Question

For the certain bivariate data on 5 pairs of observations given

∑x = 20, ∑y = 20, ∑x² = 90, ∑y² = 90, ∑xy = 76

Calculate: 

1. cov(x, y)
2. b_yx and b_xy
3. r

Answer 1

Given, ∑x = 20, ∑y = 20, ∑x² = 90, ∑y² = 90, ∑xy = 76, n = 5

Now, 

`bar x = (sumx)/n = 20/5` = 4

`bar y = (sumy)/n = 20/5` = 4

(i) cov(x, y) = `1/n sumxy - barx bary`

= `1/5 xx 76 - 4 xx 4`

= 15.2 – 16

= - 0.8

(ii) b_yx = `(sumxy - nbarxbary)/(sumx^2 - nbarx^2)`

= `(76 - 5 xx 4 xx 4)/(90 - 5(4)^2)`

= `(76 - 80)/(90 - 80)`

= `-(4)/10`

= - 0.4

b_xy = `(sumxy - nbarxbary)/(sumy^2 - nbary^2)`

= `(76 - 5 xx 4 xx 4)/(90 - 5(4)^2)`

= `(76 - 80)/(90 - 80)`

= - 0.4

(iii) r = `+-sqrt(b_xy. b_yx)`

= `+- sqrt((-0.4)(-0.4))`

= `+- sqrt(0.16)`

= ± 0.4

Since b_yx and b_xy both are negative,

∴ r is negative.

∴ r = - 0.4