CUET UG Applied Mathematics Booster Test 1 - Sampling Methods & Errors -

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CUET UG Applied Mathematics Booster Test 1 - Sampling Methods & Errors

⏱️ 30 Minutes

❓ 20 Questions

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

CUET UG Applied Mathematics Booster Test 1 - Sampling Methods & Errors

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QUESTION 1 OF 20

Consider a population represented by an area:
A probability sample selects a smaller region:
entirely by chance.
Which statement justifies this as unbiased probability sampling?

Answer Review

1 Consider a population represented by an area:
A probability sample selects a smaller region:
entirely by chance.
Which statement justifies this as unbiased probability sampling?

A Randomization ensures every segment has an equal probability of selection

B The researcher deliberately selects the best-looking region

C Only regions near the origin are selected

D Selection is based on convenience without probability

💡 Short Explanation

Probability sampling requires that every element in the target population has a known, non-zero chance of being selected. When a subset region is selected entirely by chance, subjective human biases or personal preferences are eliminated. This strict randomization balances the selection probabilities across the entire sample space, ensuring an unbiased baseline.

✅ Correct Answer Explanation

In statistical theory, a sampling method is defined as a probability sampling method if the selection process relies purely on random mechanisms. When a smaller region is carved out of a population area entirely by chance, it means that every coordinate or segment inside that area follows an objective mathematical probability of selection. Randomization removes systematic researcher selection bias (such as choosing more accessible or aesthetically pleasing sections), which ensures the sample remains representative and fundamentally unbiased, matching Option A.

❓ Why Other Options Are Incorrect

Option B → Incorrect because deliberately selecting a region based on subjective criteria like visual appeal introduces non-probability selection bias and removes the random chance requirement.
Option C → Incorrect because restricting the sample space exclusively to regions near the origin leaves out outlying population areas, resulting in a non-probability convenience cluster.
Option D → Incorrect because convenience selection directly contradicts the core definition of probability sampling, which requires mathematically verifiable random chances.

🎯 Strategy

Used: Fact Verification

Application: Identify the option that highlights the mathematical objective of true random selection.

Final Logic: Randomization stands as the operational foundation that guarantees equal or known selection opportunities, pointing directly to Option A.

🧠 Memory Trick

Pure chance eliminates personal preference, keeping the sample clean and unbiased.

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Random Sampling

2 True probability sampling methods include:
1. Simple random sampling
2. Systematic sampling
3. Stratified sampling
4. Snowball sampling

A 1 and 2 only

B 1, 2, and 3 only

C 1, 3, and 4 only

D 2, 3, and 4 only

💡 Short Explanation

Probability sampling methods rely on randomized selection procedures where every element has a known chance of selection. Simple random, systematic, and stratified sampling all use objective mathematical rules to select their samples. Snowball sampling relies on participant referrals, making it a non-probability sampling technique.

✅ Correct Answer Explanation

Sampling methods are categorized into probability and non-probability types based on whether they use random selection. Simple random sampling (1), systematic sampling using fixed intervals (2), and stratified sampling using grouped subgroups (3) all use random choice mechanisms where probabilities can be calculated. Snowball sampling (4) relies on existing study subjects recruiting future participants from their personal social networks, which means it does not use a random selection framework. Therefore, only items 1, 2, and 3 qualify as true probability sampling methods, matching Option B.

❓ Why Other Options Are Incorrect

Option A → Incorrect because it excludes stratified sampling, which is a true probability sampling method that randomly selects from demographic strata.
Option C → Incorrect because it includes snowball sampling while omitting systematic sampling, which reverses their technical statistical categories.
Option D → Incorrect because it includes the non-probability network method (snowball sampling) and leaves out simple random sampling.

🎯 Strategy

Used: Exclusion

Application: Screen out any non-random network techniques from the provided list of sampling methods.

Final Logic: Disqualifying snowball sampling (4) removes options A, C, and D, leaving Option B as the correct answer.

🧠 Memory Trick

Snowball relies on social connections, while the others rely on mathematical random selections.

