Question 4 OR (c)
Second-degree parabolic trend of production and a forecast for 2027
9 marks
- Time to write
- About 18 minutes
- Topic
- Trend fitting
Question
Question 4 OR (c) · 9 marksThe production of a commodity in 2019-2024 is shown below:
| Year | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
|---|---|---|---|---|---|---|
| Production ('000 tonnes) | 14 | 16 | 19 | 21 | 24 | 28 |
You are required to:
Fit a parabolic trend of second degree,
Compute the trend values for each of the years 2019 through 2024.
Predict the production in the year 2027.
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).
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 2027 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, as the syllabus requires, extending the normal-equation approach used for a straight line. With six years, time is measured from the middle of 2021 and 2022 in half-year units, X = 2(year − 2021.5), so that ΣX = ΣX³ = 0 and the three normal equations become
ΣY = na + cΣX²; ΣXY = bΣX²; ΣX²Y = aΣX² + cΣX⁴.
Working table (production Y in '000 tonnes)
| Year | Y | X | X² | X⁴ | XY | X²Y | Trend T |
|---|---|---|---|---|---|---|---|
| 2019 | 14 | −5 | 25 | 625 | −70 | 350 | 14.07 |
| 2020 | 16 | −3 | 9 | 81 | −48 | 144 | 16.10 |
| 2021 | 19 | −1 | 1 | 1 | −19 | 19 | 18.49 |
| 2022 | 21 | 1 | 1 | 1 | 21 | 21 | 21.23 |
| 2023 | 24 | 3 | 9 | 81 | 72 | 216 | 24.33 |
| 2024 | 28 | 5 | 25 | 625 | 140 | 700 | 27.79 |
| Total | 122 | 0 | 70 | 1,414 | 96 | 1,450 |
- From ΣXY = bΣX²: 96 = 70b, so b = 1.3714.
- The other two equations: 122 = 6a + 70c … (1) and 1,450 = 70a + 1,414c … (2). Multiplying (1) by 70 and (2) by 6: 8,540 = 420a + 4,900c and 8,700 = 420a + 8,484c. Subtracting, 160 = 3,584c, so c = 0.0446; then 6a = 122 − 70 × 0.044643 = 118.875, so a = 19.8125.
Y = 19.8125 + 1.3714X + 0.0446X², origin: middle of 2021 and 2022 (1 January 2022); X unit: half a year; Y in '000 tonnes.
(ii) Trend values, 2019 to 2024
Putting X = −5, −3, −1, 1, 3, 5 in the equation gives the last column of the table: 2019 14.07; 2020 16.10; 2021 18.49; 2022 21.23; 2023 24.33; 2024 27.79 thousand tonnes. The trend rises by a growing amount each year (2.03, 2.39, 2.74, 3.10, 3.46), which is what the positive c expresses: production is rising at an increasing rate.
For 2027, X = 2(2027 − 2021.5) = 11: Y = 19.8125 + 1.3714 × 11 + 0.0446 × 121 = 19.8125 + 15.0857 + 5.4018 = 40.30 thousand tonnes (about 40,300 tonnes).
The forecast assumes the accelerating trend of 2019–2024 continues for three more years; a parabola extrapolated far ahead can overstate growth, so the figure should be read with care.
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 2027 | 1 |
| Total | 9 |
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