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Samacheer Kalvi 12th Statistics Chapter 2 Exercise Solutions

Tamil Nadu Class 12 Statistics Chapter 2 Sampling Techniques exercise solutions — random sampling, stratified, systematic, cluster sampling, sampling errors MCQs and answers.

Updated 18 June 20265 min read

Chapter 2 — Sampling Techniques

Key Concepts Quick Recap

Sampling MethodProcessBest Used When
Simple Random Sampling (SRS)Each member has equal probability; lottery/random number tableHomogeneous population
Stratified Random SamplingDivide into strata; random sample from eachHeterogeneous population with known subgroups
Systematic SamplingEvery kth element (k = N/n)Population arranged in a list/order
Cluster SamplingDivide into clusters; select some clusters entirelyPopulation in natural groups (villages, schools)
Purposive (Judgment) SamplingResearcher selects based on judgmentExpert opinion, qualitative research
Quota SamplingFixed quotas per subgroupMarket research, opinion polls

I. Choose the Best Answer (MCQs)

1. In systematic sampling, if population size N = 1000 and sample size n = 50, the sampling interval k is: (a) 50 (b) 20 (c) 100 (d) 10

Answer(b) 20 k = N/n = 1000/50 = 20. Select a random start between 1 and 20, then select every 20th element.

2. Stratified random sampling is preferred when: (a) Population is homogeneous (b) Population is heterogeneous with known subgroups (c) Population is very small (d) Census is possible

Answer(b) Population is heterogeneous with known subgroups Stratification ensures representation of all subgroups. More precise than SRS for heterogeneous populations. Strata: age groups, gender, income levels, regions.

3. In lottery method of sampling: (a) Larger units have higher probability (b) Each unit has equal probability of selection (c) Only a fixed interval is selected (d) Clusters are selected

Answer(b) Each unit has equal probability of selection Lottery method (SRS without replacement): write each unit on a slip, mix, draw required number blindly. Each unit has probability n/N of selection. This is the simplest form of SRS.

4. Sampling error can be reduced by: (a) Using purposive sampling (b) Increasing sample size (c) Reducing sample size (d) Using quota sampling

Answer(b) Increasing sample size Larger samples → statistic closer to parameter → smaller sampling error. Standard error (SE) = σ/√n → as n increases, SE decreases.

5. Census (complete enumeration) differs from sampling in that: (a) Census is faster (b) Census covers all units of the population (c) Census has more sampling error (d) Census uses random numbers

Answer(b) Census covers all units of the population Census: every member of the population is studied. Sampling: only a subset is studied. Census is accurate but expensive, time-consuming. Sampling is faster and cheaper but has sampling error.

6. Cluster sampling is efficient when: (a) Each cluster is heterogeneous internally (b) Population is small (c) All clusters are identical (d) Precise estimates are not needed

Answer(a) Each cluster is heterogeneous internally Cluster sampling works best when clusters are heterogeneous within but similar between clusters. Example: select 10 schools (clusters) and test all students in those schools.

7. Non-sampling errors arise due to: (a) Sample size being too small (b) Measurement errors, recording errors, non-response (c) Random variation in selection (d) Using stratified sampling

Answer(b) Measurement errors, recording errors, non-response Non-sampling errors occur even in a census. Types: response bias (incorrect answers), measurement error (wrong instrument), processing error (data entry mistakes), non-response (selected units don't participate).

II. Short Answer Questions

Q1. Compare simple random sampling and stratified random sampling.
Answer
FeatureSimple Random SamplingStratified Random Sampling
ProcessEach unit equally likely; random selectionDivide into strata; SRS within each stratum
PopulationHomogeneousHeterogeneous
PrecisionLower (for heterogeneous pop.)Higher (ensures all groups represented)
ComplexitySimpleRequires knowledge of strata
ExampleDrawing names from a boxSampling students by class (10th, 11th, 12th)
Q2. Define sampling error and give its formula.
Answer

Sampling error = difference between sample statistic and population parameter.

For sample mean: SE(x̄) = σ/√n

For sample proportion: SE(p) = √[P(1−P)/n]

Where n = sample size, σ = population SD, P = population proportion.

Sampling error DECREASES as sample size increases.

Common Mistakes

  • Confusing systematic sampling (every kth element) with stratified sampling (groups then random).
  • Thinking census has no error — census has no SAMPLING error but can have non-sampling errors.
  • Forgetting k = N/n (not n/N) in systematic sampling.

Exam Checklist

  • SRS: equal probability, lottery method, random number table.
  • Stratified: heterogeneous population, proportionate vs disproportionate allocation.
  • Systematic: k = N/n; select random start, then every kth.
  • Cluster: natural groups selected entirely; efficient when clusters are heterogeneous.
  • Sampling error = σ/√n (reduces as n increases).
  • Non-sampling error: measurement, recording, non-response (not reduced by larger sample).

Frequently Asked Questions

1. Simple Random Sampling (SRS): each element has equal probability (lottery method, random number tables). 2. Stratified Random Sampling: population divided into strata (groups); random sample from each stratum. 3. Systematic Sampling: every kth element selected (k = N/n). 4. Cluster Sampling: population divided into clusters; all elements from selected clusters included.
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