The number of students enrolled in a school is given below.
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| Students enrolled | 120 | 126 | 129 | 135 | 141 | 142 | 152 |
According to the straight-line trend fitted by the method of least squares, the number of students enrolled increases every year by
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 5
Let the trend line be \(y = a + bX\) with X = year − 2021, the middle year as origin, so X = −3 to 3 and \(\sum X = 0\). The trend value goes up by \(b\) every year, where
\( \displaystyle b = \frac{\sum Xy}{\sum X^2} \)
The sums are
\(\sum Xy = -360 - 252 - 129 + 0 + 141 + 284 + 456 = 140\)
\(\sum X^2 = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28\)
\( \displaystyle b = \frac{140}{28} = 5 \)
So the slope of the trend line is 5: the enrolment increases by 5 students every year.
Why the other options are incorrect
Option (1) 20 — Incorrect. It is \(\sum Xy\) divided by n = 7; the divisor must be \(\sum X^2\) = 28.
Option (2) 5.33 — Incorrect. It is (152 − 120) ÷ 6, which uses only the first and the last year, not the least-squares line.
Option (4) 4 — Incorrect. The slope is 140 ÷ 28 = 5.