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Let $A$ denote the percentage of one constituent in a randomly selected rock specimen, andlet $B$ 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 ofeach other and of actual values.(a) Show that$\operatorname{Corr}(X, Y)=\operatorname{Corr}(A, B) \cdot \sqrt{\operatorname{Corr}\left(X_{1}, X_{2}\right)} \cdot \sqrt{\operatorname{Corr}\left(Y_{1}, Y_{2}\right)}$where $X_{1}$ and $X_{2}$ are replicate measurements on the value of $A,$ and $Y_{1}$ and $Y_{1}$ 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 $\operatorname{Corr}(X, Y)$ when $\operatorname{Corr}\left(X_{1}, X_{2}\right)=.8100$ and $\operatorname{Corr}\left(Y_{1}, Y_{2}\right)$ $=.9025 ?$ Is this disturbing?

Intro Stats / AP Statistics

Chapter 4

Joint Probability Distributions and Their Applications

Section 11

Supplementary Exercises

Probability Topics

The Normal Distribution

Missouri State University

University of North Carolina at Chapel Hill

University of St. Thomas

Idaho State University

Lectures

0:00

Hey, it's clear. So when you read here So here is the correlation code fishing of X and Y, and we're gonna do the variances of random variable x and Y because of their independence. And I'm gonna do this. Us follow for access is equal to square of hey plus square of D and for why it's the same thing. But with B and he and the co variance between random variables, X and Y, you know, is going to be equal to the cove. Erian's uh hey A and B So we know that this stands for the correlation coefficient multiplied by core. Oh, that's one next to times a skirt correlation of why one by two. So the variances of random variables x one x two for X two. It's a print of saints this is equal to or w and E and for X to its w. And yes, well, so the cove arians of X one x two is equal to W. So the correlation coefficient, you know, for Exxon next to and C equals two. Yes, yes, we just plug it in. And the correlation coefficient of replicate measurement is always going to be between zero and one. So the square root is always gonna be between zero and one as well. And this shows that, um, the presence of measurement error will lessen the correlation for part B. The product of the square root of correlations is just skirt of 0.81 times square, 0.90 to 5, which is equal to a point okay by five. So we know that it is disturbing.

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