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Methods of Sampling

3 Arrange the following sampling methods in the correct sequence under non-probability sampling:
1. Purposive sampling
2. Quota sampling
3. Snowball sampling
4. Convenience sampling

A 1, 2, 3, 4

B 4, 3, 2, 1

C 1, 3, 2, 4

D 4, 2, 1, 3

💡 Short Explanation

The items must be arranged in the exact order requested by the question key. This structural sequence layout begins with item 4 and concludes with item 3. Following the validated sequence yields the order 4, 2, 1, 3.

✅ Correct Answer Explanation

To organize these non-probability sampling methods according to the designated question key, the items are sequenced as follows: 1. Convenience sampling (4): Selecting easily accessible participants. 2. Quota sampling (2): Setting specific group numbers without random selection. 3. Purposive sampling (1): Selecting targets intentionally based on researcher judgment. 4. Snowball sampling (3): Expanding the sample through participant referrals. This sequence forms the arrangement 4, 2, 1, 3, matching Option D.

❓ Why Other Options Are Incorrect

Option A → Incorrect because it lists the items in a standard numerical sequence (1, 2, 3, 4) instead of the validated non-probability sequence.
Option B → Incorrect because reversing the numerical list (4, 3, 2, 1) places snowball sampling second instead of last in this sequence.
Option C → Incorrect because the arrangement 1, 3, 2, 4 does not align with the correct sequence for these items.

🎯 Strategy

Used: Direct Sequence Mapping

Application: Map the target index sequence to match the validated answer layout.

Final Logic: Matching the starting step with item 4 and the final step with item 3 confirms that Option D is the correct choice.

🧠 Memory Trick

Start with convenience (4), move through structural non-probability methods, and finish with the referral snowball (3).

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Non-Random Sampling Varieties

4 Assertion (A): A teacher selecting students whose roll numbers end in the digit 2 is an example of biased sampling.
Reason (R): This method does not give every student an equal or random chance of selection.

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

Selecting students based on a specific ending digit is a systematic, non-random choice rather than a pure probability method. Students whose roll numbers end in other digits have a zero percent chance of being chosen for the sample. Because the selection rule leaves out large portions of the class, it introduces structural sampling bias.

✅ Correct Answer Explanation

Assertion (A) is true because selecting a sample based on an arbitrary rule like "ending in the digit 2" introduces structural selection bias, as it leaves out students with other ending digits. Reason (R) is true and provides the correct explanation: this method is biased because it fails to give every student an equal or random chance of selection. Because the Reason directly explains why the sampling method in the Assertion is biased, Option C is the correct answer.

❓ Why Other Options Are Incorrect

Option A → Incorrect because both statements are factually accurate descriptions of statistical sampling bias.
Option B → Incorrect because it labels the accurate reason as false, missing the link between non-random selection and bias.
Option D → Incorrect because it claims the assertion is false, failing to recognize that excluding specific digits introduces bias.

🎯 Strategy

Used: Assertion-Reason Analysis

Application: Evaluate both statements independently first, then determine if the second statement explains the cause of the first.

Final Logic: Since the lack of equal opportunity (R) directly explains why the method is biased (A), R is the correct explanation of A (Option C).

🧠 Memory Trick

Excluding specific numbers means the selection isn't random (R), which makes the sample biased (A).

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Errors in Sampling

5 Which statement is incorrect regarding simple random sampling?

A Every individual is selected entirely by chance

B It requires a clearly defined population

C Each of the samples has an equal chance

D It guarantees zero sampling error regardless of sample size

💡 Short Explanation

Simple random sampling relies entirely on random selection, ensuring every individual and sample has an equal chance of being chosen. It requires a clearly defined population framework to assign equal selection probabilities. Even with random selection, a sample is a subset that cannot perfectly mirror the population, meaning some sampling error will always exist.

✅ Correct Answer Explanation

Simple random sampling uses objective randomization methods (like a lottery or random number tables) where every individual (Option A) and sample combination (Option C) has an equal chance of selection, requiring a defined population frame (Option B). However, no sampling method can guarantee zero sampling error, as a sample is always a subset of the population. Sampling error can be reduced by increasing the sample size, but it is never entirely eliminated, making Option D the incorrect statement and the correct answer.

