Chapter 5, Problem 94SE

Let A denote the percentage of one constituent in a randomly selected rock specimen, and let B denote the percentage of a second constituent in that same specimen. Suppose D and E are measurement errors in determining the values of A and B so that measured values are X = A + D and Y = B + E, respectively. Assume that measurement errors are independent of one another and of actual values.

a. Show that

     $Corr(X,Y)=Corr(A,B)\cdot \sqrt{Corr(X_1,X_2)}\cdot \sqrt{Corr(Y_1,Y_2)}$

where X₁ and X₂ are replicate measurements on the  value of A, and Y₁ and Y₂ are defined analogously  with respect to B. What effect does the presence of  measurement error have on the correlation?

b. What is the maximum value of Corr(X, Y) when Corr(X₁, X₂)  = .8100 and Corr(Y₁, Y₂) = .9025? Is this disturbing?