Quiz 3 - Solutions

Quiz 3 - Solutions - STATISTICS 321 Spring 2009 Quiz 3 (20...

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STATISTICS 321 – Spring 2009 Name: Quiz 3 (20 points) – This quiz has 2 sides. 1. Let X 1 ,X 2 ,...,X n be a random sample from a distribution with unknown mean μ and variance σ 2 = 2 . (a) We propose 2 point estimators ˆ μ 1 and ˆ μ 2 to estimate μ , where ˆ μ 1 = X 1 + X 2 + ... + X n n and ˆ μ 2 = X 1 + X 2 2 Which of the two: ˆ μ 1 or ˆ μ 2 will have the smaller Mean Squared Error (MSE), and why? (Recall that MSE = Bias 2 + Variance. Also, you don’t need to do any calculations. ) Bias(ˆ μ 1 ) = E μ 1 ) - μ = ( μ + ... + μ ) /n - μ = 0 Bias(ˆ μ 2 ) = E μ 2 ) - μ = ( μ + μ ) / 2 - μ = 0 Var(ˆ μ 1 ) = Var( X 1 + X 2 + ... + X n ) /n 2 = 2 n 2 = σ 2 n = 2 /n Var(ˆ μ 2 ) = Var( X 1 + X 2 ) / 2 2 = 2 σ 2 2 2 = σ 2 2 = 2 / 2 = 1 Thus, ˆ μ 1 has a smaller Mean Squared Error (MSE), provided n > 2 (b) When will the Mean Squared Error of a point estimator be negative? Why? Mean Squared Error (MSE) is never negative, since it is a sum of 2 non–negative terms.
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Quiz 3 - Solutions - STATISTICS 321 Spring 2009 Quiz 3 (20...

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