Worksheet · Grade 12 · Applied Mathematics
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6 questions · 6 marks
The general tendency of a time series to increase, decrease or remain steady over a long period of time is called
Answer: (b) secular trend
Secular trend is the long-term tendency of the variable; seasonal changes repeat within a year, cyclical ones follow business cycles lasting more than a year, and irregular ones come from sudden events.
A set of values of a variable recorded at successive points or periods of time and arranged in chronological order is called a ______.
Answer: time series
A time series is chronological data: the values of a variable arranged in the order of the time at which they occur.
A rise in the sale of woollen clothes every winter is an example of cyclical variation.
Answer: False
Corrected statement: a rise in the sale of woollen clothes every winter is an example of seasonal variation, because it repeats every year with the seasons; cyclical variations last more than a year and follow business cycles.
Name the component of a time series that is caused by sudden, unforeseen events such as floods, earthquakes or strikes.
Answer: Irregular (random) variation.
When 5-yearly moving averages are worked out, the first moving average is the mean of the values of years 1 to 5. Against which year is it placed?
Answer: (c) Year 3
A moving average for an odd period is placed against the middle year of its period, and the middle of years 1 to 5 is year 3.
A newspaper lists the monthly rainfall of a city for 2024, month by month from January to December. Is this a time series? Give a reason.
Answer: Yes. It gives the values of one variable (rainfall) recorded period after period and arranged in chronological order, month by month.
Build on it
8 questions · 12 marks
Assertion (A): Cyclical variations are more difficult to forecast than seasonal variations.
Reason (R): Cyclical variations are found in business and economic data such as prices, production and employment.
Answer: (b) Both A and R are true, but R is NOT the correct explanation for A
Both are true. A holds because cyclical movements have no fixed period (one cycle may last several years) while seasonal ones repeat every year; R only says where cyclical variations are found, so it does not explain A.
Assertion (A): Irregular variations in a time series cannot be predicted from its past behaviour.
Reason (R): Irregular variations repeat at regular, fixed intervals of time, like the seasons.
Answer: (c) A is true, but R is false
A is true: irregular variations come from unforeseen events such as floods or strikes. R is false: it is seasonal variations that repeat regularly; irregular ones follow no pattern.
The straight-line trend of a firm's sales (in ₹ lakh) is \(y = 40 + 3X\), with origin at 2022 and \(X\) in years. Match each item in Column A with its value in Column B.
| Column A | Column B |
|---|---|
| (i) Trend value for 2020 | (P) ₹ 49 lakh |
| (ii) Trend value for 2025 | (Q) ₹ 40 lakh |
| (iii) Increase in the trend value every year | (R) ₹ 34 lakh |
| (iv) Trend value for the year taken as origin | (S) ₹ 3 lakh |
Answer: (i)-R, (ii)-P, (iii)-S, (iv)-Q
2020 has \(X = -2\), so \(y = 40 - 6 = 34\); 2025 has \(X = 3\), so \(y = 40 + 9 = 49\); the slope 3 is the yearly change; at the origin \(X = 0\), so \(y = 40\).
The production (in thousand units) of a factory is:
2015: 20
2016: 24
2017: 22
2018: 26
2019: 28
2020: 25
2021: 29
2022: 27
2023: 31
Find the 5-yearly moving averages, and state the years for which no trend value is obtained.
Answer: 5-yearly moving totals:
2015 to 2019: \(20 + 24 + 22 + 26 + 28 = 120\), placed at 2017
2016 to 2020: \(24 + 22 + 26 + 28 + 25 = 125\), placed at 2018
2017 to 2021: \(22 + 26 + 28 + 25 + 29 = 130\), placed at 2019
2018 to 2022: \(26 + 28 + 25 + 29 + 27 = 135\), placed at 2020
2019 to 2023: \(28 + 25 + 29 + 27 + 31 = 140\), placed at 2021
Moving averages (each total divided by 5): 2017: 24, 2018: 25, 2019: 26, 2020: 27, 2021: 28 (thousand units).
