Question

the table shows the number of flowers in four bouquets and the total cost of each bouquet. what is the correlation coefficient for the data in the table? cost of bouquets number of flowers in the bouquet total cost 8 $12 12 $40 6 $15 20 $20 -0.57 -0.28 0.28 0.57

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 number of flowers and $y$ be the total cost.
For the given data:
$n = 4$
$\sum x=8 + 12+6 + 20=46$
$\sum y=12 + 40+15 + 20=87$
$\sum xy=(8\times12)+(12\times40)+(6\times15)+(20\times20)=96 + 480+90 + 400=1066$
$\sum x^{2}=8^{2}+12^{2}+6^{2}+20^{2}=64 + 144+36 + 400=644$
$\sum y^{2}=12^{2}+40^{2}+15^{2}+20^{2}=144 + 1600+225 + 400=2369$

Step2: Calculate the numerator

$n(\sum xy)-(\sum x)(\sum y)=4\times1066-46\times87=4264 - 4002=262$

Step3: Calculate the first - part of the denominator

$n\sum x^{2}-(\sum x)^{2}=4\times644-46^{2}=2576 - 2116 = 460$

Step4: Calculate the second - part of the denominator

$n\sum y^{2}-(\sum y)^{2}=4\times2369-87^{2}=9476 - 7569=1907$

Step5: Calculate the denominator

$\sqrt{(n\sum x^{2}-(\sum x)^{2})(n\sum y^{2}-(\sum y)^{2})}=\sqrt{460\times1907}=\sqrt{877220}\approx936.6$

Step6: Calculate the correlation coefficient

$r=\frac{262}{936.6}\approx0.28$

Answer:

$0.28$