Consider the following data of expenses (in lakhs) of an organization year wise
| Year | 2001 | 2002 | 2003 | 2004 | 2005 |
| Expenses (Rs. lakh) | 160 | 185 | 220 | 300 | 510 |
Then expected expenses trends for the year 2006 using method of least square is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → Rs. 519.5 lakh
Fit the trend line \(y = a + bt\) by the method of least squares. With 5 years (odd), take the middle year 2003 as origin, \(t = \text{year} - 2003\), so \(t = -2, -1, 0, 1, 2\) and \(\sum t = 0\).
\(\sum y = 160 + 185 + 220 + 300 + 510 = 1375\)
\(\sum t^2 = 4 + 1 + 0 + 1 + 4 = 10\)
\(\sum ty = -320 - 185 + 0 + 300 + 1020 = 815\)
Since \(\sum t = 0\), the normal equations give
\( \displaystyle a = \frac{\sum y}{n} = \frac{1375}{5} = 275 \)
\( \displaystyle b = \frac{\sum ty}{\sum t^2} = \frac{815}{10} = 81.5 \)
The trend line is \(y = 275 + 81.5t\). For 2006, \(t = 3\):
\(y = 275 + 81.5 \times 3 = 275 + 244.5 = 519.5\)
So the expected expenses for 2006 are Rs. 519.5 lakh.
CBSE Applied Mathematics (Class XII):
Index Numbers and Time Based Data
