Question:
The following two statements are about the use of stratified sampling for finite population:Assertion (I): Sampling error of an estimator can always be reduced by using stratified sampling following principle of stratification.Reason (II): Stratified sampling under optimal allocation for a fixed sample size reduces the mean square error of the estimator.Which of the following is the correct explanation?
The following two statements are about the use of stratified sampling for finite population:
Assertion (I): Sampling error of an estimator can always be reduced by using stratified sampling following principle of stratification.
Reason (II): Stratified sampling under optimal allocation for a fixed sample size reduces the mean square error of the estimator.
Which of the following is the correct explanation?
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Both statements are individually correct results of sampling theory, but check whether the specific "optimal allocation" claim in (II) is really what makes the general claim in (I) true.
Updated On: Jul 4, 2026
- Both (I) and (II) are true but (II) is not the correct explanation for (I)
- Both (I) and (II) are true but (II) is the correct explanation for (I)
- Only (I) is true but (II) is false
- Both (I) and (II) are false
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The Correct Option is A
Solution and Explanation
Step 1: Examine Assertion (I). A classical result of sampling theory (Cochran) states that if strata are formed sensibly, following the principle of stratification (making strata internally as homogeneous as possible with respect to the study variable), the variance of the stratified estimator with proportional allocation is never larger than the variance under simple random sampling of the same total size, and is usually smaller. So Assertion (I) is true.
Step 2: Examine Reason (II). Under Neyman (optimal) allocation, for a fixed total sample size \(n\), the sample sizes \(n_h\) in each stratum are chosen proportional to \(N_h \sigma_h\), and this allocation is proven to give the minimum possible variance among all stratified allocations, which in turn is at most the variance under proportional allocation. So Reason (II) is also a true and well-established statement.
Step 3: Check if (II) explains (I). Assertion (I) is a general statement about following the principle of stratification, without specifying the allocation method; it holds even under simple proportional allocation. Reason (II) is a narrower, specific result that applies only to optimal (Neyman) allocation. Since (I)'s general claim is not conditioned on using optimal allocation, (II) does not serve as the direct cause or explanation of (I); it is a related but separate, stronger result.
Final Answer: Both (I) and (II) are true, but (II) is not the correct explanation for (I).
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