Time Series and Trend for CUET Applied Maths

AnswerB · India's rice output for each year from 2015 to 2025

1. A time series records one quantity at successive points of time, in time order.

2. Rice output year by year is such a record. The others compare different students, players or states at a single time.

AnswerD · Time

1. The values are recorded against time, which runs on regardless.

2. So time is the independent variable, and the value recorded depends on it.

AnswerC · Both I and II

1. I: chronological order is what makes data a time series. True.

2. II: 'univariate' means a single variable, such as yearly sales. True.

AnswerB · The marks of 40 students in one test

1. A time series records one quantity at successive points of time.

2. The test marks are 40 different students at one moment, not one quantity over time.

AnswerB · On the horizontal axis

1. Time is the independent variable, so it goes along the horizontal axis.

2. The quantity measured goes on the vertical axis.

AnswerD · The values of one variable recorded over time

1. 'Univariate' means one variable; a time series records it at successive times.

2. So a univariate time series is one variable, such as monthly sales, followed through time.

AnswerA · Only I

1. I: the pattern of the past, such as a trend, is projected forward. True.

2. II: a time series is defined by its time order; shuffling it loses the trend and seasons. False.

AnswerA · Seasonal variation

1. A pattern that repeats within a year, tied to the seasons, festivals or customs, is seasonal variation.

2. Winter demand for woollens comes back every year.

AnswerA · Secular trend

1. The secular trend is the general tendency of a series to rise or fall over a long period.

2. Steady growth over fifty years is a long-term rise.

AnswerC · Irregular variation

1. A sudden, unpredictable one-off change is irregular variation.

2. So this is irregular variation.

AnswerD · Seasonal variation

1. A pattern that repeats regularly within a fixed short period is seasonal variation.

2. So this is seasonal variation.

AnswerA · Secular trend

1. A steady movement in one direction over a long period is the secular trend.

2. So this is secular trend.

AnswerA · Both A and R are true, and R is the correct explanation of A.

1. R is the description of irregular variation. True.

2. An earthquake is unpredictable and does not repeat on a schedule, so A is true because of R.

AnswerA · Seasonal variation

1. It comes back at the same time every year.

2. A movement that repeats within each year is seasonal.

AnswerD · Secular trend

1. It is the general direction over a long period.

2. That is the secular trend.

AnswerC · Irregular variation

1. It was caused by a one-off event that could not have been foreseen from the pattern.

2. That is irregular variation.

AnswerB · Irregular variation

1. Irregular variation comes from unforeseen, one-off events such as floods, strikes or wars.

2. It follows no pattern and cannot be predicted.

AnswerD · Cyclical variation

1. Cyclical variation is a wave-like rise and fall lasting more than a year, as in the business cycle.

2. Booms and slumps over 7 to 10 years are exactly that.

AnswerC · A-II, B-IV, C-I, D-III

1. A yearly peak is seasonal; a 30-year rise is the trend.

2. A sudden strike is irregular; a rise and fall over a decade is cyclical.

AnswerA · Only I

1. I: seasons, festivals and harvests come round every year. True.

2. II: cycles last more than a year, and their length is not fixed. False.

AnswerB · A rising trend with seasonal variation

1. Across the three years the sales climb: each quarter is higher than the same quarter a year before. That is an upward trend.

2. Within each year the same pattern repeats, a peak in the second quarter and a dip in the fourth: seasonal variation.

AnswerD · Seasonal variation

1. A pattern that repeats regularly within a fixed short period is seasonal variation.

2. So this is seasonal variation.

AnswerA · Secular trend

1. A steady movement in one direction over a long period is the secular trend.

2. So this is secular trend.

AnswerB · Cyclical variation

1. Rises and falls that repeat over several years are cyclical variation.

2. So this is cyclical variation.

AnswerC · I, II and IV only

1. The trend is the long-run, smoothed direction of a series, upward, downward or flat.

2. III describes seasonal variation, not the trend.

AnswerA · Cyclical variation

1. It lasts more than a year and follows the ups and downs of the economy.

2. Booms and slumps of this kind are cyclical variation.

AnswerA · $15$

1. The first 3-year moving average is the mean of the first three values.

2. $\frac{12 + 15 + 18}{3} = 15$, placed against the second year.

AnswerB · 2020

1. A moving average of an odd number of years is placed against the middle year of its group.

2. The middle of 2019, 2020 and 2021 is 2020.

AnswerC · $15$

1. $\frac{12 + x + 18}{3} = 15$, so $30 + x = 45$.

2. $x = 15$.

AnswerB · $23$

1. The first 3-year moving average uses $20, 26, 23$.

2. $\frac{20 + 26 + 23}{3} = 23$.

AnswerD · 2018

1. A moving average is placed against the middle of its period.

2. The middle of 2016 to 2020 is 2018.

AnswerC · $28$

1. The last 3-year moving average uses $23, 29, 32$.

2. $\frac{23 + 29 + 32}{3} = 28$.

AnswerD · Seasonal variation

1. Both years peak in the second quarter and dip in the third, the same pattern inside each year.

2. A pattern that repeats within a year is seasonal variation.

AnswerB · $70$

1. The first moving total adds the four quarters of year 1.

2. $12 + 30 + 8 + 20 = 70$.

