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ch14estimationofmean,varianceandproportion

# 2 insteadof 2 d 2 xi x 2 3 n

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Unformatted text preview: 7 This is a downward biased estimate for μ. This ˆ In the long run, µ underestimates μ, thus it is NOT fair. ˆ C. Unbiased Estimate for σ² x1 , x2 ,......., xn Let a sample be taken from a population. To estimate σ², we use ∑ ( xi − x ) 2 2 2 ˆ σ =s = n −1 .....(2) Instead of σ 2 = d 2 = ∑ ( xi − x ) 2 .....(3) ˆ ˆ n This is because d² gives a downward biased estimate forσ² while s² gives an unbiased estimate for σ². The rigorous proof of this fact is NOT our main concern here, but it would be useful to look at some intuitive reasons. RULE: ˆ ˆ E (θ ) = θ θ If then is an UNBIASED estimator of θ ˆ Example: E ( X ) = µ ⇒ µ = X X The sample mean ( ) is an UNBIASED estimator of µ the population mean ( ) Proof: n X 1 + X 2 + ....... + X n n n X 1 + X 2 + .......
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