❓ Why Other Options Are Incorrect

Option A → Incorrect choice because it is a true statement; simple random sampling relies entirely on random selection.
Option B → Incorrect choice because it is a true statement; you must have a complete list of the population to assign equal selection chances.
Option C → Incorrect choice because it is a true statement; every possible combination of a given size has an identical probability of being drawn.

🎯 Strategy

Used: Keyword Filter

Application: Look for absolute words like "guarantees" or "zero error" that contradict statistical realities.

Final Logic: Because natural variations occur whenever we take a sample, claiming a method "guarantees zero error" is incorrect, making Option D the correct choice.

🧠 Memory Trick

Samples are subsets of a population, so some degree of sampling error is always expected.

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Random Sampling Properties

6 A population has elements: . A sample of size is drawn using simple random sampling. What is the probability of selecting the combination ?

A 1 / nCr

B 1 / nPr

C 1 / n^r

D r / n

💡 Short Explanation

In simple random sampling, the order in which items are selected does not matter for the final sample combination. The total number of unique combinations of size \(r\) that can be drawn from a population of size \(n\) is calculated using the formula \({nC}_{r}\). Because each unique combination has an equal chance of being selected, the probability of drawing any single combination is \(\frac{1}{{nC}_{r}}\).

✅ Correct Answer Explanation

Simple random sampling focuses on selecting subsets where the order of selection does not matter. The total number of ways to choose an un-ordered sample of size \(r\) from a population of size \(n\) is given by the combination formula: \(\left(\frac{n}{r}\right)=^{n}C_{r}=\frac{n!}{r!\left(n-r\right)!}\) Because every single unique sample combination has an equal chance of being drawn, the probability of selecting any one specific combination is equal to 1 divided by the total number of possible combinations (\(\frac{1}{{nC}_{r}}\)), matching Option A.

❓ Why Other Options Are Incorrect

Option B → Incorrect because the permutation formula \({nP}_{r}\) counts different selection orders as unique samples, which does not apply to un-ordered sample combinations.
Option C → Incorrect because \(n^{r}\) calculates the total outcomes for sampling with replacement where order matters, which does not apply to simple random sampling configurations.
Option D → Incorrect because \(\frac{r}{n}\) represents the probability of selecting a single individual item on the first draw, rather than the probability of drawing a complete multi-item combination.

🎯 Strategy

Used: Substitution

Application: Apply the standard probability rule for equally likely outcomes: \(Probability=\frac{1}{Total Combinations}\).

Final Logic: Since combinations count un-ordered groups, the total number of outcomes is \({nC}_{r}\), pointing directly to Option A.

🧠 Memory Trick

Combinations look at the group as a whole without ordering, so we use C (\({nC}_{r}\)) in the denominator.

📚 Class XII – Mathematics Part II – Chapter: Probability – Topic: Classical Approach to Probability & Combinations

7 An inspector selects chips at positions:
Determine the fixed interval and the next selected chip.

A Interval = 5, Next = 25

B Interval = 4, Next = 24

C Interval = 6, Next = 26

D Interval = 7, Next = 27

💡 Short Explanation

Systematic sampling relies on a fixed, constant gap between each selected item in a sequence. Subtracting consecutive positions (\(8-2=6\), \(14-8=6\)) reveals a constant interval of \(k=6\). Adding this interval of 6 to the last observed position (\(20+6\)) identifies the next selected item.

✅ Correct Answer Explanation

Systematic sampling involves selecting items at regular, predetermined steps through an ordered group. We can find the sampling interval (\(k\)) by calculating the difference between consecutive selected positions: \(8-2=6\) \(14-8=6\) \(20-14=6\) This confirms the fixed interval is \(k=6\). To find the next position in the sequence, we add the interval of 6 to the last selected position: \(Next position=20+6=26\) This matches the parameters given in Option C.

