Partial correlation: The partial correlation between X 1 and Y ‘‘controlling for’’ X 2 through X k is defined as the simple correlation between the residuals E Y/2 ... k and E 1/2 ... k , given in the...

Partial correlation: The partial correlation between X₁
and Y ‘‘controlling for’’ X₂
through X_k
is defined as the simple correlation between the residuals E_{Y/2 ... k}
and E_{1/2 ... k ,}
given in the previous exercise. The partial correlation is denoted r_Y1/2...k
.

(a) Using the Canadian occupational prestige data, calculate the partial correlation between prestige and education, controlling for income and percentage women (see the previous exercise).

(b) In light of the interpretation of a partial regression coefficient developed in the previous exercise, why is r_{Y1=2 ... k}
= 0 if and only if B₁
(from the multiple regression of Y on X₁
through X_k
) is 0?