UG Applied Mathematics Booster Test 2 - Time Series Analysis -

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UG Applied Mathematics Booster Test 2 - Time Series Analysis

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

UG Applied Mathematics Booster Test 2 - Time Series Analysis

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QUESTION 1 OF 20

In sequential data forecasting analysis, if past quantifiable information is not available or applicable, which alternative approach is generally applied?

Answer Review

1 In sequential data forecasting analysis, if past quantifiable information is not available or applicable, which alternative approach is generally applied?

A Quantitative method

B Qualitative method

C Secular trend scaling

D Method of least squares

💡 Short Explanation

Qualitative methods use expert judgment Used when numerical data is unavailable Common in uncertain forecasting situations

✅ Correct Answer Explanation

Qualitative forecasting methods are used when historical numerical data is unavailable or unreliable. These methods depend on expert opinions, market surveys, and judgment rather than mathematical calculations. Option A is incorrect because quantitative methods require past numerical data. Option C is not a standard forecasting method. Option D is a quantitative statistical technique. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Quantitative forecasting requires measurable historical data.
Option C → "Secular trend scaling" is not a recognized forecasting approach.
Option D → Least squares requires numerical observations.

🎯 Strategy

Used

Elimination

Application:

Remove methods that require numerical historical data.

Final Logic:

Without quantifiable data, qualitative forecasting is preferred.

🧠 Memory Trick

"No Data → Use Judgment"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Forecasting Methods –

2 A continuous time-based observation follows the increasing function
\(y=6x\)
Find the area bounded by this time series curve, the x-axis, and the interval between \(x=1\) and \(x=3\).

A 12

B 18

C 24

D 30

💡 Short Explanation

Use definite integration Integrate function over interval Compute bounded area

✅ Correct Answer Explanation

Area under the curve: \(\int_{1}^{3}\,6x dx\) \(\int_{1}^{3}\,6x dx=[3x^{2}]_{1}^{3}=3(9)-3(1)=27-3=24\) Thus, the bounded area equals 24 square units.

❓ Why Other Options Are Incorrect

Option A → Incomplete integration.
Option B → Arithmetic mistake.
Option D → Overestimated area.

🎯 Strategy

Used

Substitution

Application:

Apply definite integral formula directly.

Final Logic:

Area equals integration of function over given interval.

🧠 Memory Trick

"Area = Integral of Curve"

📚 Class XII – Applied Mathematics – Chapter: Integral Calculus – Topic: Definite Integration –

3 A time series variation records specific changes structured as a vector
\(\vec{A}=4\hat{i}-3\hat{j}\)
What is the unit vector representing the standardized direction of this data variation?

A \(\frac{3}{5}\hat{i}-\frac{4}{5}\hat{j}\)

B \(\frac{1}{5}\hat{i}-\frac{2}{5}\hat{j}\)

C \(\frac{2}{5}\hat{i}+\frac{3}{5}\hat{j}\)

D \(\frac{4}{5}\hat{i}-\frac{3}{5}\hat{j}\)

💡 Short Explanation

Unit vector has magnitude 1 Divide vector by magnitude Use vector normalization

✅ Correct Answer Explanation

Magnitude: \(∣\vec{A}∣=\sqrt{4^{2}+(-3)^{2}}\) \(∣\vec{A}∣=\sqrt{4^{2}+(-3)^{2}}=\sqrt{16+9}=5\) Unit vector: \(\frac{\vec{A}}{∣\vec{A}∣}\) \(\frac{4\hat{i}-3\hat{j}}{5}=\frac{4}{5}\hat{i}-\frac{3}{5}\hat{j}\) Thus, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Components reversed.
Option B → Incorrect normalization.
Option C → Wrong sign for j-component.

🎯 Strategy

Used

Substitution

Application:

Compute magnitude and divide components.

Final Logic:

Unit vector = vector ÷ magnitude.

🧠 Memory Trick

"Unit Vector = Divide by Length"

📚 Class XI – Mathematics – Chapter: Vector Algebra – Topic: Unit Vector –

4 Cross-sectional data tracks a variable whose instantaneous rate of change across a specific cross-section is
\(f^{'}(x)=6x\)
If the initial cross-sectional baseline is \(f(0)=2\), what is the original function \(f(x)\)?

