CUET UG Applied Mathematics Booster Test 1

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CUET UG Applied Mathematics Booster Test 1 - Population & Sample

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 1 - Population & Sample

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QUESTION 1 OF 20

Which of the following describe a parameter in relation to a population?
Statements:
I. It is a numerical value computed from all population observations.
II. It is a characteristic strictly belonging to a sample.
III. It is commonly denoted by Greek symbols (e.g., μ, σ).

Answer Review

1 Which of the following describe a parameter in relation to a population?
Statements:
I. It is a numerical value computed from all population observations.
II. It is a characteristic strictly belonging to a sample.
III. It is commonly denoted by Greek symbols (e.g., μ, σ).

A I and II

B II and III

C I and III

D I, II, and III

💡 Short Explanation

A parameter is a numerical characteristic of an entire population. It is usually represented by Greek letters, such as μ (population mean) and σ (population standard deviation).

✅ Correct Answer Explanation

A parameter summarizes a characteristic of the entire population, not just a sample. Statement I: ✔ True A parameter is calculated using all observations in the population. Example: The mean height of every student in a school is a population parameter. Statement II: ✘ False A characteristic of a sample is called a statistic, not a parameter. Statement III: ✔ True Population parameters are commonly represented using Greek symbols, such as: μ = Population mean σ = Population standard deviation σ² = Population variance Therefore, the correct combination is I and III, making Option C the correct answer.

❓ Why Other Options Are Incorrect

Option A) I and II → Incorrect because Statement II refers to a sample statistic, not a population parameter.
Option B) II and III → Incorrect because Statement II is false.
Option D) I, II, and III → Incorrect because Statement II is false.

🎯 Strategy

Used

Conceptual Recall

Application: Distinguish between a population parameter and a sample statistic, and recall the standard notation used in statistics.

Final Logic: Since Statements I and III are true, the correct answer is Option C.

🧠 Memory Trick

Statistic = Sample = Roman letters (e.g., x̄, s)

📚 Class XI – Statistics – Topic: Population, Sample, Parameter, and Statistic

2 In a continuous setting, the population size N is modeled by:
\(N=\int_{0}^{2}\,3x^{2}dx\)

A 4

B 8

C 16

D 6

💡 Short Explanation

Integrate polynomial Apply limits 0 to 2 Compute definite integral

✅ Correct Answer Explanation

\(\int 3x^{2}dx=x^{3}\). Evaluating from 0 to 2: \(2^{3}-0=8\)

❓ Why Other Options Are Incorrect

Option A → Incorrect evaluation
Option C → Overestimation
Option D → Wrong integration result

🎯 Strategy

Used: Substitution

Application: Direct integral evaluation

Final Logic: x³ evaluated at bounds

🧠 Memory Trick

"3x² → x³ shortcut"

📚 Class XII – NCERT Mathematics – Chapter: Integrals – Topic: Definite Integrals –

3 Match List I with List II:

List I	List II
1. μ	a. Population mean
2. x̄	b. Sample mean
3. σ	c. Population standard deviation
4. S	d. Sample standard deviation

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

μ = population mean x̄ = sample mean σ = population SD S = sample SD

✅ Correct Answer Explanation

Standard statistical notation mapping directly defines population and sample measures.

❓ Why Other Options Are Incorrect

Option B → Mislabels mean/SD
Option C → Completely reversed mapping
Option D → Incorrect swapping

🎯 Strategy

Used: Option Grouping

Application: Symbol-definition pairing

Final Logic: Standard statistical notation

🧠 Memory Trick

"Greek = population, Latin = sample"

📚 Class XI – NCERT Statistics – Chapter: Statistics – Topic: Symbols in Statistics –

4 Arrange progression from broadest to most specific:
I. Statistic
II. Sample
III. Parameter
IV. Population

A I, II, III, IV

B II, I, IV, III

C III, IV, I, II

D IV, III, II, I

💡 Short Explanation

Population is largest set Sample is subset Parameter describes population Statistic describes sample

✅ Correct Answer Explanation

Population → Parameter → Sample → Statistic represents hierarchy from general to specific inference levels.

