Q. 4.143 - Elementary Statistics

Question

Determine the linear correlation coefficient by using 

a. definition 4.8 on page 183.

b. Formula 4.3 on page 185.

Compare your answers in parts (a) and (b).

Step-by-Step Solution

Answer

The linear correlation coefficient of the given set of data is, $-0.685$.

Step 1 . Given information

Step 2 . Table for obtaining the linear correlation coefficient is,

$x$	$y$	$xy$	$x^2$	$y^2$
$0$	$4$	$0$	$0$	$16$
$2$	$2$	$4$	$4$	$4$
$2$	$0$	$0$	$4$	$0$
$5$	$-2$	$-10$	$25$	$4$
$6$	$1$	$6$	$36$	$1$
$\sum x=15$	$\sum y=5$	$\sum xy=0$	$\sum x^2=69$	$\sum y^2=25$

Step 3 . The linear correlation coefficient of given set of data is,

$$\begin{gathered} r=\frac{\sum x_i{y}_i-{\displaystyle \frac{\left(\sum x_i\right)\left(\sum y_i\right)}{n}}}{\sqrt{\left[\sum x_i^2-{\displaystyle \frac{{\left(\sum x_i\right)}^2}{n}}\right]\left[\sum y_i^2-\frac{{\left(\sum y_i\right)}^2}{n}\right]}} \\ =\frac{0-{\displaystyle \frac{\left(15\right)\left(5\right)}{5}}}{\sqrt{\left[69-{\displaystyle \frac{{\left(15\right)}^2}{5}}\right]\left[25-{\displaystyle \frac{{\left(5\right)}^2}{5}}\right]}} \\ =-0.685 \end{gathered}$$