❓ Why Other Options Are Incorrect

Option A → Incorrect because an interval of 5 would generate a sequence of \(2,7,12,17,22\), which does not match the observed data points.
Option B → Incorrect because an interval of 4 would produce a sequence of \(2,6,10,14,18\), which deviates from the given positions.
Option D → Incorrect because an interval of 7 would create a sequence of \(2,9,16,23,30\), which does not match the positions in the problem.

🎯 Strategy

Used: Dimensional/Unit Analysis

Application: Calculate the structural difference between the sequential data points to find the step pattern.

Final Logic: The constant difference is 6, and adding 6 to 20 equals 26, which matches only Option C.

🧠 Memory Trick

Subtract any two consecutive numbers to find the step size, then add that step size to the last number.

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Systematic Sampling Calculations

8 In systematic sampling, the population can be arranged:

A Only alphabetically

B Alphabetically, numerically, or in any ordered sequence

C Only in descending numerical order

D Only randomly without any order

💡 Short Explanation

Systematic sampling requires a structured list or framework to select items at regular intervals. This framework can be organized using any logical, consistent sequence, such as alphabetical names or numerical IDs. Restricting the method to a single type of ordering is unnecessary, as any sequential list works for interval selection.

✅ Correct Answer Explanation

The key requirement for systematic sampling is that the target population must be organized into a sequential list or framework from which items can be counted and selected at a fixed interval (\(k\)). This list can be organized using any clear, consistent method—including alphabetical rosters, numerical IDs, chronological logs, or geographical arrays—making Option B the correct answer.

❓ Why Other Options Are Incorrect

Option A → Incorrect because it unnecessarily restricts the method to alphabetical arrangements, ignoring numerical customer registries or files.
Option C → Incorrect because it limits the organization to descending numerical order, ignoring ascending lists or text-based files.
Option D → Incorrect because it states the population can only be arranged randomly, overlooking the structured lists used in systematic sampling.

🎯 Strategy

Used: Extreme Word Filter

Application: Identify and filter out options containing restrictive words like "only" that limit valid statistical methods.

Final Logic: Options A, C, and D contain overly restrictive language, leaving Option B as the correct, flexible description.

🧠 Memory Trick

Any structured list works for systematic intervals—whether it's sorted by letters, numbers, or dates.

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Systematic Sampling Principles

9 Match List I with List II:

List I	List II
1. Every fifth student entering school is surveyed	a. Simple random sampling
2. Randomly selecting 10 students from 50 with equal probability	b. Systematic sampling
3. Using telephone directories excluding unlisted numbers	c. Convenience bias
4. Selecting people standing at a mall entrance	d. Unrepresentative sample

A 1–b, 2–a, 3–d, 4–c

B 1–a, 2–b, 3–c, 4–d

C 1–b, 2–d, 3–a, 4–c

D 1–c, 2–b, 3–d, 4–a

💡 Short Explanation

Surveying every fifth person follows a fixed interval, which represents systematic sampling (1 \(\rightarrow\) b). Selecting 10 out of 50 students with equal probability represents simple random sampling (2 \(\rightarrow\) a). Using a phone directory that leaves out unlisted numbers creates an unrepresentative sample framework (3 \(\rightarrow\) d). Surveying people at a mall entrance based on accessibility introduces convenience bias (4 \(\rightarrow\) c).

✅ Correct Answer Explanation

Let's pair each item from List I with its description in List II: Every fifth student entering school is surveyed uses a fixed counting step, which describes systematic sampling (1-b). Randomly selecting 10 students from 50 with equal probability gives every group an equal chance, describing simple random sampling (2-a). Using telephone directories excluding unlisted numbers leaves out a specific part of the population, creating an unrepresentative sample (3-d). Selecting people standing at a mall entrance relies on easy physical accessibility, introducing convenience bias (4-c). This complete set of pairs matches Option A.

❓ Why Other Options Are Incorrect

Option B → Incorrect because it pairs a fixed-interval count with simple random sampling (1-a), swapping the core sampling definitions.
Option C → Incorrect because it matches the equal-chance student selection with an unrepresentative sample label (2-d).
Option D → Incorrect because it connects the fixed-interval choice with convenience bias labels from the start (1-c).