A \(3x^{2}+2\)

B \(2x^{2}+3\)

C \(6x^{2}+2\)

D \(3x+2\)

💡 Short Explanation

Integrate derivative Add integration constant Use initial condition

✅ Correct Answer Explanation

Integrate: \(f^{'}(x)=6x\) \(f(x)=\int 6x dx=3x^{2}+C\) Use condition: \(f(0)=23(0)^{2}+C=2C=2\) Hence: \(f(x)=3x^{2}+2\) Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect coefficient and constant.
Option C → Derivative becomes \(12x\).
Option D → Derivative becomes constant 3.

🎯 Strategy

Used

Substitution

Application:

Integrate and apply boundary condition.

Final Logic:

Antiderivative plus initial condition gives original function.

🧠 Memory Trick

"Integrate Then Add Constant"

📚 Class XII – Mathematics – Chapter: Differential Equations – Topic: Formation from Derivative –

5 A graph displays pooled data combining 2 unique regions (cross-sectional) tracked annually over 4 years (time series). How many distinct data points are plotted on the scatter plot assuming complete records?

A 6

B 8

C 16

D 2

💡 Short Explanation

Pooled data combines entities and time Multiply regions by years Total observations are counted

✅ Correct Answer Explanation

Number of regions: \(2\) Number of years: \(4\) Total data points: \(2\times 4=8\) Thus, 8 observations are plotted.

❓ Why Other Options Are Incorrect

Option A → Incorrect multiplication.
Option C → Overcounted observations.
Option D → Only counts regions.

🎯 Strategy

Used

Substitution

Application:

Multiply cross-sectional units by time periods.

Final Logic:

Total pooled observations = entities × time intervals.

🧠 Memory Trick

"Pooled Data = Cross × Time"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Pooled Data –

6 In mixed datasets, assume a time-series anomaly has probability \(P(A)=0.4\) and a cross-sectional error has probability \(P(B)=0.5\). If these data errors are completely independent events, what is the probability that both occur simultaneously?

A 0.90

B 0.10

C 0.20

D 0.45

💡 Short Explanation

Independent probabilities multiply Use intersection rule Simultaneous occurrence required

✅ Correct Answer Explanation

For independent events: \(P(A\cap B)=P(A)\times P(B)\) \(0.4\times 0.5=0.20\) Thus, the probability equals 0.20.

❓ Why Other Options Are Incorrect

Option A → Added probabilities incorrectly.
Option B → Incorrect multiplication.
Option D → Not based on independence rule.

🎯 Strategy

Used

Substitution

Application:

Use independent event probability formula.

Final Logic:

Independent simultaneous events multiply.

🧠 Memory Trick

"Independent → Multiply"

📚 Class XI – Mathematics – Chapter: Probability – Topic: Independent Events –

7 Arrange the fundamental steps to execute a standard quantitative forecasting method:
1. Apply a mathematical forecasting tool (e.g., Least Squares)
2. Gather sequential past quantifiable data
3. Formulate expected predictions based on the equation
4. Extract the mathematical relationship between the variables

A 1, 2, 3, 4

B 2, 4, 1, 3

C 4, 2, 1, 3

D 2, 1, 4, 3

💡 Short Explanation

Start with data collection Identify relationships Apply forecasting model

✅ Correct Answer Explanation

Correct sequence: 1. Gather data 2. Extract relationship 3. Apply forecasting tool 4. Make predictions Hence: \(2\rightarrow 4\rightarrow 1\rightarrow 3\) Thus, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Starts without data collection.
Option C → Relationship cannot be extracted before data collection.
Option D → Tool applied before identifying relationships.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Follow logical forecasting workflow.

Final Logic:

Data collection must precede modeling.

