Question

Estimate the production for the year 2010 and 2012 from the following data.

Answer 1

Let X and Y be year and production.

Since the known values of ‘Y’ are four, the fourth order leading differences will be 

zero, i.e. (y- 1)⁴ = 0 or Δ⁴ = 0

(y – 1 )⁴ = y₄ – 4y₃ + 6y₂ – 4y₁ + y₀ = 0 (i) 

and the second equation can be obtained by increasing the suffixes of each term of ‘Y’ by one, keeping the. coefficients same;

we get

y₅ – 4y₄ + 6y₃ – 4y₂ + y₁ = 0 ……(ii)

From equation (i) 52 – 4y₃ + 6 (28) – 4(18) + 8 = 0

52 – 4y₃ + 168 -72 + 8 = 0

So, by simplifying, – 4y₃ = – 156

∴ y₃ = 39 ie. Production for the year 2010 is 139 (tons).

From (ii) y₅ – 4y₄ + 6y₃ – 4y₂ + y₁ = 0; Here y₃ = 39

y₅ – 4 x 52 + 6 x (39) – 4 x 28 + 18 = 0

y₅ – 208 + 234 – 112 + 18 = 0

∴ y₅ = 68 i.e. the production for the year 2012 is 68(tons)