Question

In case of Simple Random Sampling with Replacement, the sample mean:

• A. Under-estimates the population mean
• B. Is an unbiased estimator of the population mean
• C. Is a consistent estimator
• D. Over-estimates the population mean

Hint:

Exam Tip:
In SRSWR:
• Sample mean is an unbiased estimator of population mean.
• Sample mean is consistent.
• Variance of sample mean is \(\frac{\sigma^2}{n}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In Simple Random Sampling with Replacement (SRSWR), each unit has an equal chance of being selected, and the sample mean is used to estimate the population mean.

Step 2: Key Properties:
The sample mean \(\bar{x}\) in SRSWR is:
• Unbiased: \(E(\bar{x}) = \mu\).
• Consistent: As sample size increases, \(\bar{x}\) converges to \(\mu\) in probability. Both (B) and (C) are true.
But the question might be asking for the most appropriate or the primary property.
The most fundamental property is that the sample mean is unbiased.
If we have to choose one, option (B) is the most direct.

Step 3: Analyzing the Options:

• (A) Under-estimates: False.
• (B) Is an unbiased estimator: True.
• (C) Is a consistent estimator: True, but not the most fundamental.
• (D) Over-estimates: False. Both B and C are correct.
But since the question likely expects the primary property, option (B) is the best answer.

Step 4: Final Answer:
Therefore, option (B) is correct.