For the given figures of production of a sugar factory, the estimate of the production for 1976 using straight line trend with origin at the year 1972 by the least squares method (∑x = 0, ∑x 2= 28, ∑xy = 56) is: Year Production(‘000 tons) (year) 1969 76 1970 87 1971 95 1972 81 1973 91 1974 96 1975 90
96
Step 1 — Trend equation: With origin at 1972 and \(\sum X = 0\), the least-squares straight line \(Y_c = a + bX\) has
\[a = \frac{\sum Y}{n},\quad b = \frac{\sum XY}{\sum X^2}\]
Step 2 — Compute a and b: \(\sum Y = 76+87+95+81+91+96+90 = 616\), \(n=7\).
\[a = \frac{616}{7}=88,\quad b = \frac{56}{28}=2\]
Step 3 — Estimate for 1976: \(X_{1976}=1976-1972=4\).
\[Y_c = 88 + 2(4) = 96\]
Therefore the estimated production is 96 ('000 tons).
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