Question

Find correlation coefficient from the following data. `["Given:" sqrt(3) = 1.732]`

x	3	6	2	9	5
y	4	5	8	6	7

Answer 1

xi	yi	xi2	yi2	xiyi
3	4	9	16	12
6	5	36	25	30
2	8	4	64	16
9	6	81	36	54
5	7	25	49	35
25	30	155	190	147

From the table, we have

n = 5, `sum"x"_"i"` = 25, `sum"y"_"i"` = 30, `sum"x"_"i"^2` = 155, `sum"y"_"i"^2` = 190, `sum"x"_"i""y"_"i"` = 147

`bar"x"=(sum"x"_"i")/"n"`

= `25/5`

= 5

`bar"y"=(sum"y"_"i")/"n"`

= `30/5`

= 6

Since, Cov (x, y) = `1/"n"sum"x"_"i""y"_"i"-bar"x"bar"y"`

∴ Cov (x, y) = `1/5xx147-(5xx6)`

= 29.4 − 30

 = − 0.6

`sigma_"x"^2=(sum"x"_"i"^2)/"n"-(bar"x")^2`

= `155/5-(5)^2`

 = 31 − 25

∴ `sigma_"x"^2` = 6

∴ σ_x = `sqrt6`

`sigma_"y"^2=(sum"y"_"i"^2)/"n"-(bar"y")^2`

= `190/5-(6)^2`

= 38 − 36

∴ `sigma_"y"^2` = 2

∴ σ_{y =} `sqrt2`

∴ σ_xσ_y = `sqrt6sqrt2=sqrt12`

= `2sqrt3`

= 2(1.732) = 3.464

Thus, the correlation coefficient between x and y is

r = `("Cov"("x,y"))/(sigma_"x"sigma_"y")`

= `(-0.6)/3.464`

= − 0.1732