothesis determining the lack of linear correlation between an analysed features of a population and it is based on the Pearson's linear correlation coefficient calculated for the sample. The closer to 0 the value of coefficient $r_p$is, the weaker dependence joins the analysed features.

Basic assumptions:

Hypotheses:

\begin{array}{cl} \mathcal{H}_0: & R_p = 0, \\ \mathcal{H}_1: & R_p \ne 0. \end{array}

The test statistic is defined by: \begin{displaymath} t=\frac{r_p}{SE}, \end{displaymath}

where $\displaystyle SE=\sqrt{\frac{1-r_p^2}{n-2}}$.

The value of the test statistic can not be calculated when $r_p=1$ or $r_p=-1$ or when $n<3$.

The test statistic has the t-Student distribution with $n-2$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$:

\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}

EXAMPLE cont. (age-height.pqs file)