Question

in his yard is as follows. 1.6 4.8 | | Price (£'000), Y Age (years), x 8 5 9.6 4 11.2 3 9 7 (i) Determine the equation of the regression line of "Price" on "age". (ii) Determine the coefficient of correlation between Price and Age. Comment on the association between them. (iii) Determine the best estimate of the price of a car whose age is 6.5 years.

Answer 1

X	Y	XY	X²	Y²
9	1.6	14.4	81	2.56
7	4.8	33.6	49	23.04
5	8	40	25	64
4	9.6	38.4	16	92.16
3	11.2	33.6	9	125.44
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
28	35.2	160	180	307.2

Sample size, n =	5
x̅ = Ʃx/n = 28/5 =	5.6
y̅ = Ʃy/n = 35.2/5 =	7.04
SSxx = Ʃx² - (Ʃx)²/n = 180 - (28)²/5 =	23.2
SSyy = Ʃy² - (Ʃy)²/n = 307.2 - (35.2)²/5 =	59.392
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 160 - (28)(35.2)/5 =	-37.12

i) Slope, b = SSxy/SSxx = -37.12/23.2 = -1.6

y-intercept, a = y̅ -b* x̅ = 7.04 - (-1.6)*5.6 = 16

Regression equation :

ŷ = 16 + (-1.6) x

ii) Correlation coefficient, r = SSxy/√(SSxx*SSyy) = -37.12/√(23.2*59.392) = -1.0

There is a prefect negative relationship between Price and Age.

iii) Predicted value of y at x = 6.5

ŷ = 16 + (-1.6) * 6.5 = 5.6