The number of landline telephone connections (in lakh) in a district is given below.
| Year | 2020 | 2021 | 2022 | 2023 | 2024 |
| Connections (in lakh) | 48 | 45 | 41 | 38 | 33 |
Using a straight-line trend fitted by the method of least squares, the trend value of the connections (in lakh) for the year 2026 is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 26.2
Let the trend line be \(y = a + bX\) with X = year − 2022, the middle year as origin. Then X = −2, −1, 0, 1, 2 and \(\sum X = 0\), so
\( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum y = 48 + 45 + 41 + 38 + 33 = 205\)
\(\sum Xy = -96 - 45 + 0 + 38 + 66 = -37\)
\(\sum X^2 = 4 + 1 + 0 + 1 + 4 = 10\)
So
\( \displaystyle a = \frac{205}{5} = 41 \)
\( \displaystyle b = \frac{-37}{10} = -3.7 \)
The trend line is \(y = 41 - 3.7X\); the negative slope shows a falling trend. For 2026, X = 4:
\(y = 41 - 3.7 \times 4\)
\(= 41 - 14.8\)
\(= 26.2\)
Why the other options are incorrect
Option (1) 55.8 — Incorrect. It comes from taking b as +3.7; the series is falling, so b = −3.7.
Option (2) 33.6 — Incorrect. It is the trend value for 2024 (X = 2), not for 2026.
Option (3) 22.5 — Incorrect. It is the trend value for 2027 (X = 5), not for 2026.