Question

Consider the following data of a bivariate distribution:

• The mean of the variablesx and y are 25 and 30 respectively.
• The regression coefficient of x on y is 0.4 and the regression coefficient of y on x is 1.6.

(a) Find the lines of best fit for the bivariate distribution.   [2]

(b) Estimate the value of y when x = 60.   [1]

(c) What is the coefficient of correlation between x and y?   [1]

Answer 1

(a) 

Regression of y on x: y − 30

= 1.6(x − 25)

⇒ y = 1.6x − 10.

Regression of x on y: x − 25

= 0.4(y − 30)

⇒ x = 0.4y + 13.

(b) Estimate y when x = 60 (use y on x): y = 1.6(60) − 10 = 96 − 10 = 86.

(c) Coefficient of correlation r 

r^2 = (regression coeff. of x on y) × (regression coeff. of y on x)

= 0.4 × 1.6

= 0.64,

so r = ±√0.64

= ±0.8.

Both regression slopes are positive, hence r = +0.8.