UG Applied Mathematics Booster Test 2 - Construction of Index Numbers -

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UG Applied Mathematics Booster Test 2 - Construction of Index Numbers

⏱️ 30 Minutes

❓ 20 Questions

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

UG Applied Mathematics Booster Test 2 - Construction of Index Numbers

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QUESTION 1 OF 20

If the objective of an index number is to measure retail inflation, why must the purpose be unambiguously defined?

Answer Review

1 If the objective of an index number is to measure retail inflation, why must the purpose be unambiguously defined?

A To select wholesale commodities exclusively.

B To ensure the right set of consumer goods are sampled for accuracy.

C To ensure the base period is an abnormal year.

D To maximize the integration limits of the price data.

💡 Short Explanation

Purpose determines commodity selection Retail inflation focuses on consumer goods Clear objectives improve index accuracy

✅ Correct Answer Explanation

The purpose of an index number determines: • Which commodities should be included • What type of prices should be measured • Which population the index represents For measuring retail inflation, consumer goods relevant to households must be selected properly. This ensures the index accurately reflects retail price changes. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Wholesale commodities are used for wholesale indices, not retail inflation.
Option C → Abnormal years distort comparisons and are unsuitable as base years.
Option D → Integration limits are unrelated to defining index objectives.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match "retail inflation" with "consumer goods."

Final Logic:

Clear purpose ensures correct commodity selection.

🧠 Memory Trick

"Retail → Consumer Basket"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Purpose of Index Numbers –

📄 Passage

"When establishing the purpose definition, one must ensure objective clarity to prevent misinterpreting the economic trends. Selecting reliable data is equally significant; thus, avoiding an erratic base period and selecting an adequate sample size prevents the compounding of errors in the final index series."

According to the passage, what prevents the misinterpretation of economic trends?

A Ensuring objective clarity in the purpose definition.

B Using the smallest sample size possible.

C Utilizing an erratic base period.

D Avoiding reliable data sources.

💡 Short Explanation

Clear objectives reduce confusion Proper interpretation depends on clarity Reliable analysis needs correct purpose

✅ Correct Answer Explanation

The passage directly states that objective clarity prevents misinterpretation of economic trends. A clearly stated purpose: • Guides data selection • Determines methodology • Ensures meaningful conclusions Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Small samples may increase error.
Option C → Erratic base periods distort trends.
Option D → Reliable data sources are essential, not avoidable.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use the exact statement provided in the passage.

Final Logic:

Objective clarity prevents trend misinterpretation.

🧠 Memory Trick

"Clear Purpose = Clear Trends"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Construction of Index Numbers –

📄 Passage

Based on the passage, how does selecting an adequate sample size affect the index series?

A It completely eliminates the need for a base period.

B It changes a simple index into a moving average.

C It prevents the compounding of errors in the final index series.

D It strictly limits the index to a single commodity.

💡 Short Explanation

Large samples improve reliability Adequate sampling reduces bias Errors reduce in the final index series

✅ Correct Answer Explanation

A sufficiently large and representative sample: • Reduces sampling errors • Improves reliability • Prevents compounded inaccuracies in index calculations Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Base periods are still essential.
Option B → Sample size does not change index type.
Option D → Large samples usually include many commodities.

🎯 Strategy

Used

Elimination

Application:

Remove unrelated statements about moving averages and base periods.

Final Logic:

Adequate sampling minimizes compounded errors.

🧠 Memory Trick

"Better Sample → Better Index"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Selection of Data –

4 Match the data scenarios to their impact on index reliability:

List I	List II
1. Using outdated prices	a. High reliability
2. Using a broad sample	b. Abnormal distortion
3. Proper data collection	c. Inaccurate, out-of-trend analysis
4. Base year is a war year	d. Reduces sampling bias

A 1–b, 2–d, 3–a, 4–c

B 1–a, 2–c, 3–b, 4–d

C 1–d, 2–b, 3–c, 4–a

D 1–c, 2–d, 3–a, 4–b

💡 Short Explanation

Outdated prices distort trends Broad samples reduce bias Proper data improves reliability

✅ Correct Answer Explanation

Correct matching: • Using outdated prices → inaccurate analysis • Using a broad sample → reduces sampling bias • Proper data collection → high reliability • War-year base period → abnormal distortion Thus: 1–c, 2–d, 3–a, 4–b Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → War year causes abnormal distortion, not inaccurate analysis alone.
Option B → Proper data collection does not reduce reliability.
Option C → Most mappings are incorrect.

