Question

compute the correlation coefficient.

x	10	31	25	38	-9	21	6
y	0	34	-8	25	36	3	27

send data to excel
use the ti - 84 plus calculator as needed.
the correlation coefficient is r = . round the answer to at least three decimal places.

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$, where $n$ is the number of data - points. Here $n = 7$.
Let $x=\{10,31,25,38, - 9,21,6\}$ and $y=\{0,34, - 8,25,36,3,27\}$.
First, calculate $\sum x$:
$\sum x=10 + 31+25+38+( - 9)+21+6=112$

Step2: Calculate $\sum y$

$\sum y=0 + 34+( - 8)+25+36+3+27=117$

Step3: Calculate $\sum xy$

$xy=\{10\times0,31\times34,25\times(-8),38\times25,(-9)\times36,21\times3,6\times27\}=\{0,1054, - 200,950,-324,63,162\}$
$\sum xy=0 + 1054-200 + 950-324+63+162=1695$

Step4: Calculate $\sum x^{2}$

$x^{2}=\{10^{2},31^{2},25^{2},38^{2},(-9)^{2},21^{2},6^{2}\}=\{100,961,625,1444,81,441,36\}$
$\sum x^{2}=100 + 961+625+1444+81+441+36=3688$

Step5: Calculate $\sum y^{2}$

$y^{2}=\{0^{2},34^{2},(-8)^{2},25^{2},36^{2},3^{2},27^{2}\}=\{0,1156,64,625,1296,9,729\}$
$\sum y^{2}=0 + 1156+64+625+1296+9+729=3879$

Step6: Substitute values into the formula

$n = 7$
$n\sum xy-\sum x\sum y=7\times1695-112\times117=11865 - 13104=-1239$
$n\sum x^{2}-(\sum x)^{2}=7\times3688-112^{2}=25816 - 12544 = 13272$
$n\sum y^{2}-(\sum y)^{2}=7\times3879-117^{2}=27153 - 13689 = 13464$
$\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}=\sqrt{13272\times13464}\approx\sqrt{178793208}\approx13368.33$
$r=\frac{-1239}{13368.33}\approx - 0.093$

Answer:

$-0.093$