A researcher is investigating the relationship between two variables, X and Y, using Spearman’s rank correlation. They suspect that a third variable, Z, may confound this relationship. Data from 10 subjects, ranked for variables X, Y, and Z, are collected to analyze this potential confounding effect. Answer the following parts:

(a) Compute the Spearman’s rank correlation coefficient \(\rho(X,Y)\) using the formula \(\rho(X,Y)=1-\frac{6\sum d^2}{n(n^2-1)}\) for \(n=10\) and \(\sum d^2=24\).

(b) Given that \(\rho(X,Y)\) from part (a) is approximately 0.855, and the Spearman’s correlations between X and Z, and Y and Z are \(\rho(X,Z)=0.50\) and \(\rho(Y,Z)=0.60\) respectively, calculate the partial Spearman’s rank correlation coefficient \(\rho_{XY.Z}\) controlling for Z using the formula:

\(\rho_{XY.Z}=\frac{\rho(X,Y)-\rho(X,Z)\,\rho(Y,Z)}{\sqrt{(1-\rho(X,Z)^2)(1-\rho(Y,Z)^2)}}\).

(c) Conduct a significance test for the partial correlation \(\rho_{XY.Z}\) from part (b). Use the t-test statistic \(t=\rho_{XY.Z}\sqrt{\frac{n-3}{1-\rho_{XY.Z}^2}}\) with \(n=10\). With a critical t-value \(t_{crit} \approx 2.306\) for \(df=n-3\) at the 5% significance level, state whether the partial correlation is statistically significant. Interpret your result in the context of controlling for variable Z.