Question 4 OR (d)
Second-degree parabolic trend of production and a forecast for 2026
9 marks
- Time to write
- About 18 minutes
- Topic
- Trend fitting
Question
Question 4 OR (d) · 9 marksThe production of commodities in 2017-2023 is shown below:
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|
| Production ('000 tonnes) | 11 | 13 | 16 | 17 | 20 | 23 |
You have to:
Fit a parabolic trend of the second degree.
Compute the trend values for the years from 2018 to 2023.
Predict the production in the year 2026.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- ComputeFormula first, then the working, then the answer with its unit (₹, %, units).
What the paper leaves open
As printed, the heading says 2017-2023 but the table has six years, 2018 to 2023; the third item is numbered (ii) a second time.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Working table and normal equations 3 marks 5 min
- 2 Solving for a, b and c; the trend equation 3 marks 5 min
- 3 Trend values 2 marks 4 min
- 4 Forecast for 2026 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 18 minutes for 9 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Method. A second-degree parabola Y = a + bX + cX² is fitted by least squares, using normal equations as for a straight line. The table covers six years, 2018 to 2023 (the heading's "2017-2023" is a misprint, as the paper note says). With an even number of years, X = 2(year − 2020.5), in half-year units from the middle of 2020 and 2021, so ΣX = ΣX³ = 0 and
ΣY = na + cΣX²; ΣXY = bΣX²; ΣX²Y = aΣX² + cΣX⁴.
Working table (production Y in '000 tonnes)
| Year | Y | X | X² | X⁴ | XY | X²Y | Trend T |
|---|---|---|---|---|---|---|---|
| 2018 | 11 | −5 | 25 | 625 | −55 | 275 | 11.11 |
| 2019 | 13 | −3 | 9 | 81 | −39 | 117 | 13.09 |
| 2020 | 16 | −1 | 1 | 1 | −16 | 16 | 15.26 |
| 2021 | 17 | 1 | 1 | 1 | 17 | 17 | 17.60 |
| 2022 | 20 | 3 | 9 | 81 | 60 | 180 | 20.12 |
| 2023 | 23 | 5 | 25 | 625 | 115 | 575 | 22.82 |
| Total | 100 | 0 | 70 | 1,414 | 82 | 1,180 |
- From ΣXY = bΣX²: 82 = 70b, so b = 1.1714.
- 100 = 6a + 70c … (1) and 1,180 = 70a + 1,414c … (2). Multiplying (1) by 70 and (2) by 6: 7,000 = 420a + 4,900c and 7,080 = 420a + 8,484c. Subtracting, 80 = 3,584c, so c = 0.0223; then 6a = 100 − 70 × 0.022321 = 98.4375, a = 16.4062.
Y = 16.4062 + 1.1714X + 0.0223X², origin: middle of 2020 and 2021 (1 January 2021); X unit: half a year; Y in '000 tonnes.
(ii) Trend values, 2018 to 2023
From the equation (last column of the table): 2018 11.11; 2019 13.09; 2020 15.26; 2021 17.60; 2022 20.12; 2023 22.82 thousand tonnes. The yearly rise in trend grows slowly (1.98, 2.17, 2.34, 2.52, 2.70), reflecting the small positive c.
For 2026, X = 2(2026 − 2020.5) = 11: Y = 16.4062 + 1.1714 × 11 + 0.0223 × 121 = 16.4062 + 12.8857 + 2.7009 = 31.99 thousand tonnes (about 31,990 tonnes).
The forecast assumes the gently accelerating growth of 2018–2023 continues for three more years.
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table and normal equations | 3 |
| 2 | Solving for a, b and c; the trend equation | 3 |
| 3 | Trend values | 2 |
| 4 | Forecast for 2026 | 1 |
| Total | 9 |
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