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Slides_Econ140A_Sample_Mean

Slides_Econ140A_Sample_Mean - Goals Make assumptions about...

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9/28/2011 1 Sample Mean and Related Issues Copyright Dick Startz 1 Goals Make assumptions about data generating process Derive properties (expectation, variance, distribution) of sample mean. Hypothesis testing Confidence intervals Copyright Dick Startz 2 ? ? = 𝛽? ? + 𝜀 ? , E 𝜀 ? = 0, ? ? = 𝛽 + 𝜀 ? 𝛽 = ∑? ? 𝑛 Copyright Dick Startz 3 Data Generating Process ( DGP ) ? ? = 𝜇 + 𝜀 ? E(𝜀) = 0 E(? ? ) = E(𝜇 + 𝜀 ? ) = 𝜇 + E(𝜀 ? ) = 𝜇 Copyright Dick Startz 4 Sample mean 𝜇 = 1 𝑛 ? ? 𝑛 ?=1 𝑛 = 2 ? 1 = 𝜇 + 𝜀 1 ? 2 = 𝜇 + 𝜀 2 𝜇 = ? 1 + ? 2 2 Copyright Dick Startz 5 𝜇 = ? 1 + ? 2 2 𝜇 = (𝜇 + 𝜀 1 ) + (𝜇 + 𝜀 2 ) 2 = 𝜇 + 𝜀 1 + 𝜀 2 2 Thm: The sample mean is an unbiased estimator of the expected value. Def: 𝜇 is an unbiased estimator of 𝜇 if E(𝜇 ) = 𝜇 . Copyright Dick Startz 6
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9/28/2011 2 Proof: Sample mean is unbiased 𝜇 = 𝜇 + 𝜀 1 + 𝜀 2 2 E 𝜇 = E 𝜇 + 𝜀 1 + 𝜀 2 2 = 𝜇 + E 𝜀 1 + E 𝜀 2 2 = 𝜇 + 0 + 0 2 = 𝜇 q.e.d. Copyright Dick Startz 7 ?
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