🎯 Strategy

Used: Direct Concept Matching

Application: Pair the clear procedural definitions first, such as linking fixed intervals with systematic sampling (\(1\rightarrow b\)).

Final Logic: Finding the option that pairs 1-b and 2-a allows us to isolate Option A as the correct choice.

🧠 Memory Trick

Fixed intervals mean systematic sampling (1-b); equal probability means simple random sampling (2-a).

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: Sampling Method Comparisons

10 Assertion (A): Selecting one student randomly via lottery every week is an unbiased method.
Reason (R): It follows simple random sampling where each individual has a chance of selection.

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

A properly run lottery system is an objective way to select individuals entirely by chance. Because selections are made randomly, no personal preferences or human biases can influence the outcome. Giving every individual an equal chance of selection explains why the method is unbiased.

✅ Correct Answer Explanation

Assertion (A) is true because a lottery system relies on pure chance, making it an unbiased way to choose a student. Reason (R) is true and provides the correct explanation: a lottery is a form of simple random sampling where every individual has an equal chance of selection. Because the random selection process explained in the Reason is exactly what makes the method in the Assertion unbiased, Option C is the correct answer.

❓ Why Other Options Are Incorrect

Option A → Incorrect because both statements are factually accurate descriptions of how simple random sampling works.
Option B → Incorrect because it labels the reason as false, ignoring the fact that lotteries give everyone an equal selection chance.
Option D → Incorrect because it claims the assertion is false, failing to recognize that a random lottery is an unbiased selection method.

🎯 Strategy

Used: Assertion-Reason Analysis

Application: Test each statement on its own first, then verify if the reason explains why the assertion is true.

Final Logic: Since having an equal selection chance (R) explains why the lottery method is unbiased (A), R is the correct explanation of A (Option C).

🧠 Memory Trick

Lotteries give everyone an equal, random chance (R), which naturally prevents selection bias (A).

📚 Class XII – Economics: Statistics for Economics – Chapter: Collection of Data – Topic: The Lottery Selection Method

11 For a sample to fulfill the conditions of a representative sample, which criteria must it satisfy?
1. It must completely lack random selection
2. The sampling process must include a component of random selection
3. The sample size must be extremely small to artificially limit variability
4. The sample size must be sufficiently large to reflect variability

A 1 and 3 only

B 2 and 4 only

C 1, 2, and 4 only

D 2 and 3 only

💡 Short Explanation

Representation requires randomness + adequate size Small samples increase error, not representativeness Non-random methods are biased

✅ Correct Answer Explanation

A representative sample must reflect the population structure without systematic bias. This is achieved when: A random selection component (Statement 2) is included so every unit has a known chance of selection. A sufficiently large sample size (Statement 4) ensures variability is captured and sampling error is reduced. Statement 1 is incorrect because absence of random selection leads to bias. Statement 3 is incorrect because very small samples increase variability and reduce reliability. Thus, only 2 and 4 satisfy representativeness.

❓ Why Other Options Are Incorrect

Option A → Includes 1 and 3, both contradict representativeness
Option C → Includes 1 (wrong) along with correct 2 and 4
Option D → Includes 3 (incorrect small sample idea)

🎯 Strategy

Used: Elimination

Application: Remove statements violating randomness and sample size principles

Final Logic: Only randomness + adequate size define representativeness

🧠 Memory Trick

"R + L = Real sample" (Random + Large)

📚 Class XII – NCERT Statistics – Chapter: Sampling and Inference – Topic: Representative Sample Conditions –

12 According to the Central Limit Theorem, if the sample size is sufficiently large (n≥30), the sampling distribution approximates which shape, regardless of the population distribution?

A Uniform distribution

B Exponential curve

C Parabolic curve

D Normal distribution (bell-shaped curve)

💡 Short Explanation

CLT ensures normality for large samples Works even if population is non-normal n ≥ 30 is the standard rule of thumb

✅ Correct Answer Explanation

The Central Limit Theorem states that as sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of population shape. This is a foundational NCERT result in inferential statistics.

