Let x and y be the year and population.
The quadratic equation is –
y = a + bx + cΣx2
Normal equations are na + blx + clx2
= ly alx + bΣx2 + cΣx3
= Σxy aΣx2 + bΣx3 + cΣx4 = Σx2y
From (1) 5a + b.0 + c.10 = 3550
i.e. 5a + 10c = 3550 -(4)
From (2) a.0 + 6.10 + c.0 = 1410
∴ b = 141
From (3) a.10 + b.0 + c.34 = 7310
i.e. 10a + 34c =7310 (5)
By solving (4) and (5) we get a and c
(4) x2 – (5)
Substitute c = 15 in (4)
we get 5a +10(15) = 3550
∴ a = 680
The fitted quadratic trend equation is
ŷ = 680 + 141x + 15x2
Estimation of the population for the year 2011
Here x = 3, 2 = 9 ŷ 2011 = 1238 millions.