🧠 Memory Trick

"Data → Relation → Tool → Forecast"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Forecasting Procedure –

8 Match the forecasting elements:

List I	List II
1. Quantitative methods	a. Relies on strictly quantifiable data
2. Qualitative methods	b. Based on expert judgment or opinion
3. Cause-and-effect	c. Relates forecasting variables mathematically
4. Past pattern continuity	d. Primary assumption for future historic tracking

A 1–a, 2–b, 3–c, 4–d

B 1–b, 2–a, 3–d, 4–c

C 1–c, 2–d, 3–a, 4–b

D 1–d, 2–c, 3–b, 4–a

💡 Short Explanation

Quantitative uses numbers Qualitative uses opinions Cause-effect links variables

✅ Correct Answer Explanation

Correct matches: • Quantitative → quantifiable data • Qualitative → expert judgment • Cause-and-effect → mathematical relationships • Past continuity → historical continuation assumption Thus, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Reverses qualitative and quantitative meanings.
Option C → Incorrectly assigns relationships.
Option D → Completely mismatched concepts.

🎯 Strategy

Used

Option Grouping

Application:

Match standard forecasting definitions.

Final Logic:

Forecasting concepts have fixed textbook meanings.

🧠 Memory Trick

"Quantitative = Quantity"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Forecasting Methods –

9 Which of the following examples and definitions describe a univariate series?
I. A dataset where only one variable is varying over time
II. A temperature sensor measuring only the temperature of a place every second
III. A collection of data tracking how multiple variables vary simultaneously

A I and III only

B I and II only

C II and III only

D All I, II, and III

💡 Short Explanation

Univariate means one variable Single measurement changes over time Multiple variables become multivariate

✅ Correct Answer Explanation

Statement I is correct because univariate series contains only one changing variable. Statement II is also correct because only temperature is observed. Statement III is incorrect because multiple variables indicate multivariate data. Thus, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Includes multivariate statement III.
Option C → Excludes statement I.
Option D → Statement III is incorrect.

🎯 Strategy

Used

Elimination

Application:

Remove options containing multivariate definition.

Final Logic:

Univariate means exactly one variable.

🧠 Memory Trick

"Uni = One"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Univariate Series – 262

10 Identify the incorrect statement regarding a multivariate time series:

A It contains data tracking multiple variables simultaneously

B It tracks how interacting variables vary over elapsed time

C It restricts observations strictly to a single, one-dimensional dataset per time interval

D Comparing electricity usage alongside regional population growth over time is a prime example

💡 Short Explanation

Multivariate means many variables Variables interact over time Single-variable restriction is incorrect

✅ Correct Answer Explanation

Multivariate time series involves tracking multiple variables together over time. Option C incorrectly describes a univariate series because it restricts data to one variable. Options A, B, and D correctly define multivariate datasets. Thus, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Correct definition of multivariate data.
Option B → Interaction among variables is valid.
Option D → Real-world multivariable example.

🎯 Strategy

Used

Odd One Out

Application:

Identify statement describing univariate instead of multivariate data.

Final Logic:

Multivariate cannot be single-dimensional.

🧠 Memory Trick

"Multi = Many Variables"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Multivariate Series – 262

11 Assertion (A): A secular trend always follows a strict linear straight line without any curves.
Reason (R): The secular trend represents smooth, regular, and long-term variations observed over a long period.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R is the correct explanation of A

D A is false, R is true

💡 Short Explanation

Secular trend shows long-term movement Trend may be linear or curvilinear Smooth variation occurs over long periods

✅ Correct Answer Explanation

Assertion (A) is false because secular trends do not always follow a straight line. Trends may be linear or non-linear depending on data behavior. Reason (R) is true because secular trend represents long-term smooth and regular movement over years. Thus, the correct option is D.

❓ Why Other Options Are Incorrect

Option A → Reason is true, so both are not false.
Option B → Assertion is not true.
Option C → Assertion is incorrect because trends can also be curved.

🎯 Strategy

Used

Elimination

Application:

Separate the meaning of "trend" from "shape of trend."

Final Logic:

Secular trend is long-term, but not necessarily linear.

🧠 Memory Trick

"Trend ≠ Always Straight"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Secular Trend –

12 When using moving averages to isolate seasonal variability in quarterly metrics, if a time series has consecutive data points
10, 20, 30, 40,
what is the raw 4-period moving total?

A 100

B 110

C 90

D 25

💡 Short Explanation

Moving total means summation Add all four values Used before averaging

✅ Correct Answer Explanation

Raw 4-period moving total: \(10+20+30+40\) \(10+20+30+40=100\) Hence, the moving total equals 100.

