What is a generic constant $k$ in the context of significance of the correlation coefficient ? Please see $(11.4.10)$ the snippet below:
2026-04-01 23:29:12.1775086152
generic constant and Student d.f.
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They are describing various rules to decide whether you reject the null hypothesis or not. One such rule is "reject if and only if $(1-r^2)^{n/2} < 0.5$." Another rule is "reject if and only if $(1-r^2)^{n/2} < 0.2$." Different values of $k$ will give you different decision rules.
The bottom of your screenshot describes how to choose $k$ so that the Type I error (probability of rejecting when the null hypothesis is true) is $\alpha$.