Which of the following must be true for an estimator of a population parameter to be unbiased?

• A. The estimator is calculated only from the population data, not from a sample.
• B. The estimator always produces the same value as the parameter in every sample.
• C. The expected value of the estimator equals the true value of the parameter, that is, $E\left(\hat{\theta }\right)=\theta$.
• D. The estimator has the smallest possible variance among all estimators.

Answer: (C)

Verified step by step guidance

1. Understand that an estimator is a rule or formula that tells us how to calculate an estimate of a population parameter based on sample data.
2. Recall the definition of an unbiased estimator: an estimator \( \hat{\theta} \) is unbiased if its expected value equals the true parameter \( \theta \). Mathematically, this is expressed as \( \mathbb{E}(\hat{\theta}) = \theta \).
3. Recognize that the first option is incorrect because estimators are typically calculated from sample data, not the entire population, since population data is usually unknown.
4. Note that the second option is too strict; an estimator does not have to produce the exact parameter value in every sample, but on average (over many samples), it should equal the parameter.
5. Understand that the smallest variance condition relates to efficiency, not unbiasedness, so it is not a requirement for an estimator to be unbiased.