If \(y = a + b(x - 2005)\) fits the time series data
| x (year) : | 2003 | 2004 | 2005 | 2006 | 2007 |
| y (yield in tons) : | 6 | 13 | 17 | 20 | 24 |
Then the value of a + b is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 20.3
The line is fitted by the method of least squares. Let u = x − 2005, so u takes the values −2, −1, 0, 1, 2 and \(\sum u = 0\). The normal equations then reduce to
\( \displaystyle a = \frac{\sum y}{n},\qquad b = \frac{\sum uy}{\sum u^2} \)
The sums needed:
\(\sum y = 6 + 13 + 17 + 20 + 24 = 80\)
\(\sum uy = -12 - 13 + 0 + 20 + 48 = 43\)
\(\sum u^2 = 4 + 1 + 0 + 1 + 4 = 10\)
Hence
\( \displaystyle a = \frac{80}{5} = 16,\qquad b = \frac{43}{10} = 4.3 \)
\(a + b = 16 + 4.3 = 20.3\)