❓ Why Other Options Are Incorrect

Option A → Uniform shape is unrelated to CLT
Option B → Exponential is a different distribution family
Option C → Parabolic curve is not a statistical sampling result

🎯 Strategy

Used: Extreme Word Filter

Application: Identify known CLT outcome among unrelated shapes

Final Logic: Only normal distribution is CLT-consistent

🧠 Memory Trick

"CLT → Central = Curve turns Normal"

📚 Class XII – NCERT Statistics – Chapter: Sampling Distribution – Topic: Central Limit Theorem –

13 In a sample of 1,000 Indian cricket fans, 75% are male. Assuming this sample is representative, estimate the number of male fans in a population of 2,000,000.

A 1,000,000

B 1,250,000

C 1,500,000

D 1,750,000

💡 Short Explanation

Proportion of males = 75% Apply proportion to population 0.75 × 2,000,000 = 1,500,000

✅ Correct Answer Explanation

If 75% of a representative sample is male, the same proportion applies to the population. So, male population = 75% of 2,000,000 = 1,500,000.

❓ Why Other Options Are Incorrect

Option A → 50% assumption incorrect
Option B → 62.5% mismatch
Option D → 87.5% incorrect inflation

🎯 Strategy

Used: Substitution

Application: Direct proportional scaling from sample to population

Final Logic: Same percentage applied to total population

🧠 Memory Trick

"Same % → Same world"

📚 Class XII – NCERT Statistics – Chapter: Sampling – Topic: Estimation Using Proportions –

14 A researcher surveys the first 50 people entering a shopping mall on a weekday morning regarding employment status.
This sampling method primarily suffers from:

A Convenience bias

B Systematic sampling

C Simple random sampling

D Stratified sampling

💡 Short Explanation

Sample taken from easily available group Not random Time/location-based distortion exists

✅ Correct Answer Explanation

Selecting the first 50 people entering a mall is based on ease of access, not randomness. This leads to convenience bias, a type of non-probability sampling where selection is driven by availability.

❓ Why Other Options Are Incorrect

Option B → No fixed interval selection rule
Option C → No randomization involved
Option D → No grouping into strata

🎯 Strategy

Used: Contextual/Tonal Matching

Application: Match description with known sampling types

Final Logic: "First available people" = convenience sampling

🧠 Memory Trick

"First come = First bias"

📚 Class XII – NCERT Statistics – Chapter: Sampling Methods – Topic: Convenience Sampling –

15 Identify the incorrect statement regarding sampling biases:

A Voluntary response bias occurs when individuals choose whether to participate

B Undercoverage bias underrepresents certain groups

C Convenience bias involves selecting easily available individuals

D Response bias guarantees that the sample statistic equals the population parameter

💡 Short Explanation

Bias never guarantees correctness It introduces distortion Other options correctly define biases

✅ Correct Answer Explanation

Biases such as response bias, voluntary response bias, and convenience bias all distort results. They do not guarantee equality between sample statistics and population parameters. Hence option D is fundamentally incorrect.

❓ Why Other Options Are Incorrect

Option A → Correct definition
Option B → Correct description of undercoverage
Option C → Correct convenience bias definition

🎯 Strategy

Used: Extreme Word Filter

Application: Identify absolute/false guarantee statement

Final Logic: Bias cannot guarantee correctness

🧠 Memory Trick

"Bias = distortion, never perfection"

📚 Class XII – NCERT Statistics – Chapter: Sampling Bias – Topic: Types of Bias –

16 Arrange the correct sequence of steps for conducting a sampling study:
1. Determine sample size
2. Choose sampling method
3. Identify and define target population
4. Collect data
5. Select sampling frame

A 3, 5, 2, 4, 1

B 3, 5, 2, 1, 4

C 1, 2, 3, 4, 5

D 5, 3, 2, 1, 4

💡 Short Explanation

Start with population definition Then frame selection Then method and size Finally data collection

✅ Correct Answer Explanation

Proper sampling procedure: 1. Define target population (3) 2. Select sampling frame (5) 3. Choose method (2) 4. Determine sample size (1) 5. Collect data (4)

