Question 4 (a)
Straight-line trend of sales by least squares, trend values, monthly change and shift of origin
9 marks
- Time to write
- About 18 minutes
- Topic
- Trend fitting
Question
Question 4 (a) · 9 marksA company collected sales data for seven years, 2017 to 2023, from its sales department, as follows:
| Year | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|---|---|---|---|---|---|---|
| Sales (₹ crores) | 78 | 88 | 95 | 81 | 97 | 100 | 95 |
You have to:
Fit a straight-line trend by the method of least square.
Compute the trend values for each year from 2017 through 2023.
What is the sales change on a monthly basis?
Shift the origin to 2023.
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
- Compute…What is the…?Two asks in one question: answer both.
- 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 the trend equation 3 marks 5 min
- 2 Trend values for 2017 to 2023 2 marks 4 min
- 3 Monthly change conversion of the trend 2 marks 4 min
- 4 Shifting the origin to 2023 2 marks 3 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. The syllabus asks for trend fitting by least squares, shifting of the origin and conversion of an annual linear trend to a monthly basis; the DU guideline limits such conversion to linear trends. The trend Y = a + bX is fitted from the least-squares normal equations ΣY = na + bΣX and ΣXY = aΣX + bΣX². With an odd number of years (7), time is measured from the middle year, X = year − 2020, so ΣX = 0 and the equations reduce to a = ΣY / n and b = ΣXY / ΣX².
Working table (sales Y in ₹ crore)
| Year | Y | X | XY | X² | Trend T = 90.5714 + 2.75X |
|---|---|---|---|---|---|
| 2017 | 78 | −3 | −234 | 9 | 82.32 |
| 2018 | 88 | −2 | −176 | 4 | 85.07 |
| 2019 | 95 | −1 | −95 | 1 | 87.82 |
| 2020 | 81 | 0 | 0 | 0 | 90.57 |
| 2021 | 97 | 1 | 97 | 1 | 93.32 |
| 2022 | 100 | 2 | 200 | 4 | 96.07 |
| 2023 | 95 | 3 | 285 | 9 | 98.82 |
| Total | 634 | 0 | 77 | 28 |
ΣXY = (97 + 200 + 285) − (234 + 176 + 95) = 582 − 505 = 77.
a = ΣY / n = 634 / 7 = 90.5714; b = ΣXY / ΣX² = 77 / 28 = 2.75.
Y = 90.5714 + 2.75X, origin 2020 (the middle of the year, 1 July 2020); X unit: one year; Y: annual sales (₹ crore). On trend, sales rise by ₹2.75 crore a year.
(ii) Trend values, 2017 to 2023
From T = 90.5714 + 2.75X (last column of the table): 2017 ₹82.32 crore; 2018 ₹85.07; 2019 ₹87.82; 2020 ₹90.57; 2021 ₹93.32; 2022 ₹96.07; 2023 ₹98.82 crore. Each year adds b = ₹2.75 crore, and the unrounded trend values add up to ΣY = 634, as they must.
(iii) Sales change on a monthly basis
The annual trend is converted to a monthly trend by dividing a by 12 and b by 144:
- a / 12 = 90.5714 / 12 = 7.5476: the annual sales figure becomes the average sales of one month;
- b / 144 = 2.75 / 144 = 0.0191: the slope is a change in annual sales per year; dividing by 12 turns annual sales into monthly sales, and dividing by 12 again turns the time unit from a year into a month.
Monthly trend: Ym = 7.5476 + 0.0191x, origin 1 July 2020, x unit one month, Ym monthly sales (₹ crore). (Centred on the middle of July 2020 instead, the intercept is 7.5476 + 0.0191 / 2 = 7.5572.)
So monthly sales rise by about ₹0.0191 crore (₹1.91 lakh) from one month to the next.
The other reading. If sales change on a monthly basis is read as the rise in the annual sales figure for each month that passes (only the time unit converted), it is b / 12 = 2.75 / 12 = ₹0.2292 crore a month. The monthly-trend reading above is the conversion the syllabus names.
Moving the origin forward by 3 years (from 2020 to 2023) changes only the intercept: a′ = 90.5714 + 2.75 × 3 = 98.8214, the trend value of 2023.
Y = 98.8214 + 2.75X, origin 2023 (1 July 2023), X unit one year, Y annual sales (₹ crore).
Where the 9 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Working table and the trend equation | 3 |
| 2 | Trend values for 2017 to 2023 | 2 |
| 3 | Monthly change (conversion of the trend) | 2 |
| 4 | Shifting the origin to 2023 | 2 |
| Total | 9 |
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