Time-Based Data: Time Series and Trend

Which of the following is a time series?

1. aThe marks of 40 students of a class in one test
2. bThe heights of the 11 players of a cricket team
3. cThe closing price of a share on each working day of March 2026
4. dThe populations of the seven emirates of the UAE in 2025

The sale of umbrellas in a town rises every year during the monsoon months and falls afterwards. Which component of a time series does this show?

1. aSeasonal variation
2. bSecular trend
3. cCyclical variation
4. dIrregular variation

The least squares trend of a factory's yearly output (in tonnes) is \(y = 120 - 5X\), with origin at 2020 and \(X\) in years. Which statement is correct?

1. aThe actual output in 2020 was exactly 120 tonnes, and output falls by 5% each year
2. bThe trend value for 2020 is 120 tonnes, and the trend falls by 5 tonnes each year
3. cThe trend value for 2020 is 115 tonnes, and the trend falls by 5 tonnes each year
4. dThe trend value for 2020 is 120 tonnes, and the trend rises by 5 tonnes each year

The yearly sales (in ₹ lakh) of a shop are: 2018: 9, 2019: 15, 2020: 12, 2021: 15, 2022: 21, 2023: 18, 2024: 24. What is the 3-yearly moving average for 2021?

1. a14
2. b18
3. c15
4. d16

The production (in thousand units) of a firm is: 2017: 20, 2018: 24, 2019: 22, 2020: 28, 2021: 30, 2022: 28. What is the centred 4-yearly moving average for 2020?

1. a26
2. b27
3. c26.5
4. d24.75

The sales (in ₹ lakh) of a company are: 2019: 4, 2020: 7, 2021: 7, 2022: 9, 2023: 13. Which straight-line trend is obtained by the method of least squares, with origin at 2021 and \(X\) in years?

1. a\(y = 8 + 2X\)
2. b\(y = 8 + 4X\)
3. c\(y = 40 + 2X\)
4. d\(y = 2 + 8X\)

The profits (in ₹ lakh) of a firm are: 2020: 10, 2021: 14, 2022: 16, 2023: 20. A straight-line trend is fitted by least squares with origin midway between 2021 and 2022 and \(X\) in half-years, so \(X = -3, -1, 1, 3\). What is the trend value for 2024?

1. a19
2. b23
3. c21.4
4. d31

A straight-line trend is most suitable for a time series whose values

1. achange by roughly the same percentage each year
2. brise and fall in a regular pattern within each year
3. cmove up and down in waves lasting several years
4. dchange by roughly the same amount each year

When 5-yearly moving averages are found for 12 years of data, how many trend values are obtained?

1. a10
2. b8
3. c7
4. d12

The least squares trend of a company's profit (in ₹ lakh) is \(y = 50 + 6X\), with origin at 2022 and \(X\) in years. In which year does the trend value first exceed ₹ 80 lakh?

1. a2027
2. b2035
3. c2028
4. d2036

Assertion (A): The method of least squares gives a trend value for every year of the series, including the first and the last.

Reason (R): In the method of least squares, the trend values are worked out from a fitted equation, which can be used for any value of \(X\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): With 3-yearly moving averages, no trend value is obtained for the first and the last year of the series.

Reason (R): Each 3-yearly moving average is placed against the first of the three years it covers.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): A fall in a factory's output in one month because of a strike is a cyclical variation.

Reason (R): Cyclical variations are up-and-down movements lasting more than a year, linked to booms and depressions in business.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. I. Secular trend describes the general movement of a series over a long period of time.
2. II. II. Irregular variations can be forecast exactly from the past values of the series.

Which of the statements above about the components of a time series is/are correct?

1. aOnly I
2. bOnly II
3. cBoth I and II
4. dNeither I nor II

1. I. I. Moving averages of an odd period, such as 3 or 5 years, must be centred.
2. II. II. Moving averages smooth out short-term fluctuations and bring out the trend of the series.

Which of the statements above about moving averages is/are correct?

1. aOnly I
2. bOnly II
3. cBoth I and II
4. dNeither I nor II

1. I. I. The straight line fitted by least squares passes through every point of the data.
2. II. II. In the method of least squares, the sum of the deviations of the actual values from the trend values is zero.
3. III. III. In the method of least squares, the sum of the squares of these deviations is the least possible.

Which of the statements above about the method of least squares is/are correct?

1. aOnly I and II
2. bOnly III
3. cI, II and III
4. dOnly II and III

Match each component in List I with its example in List II:

1. a(i)-Q, (ii)-R, (iii)-S, (iv)-P
2. b(i)-Q, (ii)-S, (iii)-R, (iv)-P
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-P, (ii)-S, (iii)-R, (iv)-Q

The sales (in ₹ lakh) of a shop from 2018 to 2022 are 6, 8, 13, 11 and 17. A straight-line trend \(y = a + bX\) is fitted by least squares with origin at 2020 and \(X\) in years. Match List I with List II:

1. a(i)-Q, (ii)-S, (iii)-R, (iv)-P
2. b(i)-Q, (ii)-R, (iii)-S, (iv)-P
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-Q, (ii)-P, (iii)-R, (iv)-S

Farah records her bakery's monthly cake sales for 6 years. Sales jump every December, rise slowly from year to year, and fell sharply in one month when the shop was shut for repairs. To judge whether her business is really growing, what should she study?

1. aThe December sales each year, as they show the business at its best
2. bThe month when the shop was shut, as it shows the business at its worst
3. cThe sales of any one month, as each month shows the full picture
4. dThe secular trend, after smoothing out the seasonal and irregular changes

A school's enrolment from 2020 to 2024 was 300, 320, 350, 370 and 410. To plan classrooms, the principal fits a straight-line trend by least squares (origin 2022, \(X\) in years). What is the estimated enrolment for 2026?

1. a431
2. b566
3. c458
4. d485