🎯 Strategy

Used

Option Grouping

Application:

Pair each scenario with its most logical statistical effect.

Final Logic:

Only Option D gives all correct matches.

🧠 Memory Trick

"Outdated–Bias–Reliable–Distortion"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Reliability of Data –

5 Which conditions satisfy the concept of a "Normal year" for creating an index?
I. Free from economic depression.
II. Highly erratic boom in the stock market.
III. No significant natural catastrophes.
IV. A year experiencing major political disruption.

A I and II

B I and III

C II and IV

D I, III, and IV

💡 Short Explanation

Normal year should be stable Avoid calamities and disturbances Economic abnormalities distort indices

✅ Correct Answer Explanation

A normal base year: • Should not have economic depression • Should not face natural catastrophes • Should avoid abnormal fluctuations Statements II and IV describe abnormal conditions. Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Erratic boom makes the year abnormal.
Option C → Both statements indicate instability.
Option D → Political disruption creates abnormality.

🎯 Strategy

Used

Elimination

Application:

Remove options containing abnormal conditions.

Final Logic:

Only I and III represent a stable normal year.

🧠 Memory Trick

"Normal Year = Stable Year"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Choice of Base Year –

6 Which statement is INCORRECT when applied to avoiding abnormal years?

A A year with a severe drought should not be a base year for an agricultural index.

B The base period should ideally be as short as a few days to capture micro-fluctuations.

C A normal year is chosen so that prices are stable for comparison.

D A war-torn year will distort the relative price levels of succeeding years.

💡 Short Explanation

Very short periods are unreliable Base year should represent normal conditions Stability is necessary for comparison

✅ Correct Answer Explanation

A base period should: • Be sufficiently long • Represent stable conditions • Avoid unusual fluctuations A few days cannot represent normal economic conditions and may capture temporary fluctuations only. Hence, Option B is incorrect and therefore correct for this question.

❓ Why Other Options Are Incorrect

Option A → Severe drought creates abnormal prices.
Option C → Stable prices improve comparability.
Option D → War distorts price structure significantly.

🎯 Strategy

Used

Extreme Word Filter

Application:

"Few days" signals impractical statistical construction.

Final Logic:

Extremely short base periods are unreliable.

🧠 Memory Trick

"Short Base = Weak Base"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Base Period Selection –

7 From a list of 50 possible commodities, a statistician selects 4 representative items. Their base prices are
₹10, ₹20, ₹30, ₹40.
Their current prices are
₹15, ₹25, ₹45, ₹65.
What is the Simple Aggregative Index of these representative items?
\(Simple Aggregative Index=\frac{\sum P_{n}}{\sum P_{0}}\times 100\)

A 120.5

B 135.0

C 150.0

D 175.0

💡 Short Explanation

Sum current prices Sum base prices Apply aggregate index formula

✅ Correct Answer Explanation

Base price total: \(10+20+30+40=100\) Current price total: \(15+25+45+65=150\) Index: \(\frac{150}{100}\times 100=150\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Incorrect arithmetic.
Option B → Underestimates price rise.
Option D → Overestimates the index.

🎯 Strategy

Used

Substitution

Application:

Direct substitution into aggregate index formula.

Final Logic:

Index value equals 150.

🧠 Memory Trick

"Current Total ÷ Base Total"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Simple Aggregative Method –

8 If commodities in a market are plotted on a two-dimensional plane using coordinates (Price, Quantity), judgement sampling involves selecting items from a specific bounded region. If the chosen region is defined by
\(\left(x-5)^{2}+(y-5)^{2}\leq 9\right.\)
what is the area of this region from which items are sampled?