❓ Why Other Options Are Incorrect

Option B → Incorrect addition.
Option C → Partial total only.
Option D → Represents average, not total.

🎯 Strategy

Used

Substitution

Application:

Directly substitute values into moving total calculation.

Final Logic:

Moving total is obtained by summing all observations.

🧠 Memory Trick

"Moving Total = Add Terms"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Average Method –

13 A cyclical variation completes one standard oscillation over several years. If the peak economic boom occurs at year 1 with \(y=100\) and the cyclical trough occurs at year 3 with \(y=50\), what is the numerical amplitude (half the total range) of this cycle?

A 50

B 25

C 100

D 75

💡 Short Explanation

Amplitude = Half the range Range = Maximum − Minimum Divide by 2

✅ Correct Answer Explanation

Maximum value: \(100\) Minimum value: \(50\) Range: \(100-50=50\) Amplitude: \(Amplitude=\frac{100-50}{2}=25\) Thus, the amplitude is 25.

❓ Why Other Options Are Incorrect

Option A → Full range, not amplitude.
Option C → Maximum value only.
Option D → Incorrect calculation.

🎯 Strategy

Used

Substitution

Application:

Apply amplitude formula directly.

Final Logic:

Amplitude is half of total fluctuation range.

🧠 Memory Trick

"Amplitude = Half Range"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Cyclical Variations –

📄 Passage

"The moving average method helps to establish a trend line by eliminating cyclical, seasonal, and random variations present in the time series. The period of the moving average depends upon the length of the time series data. For an odd number of periods (\(n=odd\)), we add the values of the first \(n\) years and place the sum against the median year. There will be no trend value for the beginning period and the ending period. For an even number of periods, centered totals are calculated to align values."

Based on the passage, the moving average method is explicitly applied to eliminate which of the following?

A Long-term secular trends

B Permanent baseline mathematical constants

C Cyclical, seasonal, and random variations

D The fundamental median year data point

💡 Short Explanation

Moving averages smooth fluctuations Removes short-term disturbances Helps identify trend clearly

✅ Correct Answer Explanation

The passage states that moving averages eliminate cyclical, seasonal, and random variations. This smoothing process helps reveal the long-term secular trend. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Moving averages help identify secular trend, not remove it.
Option B → No such role exists.
Option D → Median year is still retained.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Identify the exact fluctuations named in the passage.

Final Logic:

Moving averages smooth short-term variations.

🧠 Memory Trick

"Moving Average Removes Noise"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Average Method –

📄 Passage

According to the passage, when calculating a 3-year (odd) moving sum, where is the result specifically recorded?

A Against the first calendar year

B Against the final tracking year

C Against a completely shifted origin year

D Against the median (middle) year

💡 Short Explanation

Odd moving averages have a center Sum is placed at middle year No centering adjustment needed

✅ Correct Answer Explanation

For odd-number moving averages like 3-year averages, the calculated total or average is placed against the median (middle) year. This ensures proper alignment with time. Therefore, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → First year is not the center.
Option B → Final year is not the placement point.
Option C → No shifted origin is required.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use the passage statement directly.

Final Logic:

Odd moving totals align with the middle period.

🧠 Memory Trick

"Odd → Middle Year"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Average Method –

16 A simplified semi-average method draws a trend line connecting average points
\(P_{1}(2,10) and P_{2}(6,20)\)
This linear trend forms a geometric trapezoid bounded by the x-axis, \(x=2\), and \(x=6\). What is the area of this region?

A 60

B 50

C 70

D 80

💡 Short Explanation

Region forms a trapezoid Use trapezium area formula Parallel sides are heights

✅ Correct Answer Explanation

Heights: \(10 and 20\) Distance between them: \(6-2=4\) Area of trapezium: \(Area=\frac{1}{2}(10+20)(6-2)=\frac{1}{2}(30)(4)=60\) Hence, the area equals 60.

❓ Why Other Options Are Incorrect

Option B → Incomplete calculation.
Option C → Arithmetic mistake.
Option D → Overestimated area.

