I am trying to understand the test for absence of correlation. I would like to understand why the test statistic is given in the following manner:
In the case of corelation it is
Could the reason be that the distribution for $b_{1}=$correlation coefficient
Is difficult to figure out, so instead $r$ is expressed in terms of t, so that the (known)distribution of t can be used?
I would really appreciate some tips. I have gone through many books and I couldn't find any explanation as to why the statstic is the way it is.
The above test statistic is actually used to test whether the true regression coefficient ($\beta_1$) is different from zero. You can still use this to test whether the correlation coefficient ($\rho$) is different from 0 by noting that $\rho > 0 \iff \beta_1 > 0$ and $\rho < 0 \iff \beta_1 < 0$.
If you want to test whether the true correlation is different from another value, say $\rho = 0.4$, you have to use a different test statistic from the one above. In this case you need an arctanh transform to obtain a usable test statistic, see https://en.wikipedia.org/wiki/Fisher_transformation. Note that this procedure relies on the assumption that the response and covariates follow a bivariate normal distribution.