For the following data
| Year | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| Profit (in lakh Rs.) | 16 | 17 | 24 | 28 | 29 | 36 |
using the method of least squares to fit a straight-line trend, the trend value of the profit (in lakh Rs.) for the year 2025 is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 39
There are 6 years (an even number), so the origin is the midpoint of the two middle years, 2021.5. To avoid fractions take
\( \displaystyle X = \frac{\text{year} - 2021.5}{0.5} \)
so X takes the values −5, −3, −1, 1, 3, 5 and \(\sum X = 0\). For the trend line \(y = a + bX\):
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum y = 16 + 17 + 24 + 28 + 29 + 36 = 150\)
\(\sum Xy = -80 - 51 - 24 + 28 + 87 + 180 = 140\)
\(\sum X^2 = 25 + 9 + 1 + 1 + 9 + 25 = 70\)
So
\( \displaystyle a = \frac{150}{6} = 25 \)
\( \displaystyle b = \frac{140}{70} = 2 \)
The trend line is \(y = 25 + 2X\). With this choice X increases by 2 each year, so for 2025
\( \displaystyle X = \frac{2025 - 2021.5}{0.5} = 7 \)
\(y = 25 + 2 \times 7 = 39\)
Why the other options are incorrect
Option (2) 32 — Incorrect. This comes from putting X = 2025 − 2021.5 = 3.5 into \(y = 25 + 2X\); with this line X must be 7 for 2025.
Option (3) 37 — Incorrect. It is the trend value for X = 6, which is not the X of any year (X is always odd here).
Option (4) 41 — Incorrect. It is the trend value for X = 8, which is not the X of any year either.