Fit a straight line to the following data and estimate the production in the year 1957:

Year	1951	1961	1971	1981	1991
Production in thousand tons	10	12	08	10	13

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written 9.3 years ago by

teamques10 ★ 70k

Let, the production be

We write X such that $ \sum X=0 $ . Here N=5 (odd)

x	$X_i$	$Y_i$	$X_i^2$	$X_iY_i$
1951	-20	10	400	-200
1961	-10	12	100	-120
1971	0	08	0	0
1981	10	10	100	100
1991	20	13	400	260
N=5	$\sum X_i=0$	$ \sum Y_i = 53$	$ \sum X_i^2=1000$	$\sum X_iY_i=40$

The equations are:-

$ \sum Y_i \;=\; Na + b \sum X_i \; \; \therefore 53=5a+b(0) \; \; a \;=\; \dfrac{53}{5} = 10.6 \\ \; \\ \; \\ \sum X_i Y_i \;=\; a \sum X_i + b \sum X_i^2 \\ \; \\ \; \\ \; \; \; \therefore 40=0a+1000b \; \; \; \therefore b\;=\; \dfrac{40}{1000} \;=\; 0.04 $

The equation of straight line is $y=a+bx_1$

$ \therefore y=10.6+0.04x_1 $

To find production in year 1957

Then in this case
$ x_i \;=\; 1957-1971 = -14$

Production in year 1957= y= 10.6+0.04(-14) = 10.04 Thousand Tones

$ y=10.6+0.04x_1 \; \; \; \; where \; \; x_i = X-1971 $

Production in year 1957=10.04 Thousand Tones

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