- 2065 Kartik · 10 marks
Forecast sales for year 2010
Short answer
Using a least-squares straight-line trend Y = 402.875 + 16.696X (X = 2×(year − 2003.5)), the forecast sales for 2010 is 619.9, approximately Rs 620 thousand.
Forecast the sales for year 2010.
| Year | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 |
|---|---|---|---|---|---|---|---|---|
| Sales Rs. '000 | 416 | 287 | 307 | 268 | 378 | 523 | 457 | 587 |
Method
Fit a straight-line trend Y = a + bX by least squares. There are 8 years (even), so take the mid-point 2003.5 as origin and use X = 2 × (year − 2003.5), giving X = −7, −5, −3, −1, 1, 3, 5, 7 (ΣX = 0). Then a = ΣY/n and b = ΣXY/ΣX².
| Year | X | Y (Rs '000) | XY | X² |
|---|---|---|---|---|
| 2000 | −7 | 416 | −2,912 | 49 |
| 2001 | −5 | 287 | −1,435 | 25 |
| 2002 | −3 | 307 | −921 | 9 |
| 2003 | −1 | 268 | −268 | 1 |
| 2004 | 1 | 378 | 378 | 1 |
| 2005 | 3 | 523 | 1,569 | 9 |
| 2006 | 5 | 457 | 2,285 | 25 |
| 2007 | 7 | 587 | 4,109 | 49 |
| Total | 0 | 3,223 | 2,805 | 168 |
a = 3,223 / 8 = 402.875 b = 2,805 / 168 = 16.696 (per 2 years, so about 8.35 per year)
Trend equation: Y = 402.875 + 16.696 X, with X = 2 × (year − 2003.5)
Forecast for 2010
X = 2 × (2010 − 2003.5) = 13 Y = 402.875 + 16.696 × 13 = 402.875 + 217.05 = 619.9
Answer: Forecast sales for 2010 = about Rs 620 thousand.
✓ Answer checked by Claude Opus 5.5 (high effort)
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