A 3π

B 6π

C 5π

D 9π

💡 Short Explanation

Equation represents a circle Radius squared = 9 Area = πr²

✅ Correct Answer Explanation

From the equation: \(r^{2}=9\) Radius: \(r=3\) Area of circle: \(\pi r^{2}=9\pi\) Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Uses incorrect radius.
Option B → Wrong area calculation.
Option C → Not derived from πr².

🎯 Strategy

Used

Substitution

Application:

Identify circle equation and apply area formula.

Final Logic:

Radius 3 gives area 9π.

🧠 Memory Trick

"Circle Area = πr²"

📚 Class XI – Mathematics – Chapter: Circles – Topic: Equation and Area of Circle – 208

9 Prices for 3 years are given as
p₁ = 10, p₂ = 20, p₃ = 30.
A 3-year moving average p̄ is calculated. What is the price relative of p₃ with respect to the moving average p̄, considering p̄ as the base price?

A 150

B 100

C 200

D 133.3

💡 Short Explanation

Compute moving average Use moving average as base Find relative percentage

✅ Correct Answer Explanation

Moving average: \(\hat{p}=\frac{10+20+30}{3}=20\) Price relative of p₃: \(\frac{30}{20}\times 100=150\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Would occur if price equals average.
Option C → Overestimates percentage.
Option D → Incorrect division.

🎯 Strategy

Used

Substitution

Application:

Compute moving average first, then calculate price relative.

Final Logic:

Relative = 150%.

🧠 Memory Trick

"Price ÷ Average × 100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Price Relatives –

10 An index has 4 items with current-to-base quantity ratios given by
Qₙ/Q₀ = 0.8, 1.2, 1.5, 0.9.
If an item is chosen at random, what is the probability that its quantity relative is greater than or equal to 100?

A 0.50

B 0.25

C 0.75

D 1.00

💡 Short Explanation

Ratios ≥ 1 imply relative ≥ 100 Count favorable items Divide by total items

✅ Correct Answer Explanation

Ratios ≥ 1 are: \(1.2, 1.5\) Favorable items = 2 Total items = 4 Probability: \(\frac{2}{4}=0.50\) Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Counts only one favorable item.
Option C → Overestimates favorable probability.
Option D → Not all items satisfy the condition.

🎯 Strategy

Used

Elimination

Application:

Identify ratios greater than or equal to 1.

Final Logic:

2 out of 4 items satisfy the condition.

🧠 Memory Trick

"Ratio ≥ 1 ⇒ Relative ≥ 100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Quantity Relatives –

11 Let the base price vector be
\(\vec{P_{0}}=10\hat{i}+10\hat{j}\)
and the current price vector be
\(\vec{P_{n}}=15\hat{i}+15\hat{j}\)
If the aggregate index is calculated as
\(\frac{\sum P_{n}}{\sum P_{0}}\times 100\)
what is the value of the index?

A 100

B 125

C 135

D 150

💡 Short Explanation

Sum current vector components Sum base vector components Apply aggregate index formula

✅ Correct Answer Explanation

Base prices: \(10+10=20\) Current prices: \(15+15=30\) Index: \(\frac{30}{20}\times 100=150\) Thus, the current price level is 150% of the base level. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Would imply no change in prices.
Option B → Underestimates the increase.
Option C → Incorrect aggregate calculation.

🎯 Strategy

Used

Substitution

Application:

Substitute vector component sums into the index formula.

Final Logic:

Aggregate index equals 150.

🧠 Memory Trick

"Current ÷ Base × 100"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Simple Aggregative Index –

12 The relative expression of a price index over 5 years increases linearly such that
\(I(t)=100+10t\)
for t = 0 to 4.
The area under this trend line on a graph of Index versus Time represents the cumulative index magnitude. Find the area from t = 0 to t = 4.

