Final answer:
E(S²) for an estimator for the variance σ² is shown to be [(n-1)/n]σ², proving that S² is a biased estimator. The bias is corrected by using (n-1) as the denominator to make an unbiased estimator for the population variance.
Step-by-step explanation:
Understanding the Biased Estimator S² for Variance σ²
When examining the estimator S², defined as the sum of squared deviations from the sample mean divided by the number of observations (n), we are looking to demonstrate that the expected value of S², denoted as E(S²), is [(n-1)/n]σ², where σ² represents the population variance. This shows that S² is a biased estimator for σ². In mathematical terms, this can be expressed as:
E(S²) = E[∑(Xi – Μ)²/n] = (n-1)/n σ²
The bias in the estimator arises because by using n as the divisor, instead of (n-1), the average of the squared deviations tends to be underestimated. Hence, to correct for this bias and provide an unbiased estimator of the population variance, the denominator is replaced with (n-1), leading to the familiar formula for the sample variance (often denoted s²).
This theoretical concept is essential for statistical analysis and is particularly relevant in the context of hypothesis testing and constructing confidence intervals where accurate estimation of the population variance is crucial.