Let \( y_t=a+b(x_i-2021) \) be the straight line trend for the following data by least squares method:
| Year\( (x_i) \) | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
| Production in Lakh tonnes\( (y_i) \) | 32 | 34 | 30 | 35 | 38 | 39 | 42 |
the trend value for the year 2025 approximately is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 42.57
Let \( u=x_i-2021 \). Then u = −3, −2, −1, 0, 1, 2, 3, so \( \sum u=0 \) and the normal equations give a and b directly (n = 7):
\( \displaystyle \sum y=32+34+30+35+38+39+42=250 \)
\( \displaystyle \sum u^2=9+4+1+0+1+4+9=28 \)
\( \displaystyle \sum uy=-96-68-30+0+38+78+126=48 \)
\( \displaystyle a=\frac{\sum y}{n}=\frac{250}{7}\approx 35.714 \)
\( \displaystyle b=\frac{\sum uy}{\sum u^2}=\frac{48}{28}\approx 1.714 \)
For 2025, u = 2025 − 2021 = 4:
\( \displaystyle y_t=35.714+1.714\times 4 \)
\( \displaystyle =35.714+6.857 \)
\( \displaystyle \approx 42.57 \)