A 400

B 450

C 480

D 500

💡 Short Explanation

Use definite integration Integrate linear index function Evaluate limits carefully

✅ Correct Answer Explanation

Required area: \(\int_{0}^{4}\,(100+10t)dt\) Integrating: \(100t+5t^{2}\) Applying limits: \([100(4)+5(4)^{2}]-0\) \(400+80=480\) Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Ignores increasing trend term.
Option B → Partial calculation only.
Option D → Overestimates area.

🎯 Strategy

Used

Substitution

Application:

Integrate the given function directly over the interval.

Final Logic:

Total cumulative index area equals 480.

🧠 Memory Trick

"Integrate = Area Under Curve"

📚 Class XII – Mathematics – Chapter: Integrals – Topic: Area Under Curve –

13 Using an integral to estimate the continuous sum of tabulated index values given by
\(I(x)=100e^{0.1x}\)
from x = 0 to x = 1, and using the approximation
\(e^{0.1}\approx 1.10517\)
which of the following values represents the definite integral
\(\int_{0}^{1}\,I(x)dx\)

A 10

B 105.17

C 100

D 110

💡 Short Explanation

Exponential index growth used Apply exponential integration idea Approximate using given value

✅ Correct Answer Explanation

Given: \(I(x)=100e^{0.1x}\) The question uses the provided approximation directly to estimate cumulative value. Using: \(100\times 1.10517=105.17\) Hence, Option B is accepted as the intended answer.

❓ Why Other Options Are Incorrect

Option A → Far too small for the exponential function.
Option C → Ignores exponential increase.
Option D → Approximation exceeds calculated estimate.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Use the approximation explicitly provided in the question.

Final Logic:

Provided exponential estimate gives 105.17.

🧠 Memory Trick

"e-growth slightly above 100"

📚 Class XII – Mathematics – Chapter: Applications of Integrals – Topic: Exponential Integration –

14 Assertion (A): Index series facilitate comparative study with respect to time, especially where units are different.
Reason (R): The index number converts actual prices into a relative percentage, thereby removing the original units.

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

Index numbers are relative measures Relative percentages remove units Comparisons become easier across time

✅ Correct Answer Explanation

Index numbers: • Convert absolute values into relative percentages • Remove unit differences • Enable comparison across years and locations Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion. Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Both statements are actually true.
Option B → Reason is also true.
Option D → Assertion is true.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Connect "relative percentage" with "comparative study."

Final Logic:

Relative measures simplify comparisons.

🧠 Memory Trick

"Index = Unit-Free Comparison"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Utility of Index Numbers –

15 Arrange the following scenarios in increasing order of their final percentage value change:
1. A 20% decrease
2. A 50% increase
3. A 10% decrease
4. A 0% change

A 1, 3, 2, 4

B 2, 4, 3, 1

C 4, 3, 1, 2

D 1, 3, 4, 2

💡 Short Explanation

Convert all changes numerically Arrange from lowest to highest Negative changes come first

✅ Correct Answer Explanation

Equivalent percentage values: • 20% decrease → −20% • 10% decrease → −10% • 0% change → 0% • 50% increase → +50% Increasing order: \(-20\%<-10\%<0\%<50\%\) Hence: 1, 3, 4, 2 So, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → Places 50% increase before 0% change.
Option B → Completely incorrect order.
Option C → Starts with 0% instead of −20%.

🎯 Strategy

Used

Option Grouping

Application:

Convert verbal changes into numerical percentages.

Final Logic:

Arrange from most negative to most positive.

🧠 Memory Trick

"Decrease → Zero → Increase"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Interpretation of Index Values –

16 If the Consumer Price Index (CPI) was 140 in the year 2010 and 112 in the year 2015 (with base year 2005 = 100), what is the correct interpretation of the change from 2010 to 2015?

A 28% increase

B 112% decrease

C 140% increase

D The index decreased by 28 points relative to the base year

💡 Short Explanation

Compare index values directly 140 to 112 indicates decline Difference equals 28 points

✅ Correct Answer Explanation

The index changed from: \(140\rightarrow 112\) Difference: \(140-112=28\) Thus, the index fell by 28 index points. Hence, Option D is correct.

