Testing the correlation coefficient
The sample correlation coefficient is r and the sample size n.
To test the hypothesis that in the population the unknown
correlation coefficient rho is r0 against the alternative rho>r0,
we form the test statistic
U=sqrt(n-3)*(Fisher(r)-Fisher(r0))
where
Fisher(r):=0.5*log((1+r)/(1-r))
is Fisher's transformation of the correlation coefficient.
If the null hypothesis is true, U is approximately N(0,1).
Hence we reject the hypothesis, if P=1-N.F(0,1,U)
is less than the risk level (say 0.05).
Assume now that n=50, r=0.81 and r0=0.7
Then U.=1.7806088746071 P.=0.03748818448581
Thus the hypothesis rho=0.7 is rejected on the risk level 0.05.