Question

calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. then calculate it again, letting row 2 represent the x - values and row 1 the y - values. what effect does switching the variables have on r? calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values. r = - 0.051 (round to three decimal places as needed.)

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n\sum xy-\sum x\sum y}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. Let $x$ be the values in Row 1 and $y$ be the values in Row 2. First, calculate the necessary sums:
Let the data points be $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$. Here $n = 7$.
$\sum x=19 + 20+40+50+57+61+77=324$
$\sum y=158+161+178+118+159+177+126 = 1077$
$\sum xy=19\times158+20\times161 + 40\times178+50\times118+57\times159+61\times177+77\times126$
$=3002+3220+7120+5900+9063+10797+9702$
$=48804$
$\sum x^{2}=19^{2}+20^{2}+40^{2}+50^{2}+57^{2}+61^{2}+77^{2}$
$=361+400+1600+2500+3249+3721+5929$
$=17760$
$\sum y^{2}=158^{2}+161^{2}+178^{2}+118^{2}+159^{2}+177^{2}+126^{2}$
$=24964+25921+31684+13924+25281+31329+15876$
$=168979$

Step2: Substitute into the formula

$r=\frac{7\times48804 - 324\times1077}{\sqrt{[7\times17760-324^{2}][7\times168979 - 1077^{2}]}}$
$=\frac{341628-348948}{\sqrt{[124320 - 104976][1182853-1159929]}}$
$=\frac{-7320}{\sqrt{(19344)(22924)}}$
$=\frac{-7320}{\sqrt{443568856}}$
$=\frac{-7320}{21061.06}$
$\approx - 0.347$

If we switch $x$ and $y$ (i.e., let Row 2 be $x$ - values and Row 1 be $y$ - values), the formula for $r$ remains the same. The correlation coefficient is symmetric with respect to $x$ and $y$. So the value of $r$ will be the same.

Answer:

The correlation coefficient $r\approx - 0.347$. Switching the variables has no effect on the value of $r$.