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