5.6 Tests

A sample coefficient \(r_s\) estimates the population coefficient \(\rho _s\), and the question is whether an observed \(r_s\) is far enough from zero to rule out \(\rho _s=0\). The critical values are tabulated for small \(n\) and approximated for large \(n\).

(i).

For \(H_0:\rho _s\leq 0\) against \(H_a:\rho _s>0\), reject \(H_0\) if \(r_s\) exceeds the critical value at level \(\alpha \).

(ii).

For \(H_0:\rho _s\geq 0\) against \(H_a:\rho _s<0\), reject \(H_0\) if \(r_s<0\) and \(\left |r_s\right |\) exceeds the critical value at level \(\alpha \).

(iii).

For \(H_0:\rho _s=0\) against \(H_a:\rho _s\neq 0\), reject \(H_0\) if \(\left |r_s\right |\) exceeds the critical value at level \(\alpha /2\).

The halving belongs to the two-sided test alone. A one-sided test spends its whole \(\alpha \) in one tail, and using \(\alpha /2\) there makes the test conservative without cause.

For moderate \(n\), roughly \(20<n<40\), \[t=r_s\sqrt {\dfrac {n-2}{1-r_s^{2}}}\ \thicksim \ t_{n-2},\] referred to one- or two-tailed critical values of the \(t\) distribution on \(n-2\) degrees of freedom. An equivalent form sometimes quoted is \[F=\dfrac {1+\left |r_s\right |}{1-\left |r_s\right |}\ \thicksim \ F_{(n-2,\,n-2)} .\] For \(n\geq 40\) the simpler large-sample result \[Z=r_s\sqrt {n-1}\ \thicksim \ N(0,1)\] is adequate.