Question

a) state the random variables. rv x = the length of metacarpal of a randomly selected set of skeletal remains rv y = the height of a randomly selected set of skeletal remains b) the symbol and value of the correlation coefficient are as follows: round final answer to 3 decimal places. r = interpret this value: • there is a strong positive linear relation between length of metacarpal and height for sets of skeletal remains. c) the symbol and value of the coefficient of determination are as follows: round final answer to 3 decimal places. r² =

Explanation:

Step1: Recall correlation - coefficient formula

The formula for the correlation coefficient $r$ is $r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^{2}-(\sum x)^{2}][n\sum y^{2}-(\sum y)^{2}]}}$. First, calculate the necessary sums:
Let $x$ be the length of metacarpal and $y$ be the height. $n = 18$.
$\sum x=49 + 49+51+41+40+48+42+46+45+50+44+52+40+48+41+46+47+42=815$
$\sum y=185 + 170+180+162+163+171+165+169+173+181+173+176+155+174+165+173+172+175 = 3044$
$\sum x^{2}=49^{2}+49^{2}+51^{2}+41^{2}+40^{2}+48^{2}+42^{2}+46^{2}+45^{2}+50^{2}+44^{2}+52^{2}+40^{2}+48^{2}+41^{2}+46^{2}+47^{2}+42^{2}=37359$
$\sum y^{2}=185^{2}+170^{2}+180^{2}+162^{2}+163^{2}+171^{2}+165^{2}+169^{2}+173^{2}+181^{2}+173^{2}+176^{2}+155^{2}+174^{2}+165^{2}+173^{2}+172^{2}+175^{2}=518792$
$\sum xy=49\times185 + 49\times170+51\times180+41\times162+40\times163+48\times171+42\times165+46\times169+45\times173+50\times181+44\times173+52\times176+40\times155+48\times174+41\times165+46\times173+47\times172+42\times175=137997$

Step2: Calculate the correlation - coefficient $r$

$r=\frac{18\times137997 - 815\times3044}{\sqrt{(18\times37359-815^{2})(18\times518792 - 3044^{2})}}$
$=\frac{2483946-2474860}{\sqrt{(672462 - 664225)(9338256-9265936)}}$
$=\frac{9086}{\sqrt{(8237)(72320)}}$
$=\frac{9086}{\sqrt{595690840}}$
$=\frac{9086}{24406.78}$
$r\approx0.372$

Step3: Calculate the coefficient of determination $r^{2}$

The coefficient of determination $r^{2}$ is the square of the correlation - coefficient. So $r^{2}=(0.372)^{2}=0.138$

Answer:

b) $r = 0.372$
c) $r^{2}=0.138$