No trend value is obtained for the first two years (2015, 2016) and the last two years (2022, 2023). The trend rises by 1 thousand units a year.
How marks are given
A town's population rose steadily from 1.2 lakh in 1991 to 2.0 lakh in 2021, with only small ups and downs in between. Which component of a time series does this long-term rise show?
Answer: Secular trend (a rising long-term trend).
For the six years 2019 to 2024, a straight-line trend is fitted by the method of least squares with the origin midway between 2021 and 2022 and \(X\) measured in units of half a year. What is the value of \(X\) for 2024?
Answer: (d) \(5\)
2024 is 2.5 years, that is 5 half-years, after the origin, so \(X = 5\); the six years get \(X = -5, -3, -1, 1, 3, 5\).
Ravi's shop sales fell for about two years during a general business slowdown (2008 to 2010) and recovered later. They also fell sharply in one month of 2019, when a flood closed the market. Ravi says both falls are the same kind of variation. Explain why he is wrong.
Answer: The fall during 2008 to 2010 is a cyclical variation: an up-and-down movement lasting more than a year, linked to the business cycle of boom, slowdown (depression) and recovery.
The fall in one month of 2019 is an irregular variation: a sudden, short-lived change caused by an unforeseen event (the flood), which follows no pattern and cannot be predicted.
How marks are given
Changes in a time series that repeat in a regular way within a year, such as the rise in the sale of sweets every year at Diwali, are called ______ variations.
Answer: seasonal
Seasonal variations are regular movements that repeat within a period of one year, caused by climate, festivals and customs.
Stretch yourself
6 questions · 22 marks
The least squares trend of a firm's sales (in ₹ lakh) for 2019 to 2023, with origin at 2021 and \(X\) in years, is \(y = 18 + 3X\). The sales for 2019, 2020, 2022 and 2023 were 12, 13, 19 and 24, but the record for 2021 is lost. What were the sales in 2021?
Answer: (c) ₹ 22 lakh
With \(\sum X = 0\), \(a = \dfrac{\sum y}{n}\), so \(\sum y = 5 \times 18 = 90\) and the 2021 sales are \(90 - (12 + 13 + 19 + 24) = 22\). Check: \(\sum Xy = -24 - 13 + 19 + 48 = 30\) and \(b = \dfrac{30}{10} = 3\).
Meera's shop sold goods worth ₹ 18 lakh in 2023 and ₹ 21 lakh in 2024. She concludes that the secular trend of her sales is rising. Explain why this conclusion is not justified, and describe what she should do to find the secular trend.
Answer: Secular trend is the tendency of a variable to increase or decrease over a long period of time; it cannot be judged from just two years.
The rise from 2023 to 2024 may come from short-term forces, such as a boom in business (cyclical variation) or a one-time event like a single large order (irregular variation), and it could be reversed later.
She should collect yearly sales for a long period (say 8 to 10 years) and measure the trend by the moving average method or by fitting a straight line by the method of least squares; only if the trend values rise over the years is the secular trend rising.
How marks are given
The sales (in ₹ lakh) of a company are:
2016: 8
2017: 10
2018: 7
2019: 11
2020: 16
2021: 18
2022: 15
2023: 19
Find the trend values by 4-yearly centred moving averages, and name the years for which no trend value is obtained.
Answer: 4-yearly moving totals (each placed between the middle two years of its period):
2016 to 2019: 36 (between 2017 and 2018)
2017 to 2020: 44 (between 2018 and 2019)
2018 to 2021: 52 (between 2019 and 2020)
2019 to 2022: 60 (between 2020 and 2021)
2020 to 2023: 68 (between 2021 and 2022)
Centring: add two successive 4-yearly totals and divide by 8.
2018: \(\dfrac{36 + 44}{8} = 10\); 2019: \(\dfrac{44 + 52}{8} = 12\); 2020: \(\dfrac{52 + 60}{8} = 14\); 2021: \(\dfrac{60 + 68}{8} = 16\) (₹ lakh).
No trend value is obtained for 2016, 2017, 2022 and 2023.
