calculate the correlation coefficient r, letting row 1 represent the x - values and row 2 the y - values…

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 Row 1 be $x$ - values: $x_1 = 19,x_2=20,x_3 = 40,x_4=50,x_5=57,x_6=61,x_7 = 77$ and Row 2 be $y$ - values: $y_1 = 158,y_2=161,y_3 = 178,y_4=118,y_5=159,y_6=177,y_7 = 126$. Calculate $\sum x=19 + 20+40+50+57+61+77=324$, $\sum y=158 + 161+178+118+159+177+126=1077$, $\sum xy=19\times158+20\times161+40\times178+50\times118+57\times159+61\times177+77\times126=19\times158+20\times161+40\times178+50\times118+57\times159+61\times177+77\times126=3002+3220+7120+5900+9063+10797+9702=48804$, $\sum x^{2}=19^{2}+20^{2}+40^{2}+50^{2}+57^{2}+61^{2}+77^{2}=361+400+1600+2500+3249+3721+5929=17760$, $\sum y^{2}=158^{2}+161^{2}+178^{2}+118^{2}+159^{2}+177^{2}+126^{2}=24964+25921+31684+13924+25281+31329+15876=168979$.

Step2: Substitute values into formula

$r=\frac{7\times48804 - 324\times1077}{\sqrt{(7\times17760-324^{2})(7\times168979 - 1077^{2})}}$ First, $7\times48804-324\times1077=341628-348948=-7320$. Second, $7\times17760 - 324^{2}=124320 - 104976 = 19344$, $7\times168979-1077^{2}=1182853-1159929 = 22924$. Then $\sqrt{(7\times17760 - 324^{2})(7\times168979 - 1077^{2})}=\sqrt{19344\times22924}=\sqrt{443441856}\approx21058$. $r=\frac{-7320}{21058}\approx - 0.347$. If we switch $x$ and $y$ (i.e., let Row 2 be $y$ - values and Row 1 be $x$ - values), the correlation coefficient formula is symmetric in $x$ and $y$. So the value of $r$ remains the same.

Answer:

$-0.347$