Question

Fit equation of trend line for the data given below.

Year	Production (y)	x	x2	xy
2006	19	– 9	81	– 171
2007	20	– 7	49	– 140
2008	14	– 5	25	– 70
2009	16	– 3	9	– 48
2010	17	– 1	1	– 17
2011	16	1	1	16
2012	18	3	9	54
2013	17	5	25	85
2014	21	7	49	147
2015	19	9	81	171
Total	177	0	330	27

Let the equation of trend line be y = a + bx   .....(i)

Here n = `square` (even), two middle years are `square` and 2011, and h = `square`

The normal equations are Σy = na + bΣx

As Σx = 0, a = `square`

Also, Σxy = aΣx + bΣx²

As Σx = 0, b = `square`

Substitute values of a and b in equation (i) the equation of trend line is `square`

To find trend value for the year 2016, put x = `square` in the above equation.

y = `square`

Answer 1

Let the equation of trend line be y = a + bx   .....(i)

Here n = 10 (even), two middle years are 2010 and 2011, and h = 2 

The normal equations are Σy = na + bΣx

As Σx = 0,

∴ 177 = 10a + b(0)

∴ a = `177/10`

a = 17.7 

Also, Σxy = aΣx + bΣx²

As Σx = 0, 

27 = a(0) + b(330)

∴ b = `27/330`

= 0.08

b = 0.1

Substitute values of a and b in equation (i) the equation of trend line is y = 17.7 + 0.1x

To find trend value for the year 2016, put x = 11 in the above equation.

y = 17.7 + 1.1

y = 18.8