❓ Why Other Options Are Incorrect

Option A → reverse order
Option B → incorrect grouping
Option C → mismatched hierarchy

🎯 Strategy

Used: Logical Ordering

Application: Hierarchical arrangement

Final Logic: population → sample flow

🧠 Memory Trick

"Big → Small → Measure"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Data Hierarchy –

5 Assertion (A): It is nearly impossible to conduct clinical trials on an entire country's population.
Reason (R): Large populations involve constraints of time, cost, and logistics, requiring sampling.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R explains A

D A is false, R is true

💡 Short Explanation

Clinical trials are costly Full population studies impractical Sampling reduces burden

✅ Correct Answer Explanation

Large populations require sampling due to cost, time, and feasibility constraints.

❓ Why Other Options Are Incorrect

Option A → Both statements are true
Option B → Reason is valid
Option D → Incorrect relationship

🎯 Strategy

Used: Cause–Effect Matching

Application: Link practicality with sampling need

Final Logic: constraints justify sampling

🧠 Memory Trick

"Big population → sample solution"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Need for Sampling –

6 Find expected value of vector:
Let:
\(\vec{X}=[\hat{x}_{1},\hat{x}_{2},\hat{x}_{3}],with E[\hat{x}_{i}]=\mu\)
Find:
\(E[\vec{X}]\)

A \(\left[ \right]\)

B \(\left[\mu \ ,\ \mu \ ,\ \mu \right]\)

C \(\left[{\mu}_{1}\ ,\ {\mu}_{2}\ ,\ {\mu}_{3}\right]\)

D \({\mu}^{3}\)

💡 Short Explanation

Expectation distributes linearly Each component equals μ Vector expectation becomes uniform

✅ Correct Answer Explanation

Since E[x̄ᵢ] = μ, expectation of vector becomes [μ, μ, μ].

❓ Why Other Options Are Incorrect

Option A → incomplete vector
Option C → mismatched components
Option D → invalid transformation

🎯 Strategy

Used: Substitution

Application: Apply expectation operator component-wise

Final Logic: linearity of expectation

🧠 Memory Trick

"E spreads evenly"

📚 Class XII – NCERT Mathematics – Chapter: Probability – Topic: Expectation –

7 According to the Central Limit Theorem, as sample size \(n\) increases, the sampling distribution:

A Becomes flat

B Approaches a normal (bell-shaped) curve

C Exactly matches population shape

D Becomes highly skewed

💡 Short Explanation

CLT governs distribution shape Larger samples normalize distribution Approaches Gaussian form

✅ Correct Answer Explanation

Central Limit Theorem states sampling distribution tends toward normality as n increases.

❓ Why Other Options Are Incorrect

Option A → incorrect flattening
Option C → false exact match claim
Option D → opposite behavior

🎯 Strategy

Used: Theoretical Recall

Application: CLT property recognition

Final Logic: convergence to normal curve

🧠 Memory Trick

"Big n → Bell curve"

📚 Class XI – NCERT Statistics – Chapter: Sampling Distributions – Topic: Central Limit Theorem – 310

8 A 95% confidence interval represents the central area under a normal curve.
What is the combined area in the two tails?

A \(5\%\)(\(0.05\))

B \(10\%\)(\(0.10\))

C \(95\%\)(\(0.95\))

D \(2.5\%\)(\(0.025\))

💡 Short Explanation

Central area = 95% Remaining area = tails Split equally across two tails

✅ Correct Answer Explanation

Total probability is 1, so outside 95% interval is 0.05.