AnswerC · $5$

1. The groups are years 1–3, 2–4, 3–5, 4–6 and 5–7.

2. That is $7 - 3 + 1 = 5$ moving averages; the first and last years get none.

AnswerA · Only I

1. I: averaging neighbouring years cancels out much of the short-term change. True.

2. II: the first and last years have no neighbour on one side, so they get no 3-year average. False.

AnswerA · $15$

1. The first 5-year moving average uses $11, 12, 16, 17, 19$.

2. $\frac{11 + 12 + 16 + 17 + 19}{5} = 15$.

AnswerD · A 4-quarter moving average

1. A moving average removes a regular pattern when its period equals the pattern's length.

2. The pattern repeats every 4 quarters, so each 4-quarter average contains one of each quarter.

AnswerB · $47$

1. The second 3-year moving average uses $46, 43, 52$.

2. $\frac{46 + 43 + 52}{3} = 47$.

AnswerC · $5$

1. The first covers years 1 to 5, the last years 5 to 9.

2. Starting years 1 to 5: $9 - 5 + 1 = 5$ averages.

AnswerA · Only I

1. I: the middle of 4 years lies between the 2nd and 3rd, so pairs of averages are averaged again. True.

2. II: an odd period has a middle year, so no centring is needed. False.

AnswerB · $17.75$

1. The first two 4-quarter totals are 70 and 72.

2. Centred average $= \frac{70 + 72}{8} = 17.75$.

AnswerD · The third quarter of year 1

1. The first total, quarters 1–4, centres between quarters 2 and 3; the second, quarters 2–5, between 3 and 4.

2. Averaging the two lines the result up with quarter 3 of year 1.

AnswerB · $14.75$

1. The 4-year moving averages are $\frac{10 + 14 + 12 + 18}{4} = 13.5$ and $\frac{14 + 12 + 18 + 20}{4} = 16$; each falls between two years.

2. Centring averages the two: $\frac{13.5 + 16}{2} = 14.75$, placed against the third year.

AnswerB · $12.25$

1. The 4-year moving averages are $\frac{44}{4} = 11$ and $\frac{54}{4} = 13.5$; each falls between two years.

2. Centring averages them in pairs: $\frac{11 + 13.5}{2} = 12.25$, placed against the third year.

AnswerC · $54$

1. 2026 is 4 years after 2022, so $x = 4$.

2. $y = 40 + 3.5 \times 4 = 54$.

AnswerB · Sales fall by about 2.5 units a year

1. The slope is the change in $y$ for each year.

2. $-2.5$ means the trend falls by 2.5 units a year; 85 is the trend value at $x = 0$.

AnswerA · $108$

1. 2019 is 2 years before 2021, so $x = -2$.

2. $y = 120 - 12 = 108$.

AnswerC · $46$

1. The average placed against 2021 is the mean of 2020, 2021 and 2022.

2. $\frac{42 + 45 + 51}{3} = 46$.

AnswerC · $156$

1. 2026 is $x = 6$.

2. $y = 120 + 6(6) = 156$.

AnswerA · Sales rise by about 6 thousand tractors a year

1. 6 is the slope: each extra year adds 6 to $y$.

2. $y$ is in thousands, so about 6 thousand tractors a year.

AnswerC · $55$

1. The average placed against 2023 is the mean of 2022, 2023 and 2024.

2. $\frac{51 + 54 + 60}{3} = 55$.

AnswerC · $15$

1. With $x$ running $-2, -1, 0, 1, 2$, $\sum x = 0$, and the first normal equation $\sum y = na + b\sum x$ becomes $\sum y = na$.

2. $a = \frac{\sum y}{n} = \frac{75}{5} = 15$, the mean of the sales.

AnswerC · $25.8$

1. 2027 is 4 years after 2023, so $x = 4$.

2. $y = 15 + 2.7 \times 4 = 25.8$.

AnswerD · The trend rises by 8 units every year

1. In $y = a + bx$, $b$ is the change in the trend value when $x$ goes up by 1, that is per year.

2. So the trend output rises by 8 units a year; the 120 is the trend value for 2020.

AnswerD · $57.6$

1. 2026 is 4 years after 2022, so $x = 4$.

2. $y = 39.6 + 4.5 \times 4 = 57.6$ lakh rupees.

AnswerB · It makes $\sum x = 0$, so $a = \frac{\sum y}{n}$ and $b = \frac{\sum xy}{\sum x^{2}}$

1. The normal equations are $\sum y = na + b\sum x$ and $\sum xy = a\sum x + b\sum x^{2}$.

2. With $\sum x = 0$ they separate: $a = \frac{\sum y}{n}$, $b = \frac{\sum xy}{\sum x^{2}}$.

AnswerD · I, II and III only

1. I is what 'least squares' means. II and III follow from the first normal equation, $\sum y = na + b\sum x$.

2. IV is false: a falling series gives a negative slope.

AnswerD · $38$

1. With $x = -2, -1, 0, 1, 2$, $\sum x = 0$, so $a = \frac{\sum y}{n}$.

2. $a = \frac{190}{5} = 38$.

AnswerA · $19.6$

1. 2020 is 3 years before 2023, so $x = -3$.

2. $y = 25 - 5.4 = 19.6$ thousand.