Question

on a separate piece of paper, complete the following table to calculate σ(x - x̄)(y - ȳ). complete the table to calculate the sums

x	y	x - x̄	y - ȳ	(x - x̄)(y - ȳ)	(x - x̄)²	(y - ȳ)²
2	3
3	6
5	7
6	8
9	11
x̄ =	ȳ =			σ =	σ =	σ =

48
27
47
54
31

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

$\bar{x}=\frac{2 + 3+5+6+9}{5}=\frac{25}{5} = 5$
$\bar{y}=\frac{3 + 6+7+8+11}{5}=\frac{35}{5}=7$

Step2: Calculate \(x-\bar{x}\), \(y - \bar{y}\), \((x-\bar{x})(y - \bar{y})\), \((x-\bar{x})^2\) and \((y-\bar{y})^2\) for each pair

For \((x = 2,y = 3)\):
$x-\bar{x}=2 - 5=-3$
$y-\bar{y}=3 - 7=-4$
$(x-\bar{x})(y - \bar{y})=(-3)\times(-4) = 12$
$(x-\bar{x})^2=(-3)^2 = 9$
$(y-\bar{y})^2=(-4)^2 = 16$

For \((x = 3,y = 6)\):
$x-\bar{x}=3 - 5=-2$
$y-\bar{y}=6 - 7=-1$
$(x-\bar{x})(y - \bar{y})=(-2)\times(-1)=2$
$(x-\bar{x})^2=(-2)^2 = 4$
$(y-\bar{y})^2=(-1)^2 = 1$

For \((x = 5,y = 7)\):
$x-\bar{x}=5 - 5=0$
$y-\bar{y}=7 - 7=0$
$(x-\bar{x})(y - \bar{y})=0\times0 = 0$
$(x-\bar{x})^2=0^2 = 0$
$(y-\bar{y})^2=0^2 = 0$

For \((x = 6,y = 8)\):
$x-\bar{x}=6 - 5=1$
$y-\bar{y}=8 - 7=1$
$(x-\bar{x})(y - \bar{y})=1\times1 = 1$
$(x-\bar{x})^2=1^2 = 1$
$(y-\bar{y})^2=1^2 = 1$

For \((x = 9,y = 11)\):
$x-\bar{x}=9 - 5=4$
$y-\bar{y}=11 - 7=4$
$(x-\bar{x})(y - \bar{y})=4\times4 = 16$
$(x-\bar{x})^2=4^2 = 16$
$(y-\bar{y})^2=4^2 = 16$

Step3: Calculate the sums

$\sum(x-\bar{x})(y - \bar{y})=12 + 2+0+1+16=48$
$\sum(x-\bar{x})^2=9 + 4+0+1+16=30$
$\sum(y-\bar{y})^2=16 + 1+0+1+16=34$

Answer:

48