0
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
straight line • 2.5k  views
0
2

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

2