Exercise · ZW-ESTIMATOR-008 · ★★☆ Intermediate
With \(n=25\), which unbiased estimator of \(\mu\) is preferable: \(\bar X\) or just \(X_1\)?
Choose one option.
A \(\bar X\), because its variance \(\sigma^2/25\) is smaller than \(\sigma^2\) B \(X_1\), because it is simpler to compute C Neither, because both are unbiased D \(X_1\), because its variance \(\sigma^2\) is smaller Show answer A. \(\bar X\), because its variance \(\sigma^2/25\) is smaller than \(\sigma^2\)← Back to the question
Explanation Among unbiased estimators the one with smaller variance is preferred. \(\operatorname{Var}(\bar X)=\sigma^2/25<\sigma^2=\operatorname{Var}(X_1)\).
Review the concept: Estimator An estimator \(\hat\theta\) is a rule \(T\), a function of the random sample \(X_i\), that produces a value used to approximate an unknown population parameter \(\theta\).
\[\hat\theta=T(X_1,\dots,X_n)\]
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