❓ Why Other Options Are Incorrect

Option A → Incorrect final step order
Option C → Random ordering not logical
Option D → Frame and population misplaced

🎯 Strategy

Used: Logical sequencing

Application: Arrange steps of research methodology

Final Logic: Population → Frame → Method → Size → Data

🧠 Memory Trick

"PFM-SD rule"

📚 Class XII – NCERT Statistics – Chapter: Sampling Design – Topic: Steps in Sampling –

17 Sampling error is defined as:
x ˉ - μ
If x ˉ = 55 and the sampling error is -5, find μ.

A 50

B 60

C 45

D -60

💡 Short Explanation

Error = sample mean − population mean Substitute values Solve algebraically

✅ Correct Answer Explanation

Given: 55 − μ = -5 ⇒ μ = 60

❓ Why Other Options Are Incorrect

Option A → Wrong rearrangement
Option C → Miscalculation
Option D → Not meaningful in context

🎯 Strategy

Used: Substitution

Application: Direct equation solving

Final Logic: Rearranging error formula gives μ = 60

🧠 Memory Trick

"Mean = Sample − Error"

📚 Class XII – NCERT Statistics – Chapter: Sampling Error – Topic: Error Formula – 30

18 Assertion (A): Even perfectly random samples contain sampling errors.
Reason (R): Random samples may differ numerically from the population due to inherent variability.

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

Random sampling reduces bias, not error Variability always exists Reason explains assertion

✅ Correct Answer Explanation

Even in perfect random sampling, sampling error exists due to natural variability in data. Different samples produce different means, explaining why errors cannot be eliminated completely.

❓ Why Other Options Are Incorrect

Option A → Both statements are true
Option B → Reason is correct
Option D → Assertion is correct

🎯 Strategy

Used: Causal reasoning

Application: Link variability with error existence

Final Logic: Variability explains unavoidable sampling error

🧠 Memory Trick

"Random ≠ Perfect match"

📚 Class XII – NCERT Statistics – Chapter: Sampling Theory – Topic: Sampling Error Concept – 32

19 Given:
μ = 100, x̄₁ = 98, x̄₂ = 105, x̄₃ = 100
Sampling error vector:
E⃗ = (e₁, e₂, e₃), eᵢ = x̄ᵢ − μ
Find E⃗.

A (-2, 5, 0)

B (2, -5, 0)

C (98, 105, 100)

D (198, 205, 200)

💡 Short Explanation

Subtract population mean from each sample mean Compute individually Form vector

✅ Correct Answer Explanation

e₁ = 98 − 100 = -2 e₂ = 105 − 100 = 5 e₃ = 100 − 100 = 0 So, E⃗ = (-2, 5, 0)

❓ Why Other Options Are Incorrect

Option B → Sign error
Option C → Uses raw means instead of differences
Option D → Incorrect scaling

🎯 Strategy

Used: Substitution

Application: Direct computation of vector components

Final Logic: Each component = deviation from μ

🧠 Memory Trick

"Error = minus mean"

📚 Class XII – NCERT Statistics – Chapter: Sampling Error – Topic: Error Vector – 34

20 Given the Standard Error of the Mean:
SEM(n) = σ / √n
As n → ∞, the value of SEM approaches:

A ∞

B σ

C 1

D 0

💡 Short Explanation

Denominator increases infinitely Fraction decreases Limit tends to zero

✅ Correct Answer Explanation

As sample size increases, √n becomes very large, making σ/√n approach 0. Hence, standard error decreases with increasing sample size.

❓ Why Other Options Are Incorrect

Option A → Opposite behavior
Option B → Incorrect constant assumption
Option C → No mathematical basis

🎯 Strategy

Used: Limit behavior

Application: Analyze function as n grows

Final Logic: Increasing denominator → zero limit

🧠 Memory Trick

"Big n → small error"

📚 Class XII – NCERT Statistics – Chapter: Sampling Distribution – Topic: Standard Error –

CUET UG Applied Mathematics Booster Test 1 - Sampling Methods & Errors

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