Question

Fit a linear equation from the following data for variable (y) of a time series: n = 4, Σy = 270, Σty = 734

Answer 1

n = 4 is given. So t = 1, 2, 3, 4

∴ Σt = 1 + 2 + 3 + 4=10 and

Σt² = 1 + 4 + 9 + 16 = 30

Here, n = 4; Σt = 10; Σt² = 30;

Σy = 270; Σty = 734

∴ t̄ = \(\frac{Σt}{n} = \frac{10}{4} \)= 2.5

ȳ = \(\frac{Σy}{n} = \frac{270}{4}\) = 67.5

Putting n = 4, Σty = 734, Σt = 10, Σy = 270, Σt² = 30 in the formula,

a = ȳ – bt̄
Putting ȳ = 67.5, b = 11.8 and t̄ = 2.5, we get
a = 67.5 – 11.8 (2.5)
= 67.5 – 29.5
= 38

Equation of linear trend:
Putting a = 38, b = 11.8 in ŷ = a + bt, we get ŷ = 38 + 11.8t

Hence, the equation of linear trend obtained is ŷ = 38 + 11.8t.