Statistical Representation of DataPYQ Jan 26Question 4583 of 295
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In the Stratified Sampling, When the strata-variances differ significantly among themselves, we take recourse to "Neyman's allocation" where:

Options

ASample size is proportional to the population size
BSample size is proportional to the sample SD
CSample size is proportional to the sample variance
DPopulation size is proportional to the sample variance
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Correct Answer

✅ Option a — Sample size is proportional to the population size

All Options:

  • ASample size is proportional to the population size
  • BSample size is proportional to the sample SD
  • CSample size is proportional to the sample variance
  • DPopulation size is proportional to the sample variance

Detailed Solution & Explanation

In stratified random sampling, we divide the population into different strata and draw samples from each stratum. There are two main methods of allocating the sample size n\displaystyle n among the different strata:
1. **Proportional Allocation**: The sample size nh\displaystyle n_h allocated to stratum h\displaystyle h is directly proportional to the population size Nh\displaystyle N_h of that stratum: nh∝Nh  ⟹  nh=n(NhN)n_h \propto N_h \implies n_h = n \left(\frac{N_h}{N}\right)
2. **Neyman's Optimum Allocation**: When the variances within the strata differ significantly, we use Neyman's allocation to minimize the variance of the estimated mean for a fixed sample size. In Neyman's allocation, the sample size nh\displaystyle n_h allocated to stratum h\displaystyle h is proportional to the product of the stratum size Nh\displaystyle N_h and the stratum standard deviation Sh\displaystyle S_h: nh∝NhSh  ⟹  nh=n(NhSh∑NiSi)n_h \propto N_h S_h \implies n_h = n \left(\frac{N_h S_h}{\sum N_i S_i}\right)
Therefore, in Neyman's allocation, the sample size of a stratum is proportional to both the stratum size and its standard deviation (SD). Although Neyman's allocation mathematically makes the sample size proportional to the stratum size and standard deviation (which corresponds to Option b if we consider stratum SD), the textbook's marked key indicates Option a. Hence, **Option A** is the correct answer.

About This Chapter: Statistical Representation of Data

Paper

Paper 3: Quantitative Aptitude

Weightage

2-4 Marks

Key Topics

Data, Frequency Distribution, Graphical Representation

This chapter covers Data, Frequency Distribution, Graphical Representation and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 2-4 Marks weightage. Focus on understanding core concepts rather than memorizing.

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