Fit a straight line to following data and also estimate the production in 1957. | Year | 1951 | 1961 | 1971 | 1981 | 1991 || Production in Thousand Tones | 10 | 12 | 8 | 10

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written 2.9 years ago by

teamques10 ★ 64k

Since the values of X are odd and equispaced, We change X to x by the relation x= $=\dfrac{x-1971}{10}$

Let the straight line be y= a+bx.Then the normal equations are

$\sum yi=Na+b\sum xi\\[2ex] \sum xiyi=a\sum xi+b\sum xi^2$

Now the normal equation become,

53=5a $\therefore a=10.6$

$And\ 4=10b\ \therefore b=0.4$

Hence the equation of the straight line is y=10.6+0.4x

Calculations:

X	$x_i=\dfrac{X-1971}{10}$	$y_i$	$x_iy_i$	$x_i^2$
1951	-2	10	-20	4
1961	-1	12	-12	1
1971	0	8	0	0
1981	1	10	10	1
1991	2	13	26	4
	0	53	4	10

putting y=Y nad $x=\dfrac{X-1971}{10}$ the equations is,

$Y=10.6+0.4\left(\dfrac{X-1971}{10}\right)$

Y=-68.24+0.04

When X=1957, $Y= -68.24+0.04(1957)=10.04$

Hence, production in 1957 is 10.04

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