🎯 Strategy

Used

Substitution

Application:

Apply trapezium area formula directly.

Final Logic:

Average of parallel sides × distance.

🧠 Memory Trick

"Trap Area = Half × Sum × Distance"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Semi-average Method –

17 For a dataset smoothed using a 5-year moving average with the values
2, 4, 6, 8, 10, 12,
what is the second sequentially calculated moving average?

A 6

B 8

C 10

D 4

💡 Short Explanation

Second moving average uses last five values Add observations Divide by 5

✅ Correct Answer Explanation

Second 5-year group: \(4,6,8,10,12\) Average: \(\frac{4+6+8+10+12}{5}=\frac{40}{5}=8\) Thus, the second moving average is 8.

❓ Why Other Options Are Incorrect

Option A → First moving average.
Option C → Largest observation, not average.
Option D → Incorrect averaging.

🎯 Strategy

Used

Substitution

Application:

Use the second rolling window.

Final Logic:

Moving average shifts one observation forward.

🧠 Memory Trick

"Shift and Average"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Moving Average Method –

18 In an even-period smoothing matrix \(\left(n,\ 4\right)\), the first calculated 4-year moving average is 432.5 and the subsequent 4-year moving average is 437.75. What is their respective centered average?

A 434.50

B 436.00

C 435.13

D 438.10

💡 Short Explanation

Even moving averages require centering Take average of adjacent values Centered value aligns time periods

✅ Correct Answer Explanation

Centered average: \(\frac{432.5+437.75}{2}\) \(\frac{432.5+437.75}{2}=435.125\approx 435.13\) Hence, the centered moving average is 435.13.

❓ Why Other Options Are Incorrect

Option A → Incorrect averaging.
Option B → Rounded incorrectly.
Option D → Larger than both values.

🎯 Strategy

Used

Substitution

Application:

Average the two adjacent moving averages.

Final Logic:

Centering aligns even-period averages.

🧠 Memory Trick

"Even → Average Two"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Centered Moving Average –

19 When fitting a linear trend equation
\(y=a+bx\)
by shifting the origin such that \(\sum X=0\), if \(\sum Y=1065\) and the sample size is \(n=7\), what is the value of parameter \(a\)?

A 16.6

B 102.3

C 201.9

D 152.1

💡 Short Explanation

When \(\sum X=0\), use simplified formula \(a=\frac{\sum Y}{n}\) Divide total by observations

✅ Correct Answer Explanation

Formula: \(a=\frac{\sum Y}{n}\) Substitute values: \(a=\frac{1065}{7}=152.14\approx 152.1\) Thus, parameter \(a\) equals approximately 152.1.

❓ Why Other Options Are Incorrect

Option A → Incorrect division.
Option B → Arithmetic mistake.
Option C → Excessively large value.

🎯 Strategy

Used

Substitution

Application:

Directly apply least squares formula.

Final Logic:

Mean of Y-values gives \(a\) when \(\sum X=0\).

🧠 Memory Trick

"When ΣX = 0, a = Average Y"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Least Squares Method –

20 Continuing with best-fit estimations, if the centered time deviations give
\(\sum X=0,\sum XY=465,\sum X^{2}=28\)
determine the optimal value of the linear slope \(b\).

A 16.6

B 15.5

C 14.8

D 17.2

💡 Short Explanation

Use least squares slope formula Divide \(\sum XY\) by \(\sum X^{2}\) Gives trend slope

✅ Correct Answer Explanation

Formula: \(b=\frac{\sum XY}{\sum X^{2}}\) Substitute values: \(b=\frac{465}{28}=16.61\approx 16.6\) Hence, the slope \(b\) is approximately 16.6.

❓ Why Other Options Are Incorrect

Option B → Incorrect division.
Option C → Miscalculation.
Option D → Overestimated value.

🎯 Strategy

Used

Substitution

Application:

Apply least squares slope equation directly.

Final Logic:

Slope equals covariance term divided by squared deviation term.

🧠 Memory Trick

"b = XY over X²"

📚 Class XI – Statistics for Economics – Chapter: Time Series – Topic: Least Squares Method –

UG Applied Mathematics Booster Test 2 - Time Series Analysis

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