❓ Why Other Options Are Incorrect

Option B → incorrect total tail sum
Option C → central value
Option D → single tail value

🎯 Strategy

Used: Complement Rule

Application: 1 − 0.95

Final Logic: tail probability = 0.05

🧠 Memory Trick

"100 − 95 = 5"

📚 Class XI – NCERT Statistics – Chapter: Normal Distribution – Topic: Confidence Intervals – 322

9 Data: \(10,15,20,25,30\)
3-week moving averages:
\(\frac{10+15+20}{3}=15,\frac{15+20+25}{3}=20,\frac{20+25+30}{3}=25\)

A 12, 18, 22

B 10, 15, 20

C 15, 20, 25

D 20, 25, 30

💡 Short Explanation

Average consecutive triplets Sliding window method Produces smoothed series

✅ Correct Answer Explanation

(10+15+20)/3 = 15, (15+20+25)/3 = 20, (20+25+30)/3 = 25.

❓ Why Other Options Are Incorrect

Option A → incorrect arithmetic
Option B → raw data misuse
Option D → incorrect transformation

🎯 Strategy

Used: Substitution

Application: Step-by-step averaging

Final Logic: sliding mean computation

🧠 Memory Trick

"Add 3, divide 3"

📚 Class XI – NCERT Statistics – Chapter: Time Series – Topic: Moving Average – 335

10 From \(N=1000\), what is the probability a specific item is not selected on the first draw?

A \(\frac{1}{1000}\)

B \(\frac{1}{50}\)

C \(\frac{50}{1000}\)

D \(\frac{999}{1000}\)

💡 Short Explanation

Only 1 selected out of 1000 Remaining are not selected Probability = complement

✅ Correct Answer Explanation

Out of 1000 items, 999 remain unselected in first draw.

❓ Why Other Options Are Incorrect

Option A → selection probability
Option B → irrelevant fraction
Option C → incorrect ratio

🎯 Strategy

Used: Complement Principle

Application: 1 − 1/1000

Final Logic: non-selection probability

🧠 Memory Trick

"Total minus one"

📚 Class XI – NCERT Statistics – Chapter: Probability – Topic: Basic Probability –

11 From N=10, choose n=3:
(10¦3)

A 120

B 1000

C 720

D 30

💡 Short Explanation

This is a combination problem: \(^{10}C_{3}\) Order does not matter Use formula \(n!/(r!(n-r)!)\)

✅ Correct Answer Explanation

\(^{10}C_{3}=\frac{10!}{3!7!}=\frac{10\times 9\times 8}{3\times 2\times 1}=120\)

❓ Why Other Options Are Incorrect

Option B → Not related to combinations
Option C → Incorrect factorial computation
Option D → Represents a product form, not combinations

🎯 Strategy

Used: Substitution

Application: Directly apply combination formula

Final Logic: Correct value comes from standard nCr formula

🧠 Memory Trick

"nCr = factorial cut-down multiplication"

📚 Class XI – NCERT Mathematics – Chapter: Permutations and Combinations – Topic: Combination Formula – 1

12 Which mathematical statement is theoretically incorrect regarding determining sample size n and degrees of freedom Df?

A A higher degree of freedom generally implies larger sample sizes

B The formula for degrees of freedom is Df=n-1

C A sample size of n≥30 is considered sufficient to apply the Central Limit Theorem

D Decreasing the sample size n mathematically improves the accuracy of population estimates

💡 Short Explanation

Smaller samples increase error Larger samples improve accuracy Df increases with sample size

✅ Correct Answer Explanation

Smaller sample sizes increase sampling error and reduce reliability. Accuracy improves with larger n, not smaller.

❓ Why Other Options Are Incorrect

Option A → True: larger n increases df
Option B → True: df = n − 1
Option C → True: CLT threshold rule
Option D → False: contradicts statistical principle

🎯 Strategy

Used: Extreme Word Filter

Application: "improves accuracy" vs statistical reality

Final Logic: Smaller sample size worsens accuracy

🧠 Memory Trick

"More data = more accuracy"

📚 Class XI – NCERT Mathematics – Chapter: Statistics – Topic: Sampling & Degrees of Freedom – 312

13 When performing a one-sample t-test, if the recorded sample size is N, the degrees of freedom Df used is:

A N

B N-1

C N+1

D N/2

💡 Short Explanation

t-test uses sample-based estimation One parameter estimated reduces freedom Hence df = n − 1

✅ Correct Answer Explanation

One degree of freedom is lost because the sample mean is estimated from the data.

