The yearly profit (in lakh Rs.) of a firm is given below.
| Year | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| Profit (in lakh Rs.) | 21 | 24 | 30 | 34 | 35 | 36 |
Taking \( \displaystyle X = \frac{\text{year} - 2021.5}{0.5}\), the straight-line trend fitted by the method of least squares is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(y = 30 + 1.6X\)
With the origin at 2021.5, X takes the values −5, −3, −1, 1, 3, 5 and \(\sum X = 0\). For the trend line \(y = a + bX\) the normal equations give
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum y = 21 + 24 + 30 + 34 + 35 + 36 = 180\)
\(\sum Xy = -105 - 72 - 30 + 34 + 105 + 180 = 112\)
\(\sum X^2 = 25 + 9 + 1 + 1 + 9 + 25 = 70\)
So
\( \displaystyle a = \frac{180}{6} = 30 \)
\( \displaystyle b = \frac{112}{70} = 1.6 \)
The trend line is \(y = 30 + 1.6X\).
Why the other options are incorrect
Option (1) \(y = 30 + 3.2X\) — Incorrect. 3.2 = 2b is the yearly rise of the trend; in terms of this X, which goes up by 2 each year, the coefficient is b = 1.6.
Option (2) \(y = 30 + 0.8X\) — Incorrect. The coefficient is 112 ÷ 70 = 1.6, not 0.8.
Option (4) \(y = 31.6 + 1.6X\) — Incorrect. The constant term is the mean 180 ÷ 6 = 30.