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ch14estimationofmean,varianceandproportion

# 45 xi x 2 n 1 712 6145 596 6145

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Unformatted text preview: al to it’s degrees of freedom. E ( χ k2 ) = k Conclusion: E(s²)=σ² Therefore the sample variance, s², is an UNBIASED estimator of the population variance, σ². ( xi − x ) 2 ∑ i =1 n n n 2 σ 2 χ n −1 ~ n 2 σ 2 χ n −1 } = E( ) n E{ i =1 ( xi − x ) 2 ∑ n σ2 2 = E ( χ n −1 ) n σ2 = ⋅ (n − 1) n ≠σ2 Conclusion: E(s²) is NOT EQUAL to σ² Therefore this estimator is NOT an UNBIASED estimator of the population variance, σ². IT is BIASED! Example 2: For the following data 71.2, 60.0, 55.3, 65.4, 32.7, 78.6, 68.8, 59.6 estimate σ². estimate Solution: x = 61.45 ( xi − x ) 2 ∑ n −1 = (71.2 − 61.45) + ....... + (59.6 − 61.45) 8 −1 ˆ σ 2 = s2 = = 189.8171 D. Estimation of σ ˆ σ 2 = s2 While is an unbiased estimate for σ², we usually use to estimate σ. ˆ σ...
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