Question

2 mark for review the table shows several values of x and their corresponding values of y. which of the following is closest to the correlation between x and y?

x	y
2	4
3	7
4	8
5	12

Explanation:

Step1: Recall correlation 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 $n = 5$.
$\sum x=1 + 2+3+4+5=15$.
$\sum y=3 + 4+7+8+12=34$.
$\sum xy=1\times3+2\times4 + 3\times7+4\times8+5\times12=3+8+21+32+60 = 124$.
$\sum x^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=1 + 4+9+16+25 = 55$.
$\sum y^{2}=3^{2}+4^{2}+7^{2}+8^{2}+12^{2}=9+16+49+64+144 = 282$.

Step2: Calculate the numerator

$n(\sum xy)-(\sum x)(\sum y)=5\times124-15\times34=620 - 510=110$.

Step3: Calculate the denominator

First part: $n\sum x^{2}-(\sum x)^{2}=5\times55-15^{2}=275 - 225 = 50$.
Second part: $n\sum y^{2}-(\sum y)^{2}=5\times282-34^{2}=1410-1156 = 254$.
Denominator $=\sqrt{50\times254}=\sqrt{12700}\approx112.69$.

Step4: Calculate the correlation coefficient

$r=\frac{110}{112.69}\approx0.98$.

Answer:

Approximately $0.98$