The yearly output (in thousand units) of a workshop is given below. Y…

The correct answer is Option (2) → 36.9

With X = year − 2022.5 the four years have

    X = −1.5, −0.5, 0.5, 1.5

so \(\sum X = 0\) and

    \( \displaystyle a = \frac{\sum y}{n}, \quad b = \frac{\sum Xy}{\sum X^2} \)

The sums are

    \(\sum y = 20 + 23 + 27 + 30 = 100\)
    \(\sum X^2 = 2.25 + 0.25 + 0.25 + 2.25 = 5\)

    \(\sum Xy = -30 - 11.5 + 13.5 + 45 = 17\)

So

    \( \displaystyle a = \frac{100}{4} = 25 \)

    \( \displaystyle b = \frac{17}{5} = 3.4 \)

Here X goes up by 1 each year, so b = 3.4 is the yearly rise. For 2026, X = 2026 − 2022.5 = 3.5:

    \(y = 25 + 3.4 \times 3.5 = 25 + 11.9 = 36.9\)

Why the other options are incorrect

• Option (1) 33.5 — Incorrect. This is the trend value for 2025 (X = 2.5).

• Option (3) 48.8 — Incorrect. This puts X = 7, the value for 2026 when X is counted in half-years; with X = year − 2022.5 the value is 3.5.

• Option (4) 40.3 — Incorrect. This is the trend value for 2027 (X = 4.5).