100BHWS3

100BHWS3 - STAT 100B HW III solution Problem 1: Suppose X1...

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STAT 100B HW III solution Problem 1: Suppose X 1 ,...,X n f ( x ) independently. Let μ = E[ X ], and σ 2 = Var[ X ]. Let ¯ X = 1 n n X i =1 X i , and s 2 = 1 n - 1 n X i =1 ( X i - ¯ X ) 2 . Prove E[ ¯ X ] = μ , and E( s 2 ) = σ 2 . A: E[ ¯ X ] = E[( n i =1 X i ) /n ] = ( n i =1 E[ X i ]) /n = nμ/n = μ . E[ s 2 ] = E[ 1 n - 1 n X i =1 ( X i - ¯ X ) 2 ] (1) = 1 n - 1 E[ n X i =1 ( X i - μ ) 2 - n ( ¯ X - μ ) 2 ] (2) = 1 n - 1 [ n X i =1 E( X i - μ ) 2 - n E( ¯ X - μ ) 2 ] (3) = 1 n - 1 ( 2 - n ( σ 2 /n )) = σ 2 . (4) Problem 2: Suppose we observe X 1 ,...,X n N ( μ 1 2 ), and Y 1 ,...,Y m N ( μ 2 2 ), and X 1 ,...,X n , Y 1 ,...,Y m are all independent of each other. Suppose we want to test whether μ 1 = μ 2 . (1) If we known σ 2 , then what is an appropriate test statistic, and what is the distribution of this test statistic under the null hypothesis? A: The test statistic is
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This note was uploaded on 03/30/2011 for the course STAT 100B taught by Professor Wu during the Winter '11 term at UCLA.

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100BHWS3 - STAT 100B HW III solution Problem 1: Suppose X1...

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