For the following data
| Year | 2020 | 2021 | 2022 | 2023 | 2024 |
| Sales (in lakh Rs.) | 12 | 15 | 19 | 22 | 27 |
using the method of least squares to fit a straight-line trend, the trend value of the sales (in lakh Rs.) for the year 2026 is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 33.8
The trend line is \(y = a + bX\). There are 5 years (an odd number), so the origin is taken at the middle year 2022. Then X = year − 2022 takes the values −2, −1, 0, 1, 2 and \(\sum X = 0\), and the normal equations give
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum y = 12 + 15 + 19 + 22 + 27 = 95\)
\(\sum Xy = -24 - 15 + 0 + 22 + 54 = 37\)
\(\sum X^2 = 4 + 1 + 0 + 1 + 4 = 10\)
So
\( \displaystyle a = \frac{95}{5} = 19 \)
\( \displaystyle b = \frac{37}{10} = 3.7 \)
The trend line is \(y = 19 + 3.7X\). For 2026, X = 2026 − 2022 = 4:
\(y = 19 + 3.7 \times 4 = 19 + 14.8 = 33.8\)