❓ Why Other Options Are Incorrect

Option A → Incorrect; ignores estimation constraint
Option C → No statistical basis
Option D → Irrelevant to df theory

🎯 Strategy

Used: Conceptual Rule Recall

Application: Standard t-test formula recognition

Final Logic: df always reduces by 1 in one-sample t-test

🧠 Memory Trick

"t-test → take 1 away"

📚 Class XI – NCERT Mathematics – Chapter: Statistics – Topic: Degrees of Freedom – 315

📄 Passage

The difference between a population parameter and a sample statistic is known as a sampling error. Even a randomly selected sample contains sampling errors because it is not identical to the population in terms of numerical measures such as means and standard deviations.
Sampling error can be positive or negative and decreases as the sample size increases.
Mathematically:
"Sampling Error"=x ˉ-μ

Causes include:
Small sample size
Faulty selection
Natural variability in outcomes

According to the passage, how is sampling error defined?

A x̄ - μ

B μ - x̄

C x̄ + μ

D x̄ / μ

💡 Short Explanation

Sampling error = sample mean − population mean Can be positive or negative Measures deviation

✅ Correct Answer Explanation

Sampling error is defined as the difference between sample statistic and population parameter.

❓ Why Other Options Are Incorrect

Option B → Reversed sign only
Option C → Not a statistical measure
Option D → Ratio, not difference

🎯 Strategy

Used: Definition Matching

Application: Direct passage extraction

Final Logic: error = estimate − true value

🧠 Memory Trick

"Error = Sample − Population"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Sampling Error – 318

📄 Passage

Even with perfect probability sampling, error still occurs. Reason:

A Perfect selection eliminates all error

B Infinite degrees of freedom

C Small size of the randomly selected sample

D Sample statistics always equal population parameters

💡 Short Explanation

Sampling error exists naturally Finite sample size causes deviation Even random sampling is imperfect

✅ Correct Answer Explanation

Even random samples vary from population due to limited sample size and natural variability.

❓ Why Other Options Are Incorrect

Option A → False assumption
Option B → Irrelevant concept
Option D → Statistically impossible

🎯 Strategy

Used: Elimination

Application: Remove unrealistic absolute statements

Final Logic: finite samples always have variability

🧠 Memory Trick

"Random ≠ perfect"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Sampling Error – 320

16 Sampling method where students selected by last digit "2":

A Systematic probability sampling

B Stratified random sampling

C Convenience sampling

D Cluster sampling

💡 Short Explanation

Selection is rule-based, not random Not all members have equal chance Hence biased sampling

✅ Correct Answer Explanation

Selecting based on digit pattern is not random and introduces bias.

❓ Why Other Options Are Incorrect

Option A → requires interval-based selection
Option B → requires strata division
Option D → requires grouped clusters

🎯 Strategy

Used: Contextual Matching

Application: Identify lack of randomness

Final Logic: fixed digit selection = bias

🧠 Memory Trick

"Fixed rule = biased sample"

📚 Class XI – NCERT Statistics – Chapter: Sampling Methods – Topic: Non-probability Sampling – 325

17 Assertion (A): A representative sample perfectly captures every unique characteristic of all population members.
Reason (R): A representative sample only needs to reflect key features of the population in an unbiased way.

A Both A and R are false

B A is true, R is false

C Both A and R are true, and R explains A

D A is false, R is true

💡 Short Explanation

Representative sample does NOT need perfect capture It should reflect population fairly Reason is correct

✅ Correct Answer Explanation

Representative sample reflects key characteristics but does not perfectly replicate every feature.