How marks are given
The yearly milk collection (in thousand litres) of a dairy is:
2017: 39
2018: 44
2019: 43
2020: 48
2021: 49
2022: 56
2023: 57
Fit a straight-line trend \(y = a + bX\) by the method of least squares, taking 2020 as origin and \(X\) in years. Find the trend values for all the years and estimate the collection for 2026.
Answer: Take \(X = -3, -2, -1, 0, 1, 2, 3\) for 2017 to 2023, so \(\sum X = 0\).
\(Xy\): \(-117, -88, -43, 0, 49, 112, 171\); \(X^{2}\): 9, 4, 1, 0, 1, 4, 9.
\(\sum y = 336\), \(\sum Xy = 84\), \(\sum X^{2} = 28\), \(n = 7\).
\(a = \dfrac{\sum y}{n} = \dfrac{336}{7} = 48\); \(b = \dfrac{\sum Xy}{\sum X^{2}} = \dfrac{84}{28} = 3\).
Trend: \(y = 48 + 3X\) (origin 2020, \(X\) in years, \(y\) in thousand litres).
Trend values: 2017: 39, 2018: 42, 2019: 45, 2020: 48, 2021: 51, 2022: 54, 2023: 57.
For 2026, \(X = 6\): \(y = 48 + 18 = 66\) thousand litres.
How marks are given
The yearly electricity use (in million units) of a town is:
2018: 48
2019: 51
2020: 58
2021: 61
2022: 61
2023: 68
2024: 73
The straight-line trend fitted by the method of least squares is \(y = 60 + 4X\), with origin at 2021 and \(X\) in years.
(a) Find the trend value for each year.
(b) In which year was the actual use furthest below its trend value? What does the gap between an actual value and its trend value show?
(c) Estimate the use for 2027.
(d) Give one reason why the actual use in 2027 may differ from this estimate.
Answer: (a) With \(X = -3\) to \(3\): trend values 2018: 48, 2019: 52, 2020: 56, 2021: 60, 2022: 64, 2023: 68, 2024: 72.
(b) Actual minus trend: 2018: 0, 2019: \(-1\), 2020: \(+2\), 2021: \(+1\), 2022: \(-3\), 2023: 0, 2024: \(+1\) (these add up to 0). The use was furthest below the trend in 2022, by 3 million units. The gap is the part of the actual value that the long-term trend does not explain: the combined effect of the short-term components (cyclical and irregular variations).
(c) For 2027, \(X = 6\): \(y = 60 + 24 = 84\) million units.
(d) The estimate assumes that the straight-line growth of 4 million units a year will continue; a change in conditions (for example, new industries opening, or a sudden event such as a long power failure) brings cyclical or irregular changes, so the actual value may differ.
How marks are given
Sharma Motors sells electric scooters in a small town, and its owner, Mr Sharma, has kept a record of the number of scooters sold every month since 2015. Looking at the record, he notices three things. First, every year sales jump in October and November, when people buy vehicles during the festival season, and then fall back in December. Second, from 2016 to 2019 the town's factories went through a slowdown and his yearly sales kept sliding; when business picked up again, sales recovered over the next few years. Third, in spite of these ups and downs, his yearly sales have, on the whole, grown. To plan his stock, his accountant fits a straight-line trend to the yearly sales by the method of least squares and gets \(y = 450 + 30X\), where \(y\) is the number of scooters sold in a year, the origin is 2022 and \(X\) is in years. In 2024 the shop actually sold 492 scooters.
(i) Seasonal variation: it repeats every year within the year, because of the festival season.
(ii) Cyclical variation: an up-and-down movement lasting more than a year, linked to the slowdown and recovery of business.
(iii) For 2026, \(X = 4\): \(y = 450 + 30 \times 4 = 570\) scooters.
(iv) For 2024, \(X = 2\): \(y = 450 + 60 = 510\) scooters. Actual sales of 492 are 18 below the trend value, so 2024 was worse than the long-term tendency; the shortfall is due to short-term (cyclical or irregular) factors, not to the trend.