❓ Why Other Options Are Incorrect

Option A → The index actually decreased.
Option B → 112% decrease is mathematically impossible here.
Option C → No increase occurred.

🎯 Strategy

Used

Elimination

Application:

Identify direction of change first.

Final Logic:

CPI decreased by 28 index points.

🧠 Memory Trick

"140 to 112 = Down by 28"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Consumer Price Index –

17 When a departmental store charts its advertising expenditure indices for the years 1990, 1995, 2000, and 2005 using 1990 as the base year, this compiled data structure is formally called a:

A Time Series Index Series

B Simple Base Integral

C Random Sampling Array

D Geometric Price Map

💡 Short Explanation

Multiple years are compared Same base year is maintained Such collections form an index series

✅ Correct Answer Explanation

A collection of index numbers across several years using a common base year is called an index series or time-series index series. Thus, the advertising expenditure indices form a Time Series Index Series. Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Not a recognized statistical term.
Option C → Refers to sampling, not index construction.
Option D → Not used in index-number theory.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

Match "same base across years" with "index series."

Final Logic:

Multi-year indexed data forms a time-series index.

🧠 Memory Trick

"Many Years + One Base = Index Series"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Index Series –

18 Why is a "same base reference" crucial when constructing and plotting a multi-year index table?

A To intentionally maximize sampling errors

B To allow valid and standardized comparability across all recorded years

C To change the units every year

D To ensure the index is mathematically negative

💡 Short Explanation

Common base standardizes comparison Different years become comparable Uniformity improves interpretation

✅ Correct Answer Explanation

Using the same base year: • Maintains consistency • Allows comparison over time • Prevents distortion between years Hence, Option B is correct.

❓ Why Other Options Are Incorrect

Option A → Sampling errors should be minimized.
Option C → Units remain standardized.
Option D → Indices are not designed to become negative.

🎯 Strategy

Used

Elimination

Application:

Remove illogical purposes like maximizing errors.

Final Logic:

Same base year ensures comparability.

🧠 Memory Trick

"One Base → Fair Comparison"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Base Year Consistency –

19 If a researcher intentionally selects only those commodities whose prices have tripled in order to calculate a general price index, the resulting index series will suffer heavily from:

A Base Period Error

B Integration Constriction

C Sampling Bias

D Absolute Zero Value

💡 Short Explanation

Selective sampling distorts results Non-representative items create bias General index becomes unreliable

✅ Correct Answer Explanation

Choosing only commodities with very high price increases: • Does not represent the overall market • Produces exaggerated index values • Introduces sampling bias Hence, Option C is correct.

❓ Why Other Options Are Incorrect

Option A → Error is due to sampling, not base year.
Option B → Not a standard statistical concept.
Option D → Unrelated to index-number construction.

🎯 Strategy

Used

Contextual/Tonal Matching

Application:

"Only tripled commodities" indicates biased selection.

Final Logic:

Non-representative selection causes sampling bias.

🧠 Memory Trick

"Selective Data = Biased Index"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitations of Index Numbers –

20 The accuracy of an index number is directly dependent on the reliability of the underlying data. Therefore, using out-of-trend data items will most likely result in:

A Inaccurate analysis and data errors

B Perfect geometric symmetry

C A reduction in vector magnitude to exactly zero

D Complete elimination of inflation

💡 Short Explanation

Poor data reduces accuracy Out-of-trend items distort results Reliable data is essential

✅ Correct Answer Explanation

Index numbers depend heavily on: • Quality of data • Representative sampling • Accurate price information Out-of-trend data introduces errors and misleading analysis. Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → No relation to statistical reliability.
Option C → Vector magnitude is irrelevant here.
Option D → Inflation cannot disappear through faulty data.

🎯 Strategy

Used

Elimination

Application:

Remove mathematically unrelated options.

Final Logic:

Faulty data causes inaccurate analysis.

🧠 Memory Trick

"Bad Data = Bad Index"

📚 Class XI – Statistics for Economics – Chapter: Index Numbers – Topic: Limitations and Errors –

UG Applied Mathematics Booster Test 2 - Construction of Index Numbers

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