❓ Why Other Options Are Incorrect

Option A → false assertion
Option B → partially correct reasoning
Option C → mismatch in logic

🎯 Strategy

Used: Logical Consistency Check

Application: Validate definition accuracy

Final Logic: representation ≠ perfection

🧠 Memory Trick

"Representative ≠ identical"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Representativeness – 327

18 Which traits apply to unrepresentative sample?
I. It is a perfectly unbiased reflection of the population
II. The sample statistic does not accurately represent the population parameter
III. It leads to selection bias

A I and II

B II and III

C I and III

D I, II, and III

💡 Short Explanation

Not representative → bias present Does not reflect population Leads to selection bias

✅ Correct Answer Explanation

Unrepresentative samples fail to reflect population and introduce bias.

❓ Why Other Options Are Incorrect

Option A → includes false claim I
Option C → includes false claim I
Option D → includes incorrect statement I

🎯 Strategy

Used: Elimination

Application: Remove incorrect property I

Final Logic: bias + mismatch only

🧠 Memory Trick

"Unrepresentative = biased + inaccurate"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Sampling Bias – 328

19 Match the sampling methods with their corresponding characteristics.

List I	List II
1. Simple Random Sampling	a. Every individual is selected purely by chance
2. Systematic Random Sampling	b. Members are selected at fixed regular intervals
3. Lottery Method	c. Each unit has an equal probability of being selected
4. Sampling Interval	d. Every kth unit is selected after a random start

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-a, 2-d, 3-b, 4-c

💡 Short Explanation

Simple random sampling is based entirely on chance. Systematic sampling selects units at fixed intervals. Lottery method gives every unit an equal chance of selection. A sampling interval refers to selecting every kth unit after a random start.

✅ Correct Answer Explanation

Sampling methods are used to select representative units from a population. In Simple Random Sampling, every individual in the population has an equal chance of being selected. One common method is the lottery method, where each unit has an equal probability of selection. Therefore, 1 → a 3 → c In Systematic Random Sampling, a random starting point is chosen, and then every kth unit is selected at fixed intervals throughout the population. Therefore, 2 → b 4 → d Thus, the correct matching is: 1 → a 2 → b 3 → c 4 → d Hence, Option A is correct.

❓ Why Other Options Are Incorrect

Option B → Incorrect because it reverses the characteristics of simple random and systematic sampling.
Option C → Incorrect because it interchanges the definitions of the lottery method and simple random sampling.
Option D → Incorrect because the lottery method is not based on fixed intervals, and the sampling interval does not describe simple random sampling.

🎯 Strategy

Used

Option Grouping
Application
Identify the sampling method first, then associate it with its defining feature.
Final Logic
Simple Random Sampling → Chance-based selection → Equal probability.
Systematic Random Sampling → Random start → Every kth unit.

🧠 Memory Trick

"Lottery = Equal Chance"

📚 Class XI – NCERT Statistics for Economics – Chapter: Collection of Data – Topic: Sampling Methods (Simple Random Sampling and Systematic Sampling)

20 Which statement is incorrect regarding biased sampling?

A Bias occurs when certain outcomes are favored

B Convenience sampling produces biased results

C Voluntary response bias occurs when individuals choose to participate

D Biased sampling guarantees equal probability 1/N for all elements

💡 Short Explanation

Biased sampling violates randomness Unequal selection probability Not uniform distribution

✅ Correct Answer Explanation

Biased sampling does NOT ensure equal probability; that is a property of random sampling.

❓ Why Other Options Are Incorrect

Option A → correct definition
Option B → true
Option C → true

🎯 Strategy

Used: Extreme Word Filter

Application: "guarantees equal probability" is contradictory

Final Logic: bias ≠ equal chance

🧠 Memory Trick

"Bias = unequal chance"

📚 Class XI – NCERT Statistics – Chapter: Sampling – Topic: Sampling Bias –

CUET UG Applied Mathematics Booster Test 1 – Population & Sample

CUET UG Applied Mathematics Booster Test 1 - Population & Sample

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Answer Review