Question

compute the correlation coefficient.

x	-6	6	-5	25	26	40	33
y	32	6	33	29	-5	7	-10

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 $n = 7$.

Step2: Calculate $\sum x$, $\sum y$, $\sum xy$, $\sum x^{2}$, $\sum y^{2}$

• $\sum x=-6 + 6-5 + 25+26 + 40+33=119$
• $\sum y=32 + 6+33 + 29-5 + 7-10=92$
• $\sum xy=(-6)\times32+6\times6+(-5)\times33+25\times29+26\times(-5)+40\times7+33\times(-10)$

$=-192 + 36-165 + 725-130 + 280-330=224$

• $\sum x^{2}=(-6)^{2}+6^{2}+(-5)^{2}+25^{2}+26^{2}+40^{2}+33^{2}$

$=36 + 36+25+625+676+1600+1089=4087$

• $\sum y^{2}=32^{2}+6^{2}+33^{2}+29^{2}+(-5)^{2}+7^{2}+(-10)^{2}$

$=1024+36+1089+841+25+49+100=3164$

Step3: Substitute values into the formula

$n = 7$
$n\sum xy=7\times224 = 1568$
$(\sum x)(\sum y)=119\times92=10948$
$n\sum x^{2}=7\times4087 = 28609$
$(\sum x)^{2}=119^{2}=14161$
$n\sum y^{2}=7\times3164 = 22148$
$(\sum y)^{2}=92^{2}=8464$

$r=\frac{1568 - 10948}{\sqrt{(28609 - 14161)(22148 - 8464)}}$
$=\frac{-9380}{\sqrt{14448\times13684}}$
$=\frac{-9380}{\sqrt{197627432}}$
$=\frac{-9380}{14058.001}$
$\approx - 0.667$

Answer:

$-0.667$