Pearson product-moment correlation coefficient (r) and t-test on it

The index of relation between $x$ and $y$ is correlation coefficient or Pearson product moment correlation coefficient as formula below. Range of correlation coefficient is between -1 and 1.

$\displaystyle r = \frac{\sum_{i=1}^n(x_i-\bar x)(y_i - \bar y)}{\sqrt{\sum_{i=1}^n(x_i - \bar x)^2}\sqrt{\sum_{i=1}^n(y_i - \bar y)^2}}$

$\bar x$ and $\bar y$ are average of $x$ and $y$ , respectively. i is number of sample (incremental variable). n is number of sample.

Correlation coefficient (r) of 2 variables randomly extracted from population follows t-distribution. T-statistics of r is calculated formula as below and follows t-distribution with degree of freedom n-2, n is number of sample. When correlation coefficient of population is $\rho$ , null hypothesis is described that “ $\rho$ = 0″. If t-statistics calculated from number of sample (n) and correlation coefficient (r) is greater than that of significance level ( $\alpha$ ), null hypothesis is rejected.

$\displaystyle t = r\sqrt{\frac{n - 2}{1 - r^2}}$

The test of significance for this important null hypothesis H (ρ = 0) is equivalent to that for the null hypothesis H (β₁ = 0) or H (β₂ = 0). It now follows that if x and y have a joint bivariate normal distribution, then the test for the null hypothesis H (ρ = 0) is obtained by using the fact that if the null hypothesis under test is true, then
$\displaystyle F = \frac{(n-2)Z^2}{XY-Z^2} = \frac{(n-2)r^2}{1-r^2}\vspace{0.1in}\\ X = \sum(x - \bar{x})^2\vspace{0.1in}\\ Y = \sum(y - \bar{y})^2\vspace{0.1in}\\ Z = \sum(x - \bar{x})(y - \bar{y})\vspace{0.1in}\\ r^2 = \frac{Z^2}{XY}$
has the F distribution with 1, n – 2 d.f. An equivalent test of significance for the null hypothesis is obtained by using the fact that if the null hypothesis is true, then
$\displaystyle t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$
has “Student’s” distribution. with n – 2 d.f.

For any non-zero null hypothesis about ρ there is no parallelism between the correlation coefficient ρ and the regression coefficients β₁ and β₂. In fact, no exact test of significance is available for testing readily non-zero null hypothesis about ρ. Fisher has given an approximate method for such null hypothesis, but we do not consider this here.

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