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

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

and are average of and , 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 , null hypothesis is described that “ = 0″. If t-statistics calculated from number of sample (n) and correlation coefficient (r) is greater than that of significance level (), null hypothesis is rejected.

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

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

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.