Chapter 3.1, Problem 24E

(a)

To determine

To Find: change in correlation

Answer to Problem 24E

The correlation is not changed

Explanation of Solution

The correlation is not changed. Because $r$ uses the standardized values of the observations, $r$ does not change when we add or subtract any number $X,Y$ or both. Subtracting 6 inches’ from the height of men does not change the correlation between them. The correlation does not help to tell whether women tend to date men taller or shorter than themselves. Instead it says that the men that shorter women tend to date are shorter than the men the taller women tend to date.

  $\begin{array}{l}r=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n\left(\frac{X_i-\overline{X}}{S_x}\right)}\left(\frac{Y_i-\overline{Y}}{S_r}\right)\\ where\\ \overline{X}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{X}_i}\\ \overline{Y}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{Y}_i}\\ S_x=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{{\displaystyle \left(X_i-\overline{X}\right)}}^2}}\\ S_y=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{{\displaystyle \left(Y_i-\overline{Y}\right)}}^2}}\end{array}$

(b)

To determine

To Find: change in correlation if heights measured in centimeters

Answer to Problem 24E

The correlation coefficient $r$ does not change.

Explanation of Solution

The correlation coefficient $r$ does not change. Because r uses the standardized values of the observations, $r$ does not change when we change the units of measurement of $X,Y$ or both. Measuring lengths in millimeter rather than centimeter would not change the correlation between them.