ps2sol_2010 - STATS 203: HOMEWORK SOLUTION #2 THANKS TO...

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STATS 203: HOMEWORK SOLUTION #2 THANKS TO ROHAN TANDON, RONGZHI LU Question 1 We are given a one-way ANOVA model Y ij = μ + α i + ± ij , ± ij N (0 2 ) The mean sum-of-treatment-squares term is MSTR = r i =1 n i ( ¯ Y i · - ¯ Y ·· ) 2 r - 1 and we assume that each group i has n observations, i.e. n i = n . (a). From the definition of a one-way ANOVA model, we see immediately that ¯ Y i · = 1 n n X j =1 Y ij = 1 n n X j =1 ( μ + α i + ± ij ) = μ + α i + ¯ ± i · ¯ Y ·· = 1 r r X i =1 1 n n X j =1 Y ij = 1 r r X i =1 1 n n X j =1 ( μ + α i + ± ij ) = 1 r r X i =1 ( μ + α i + ¯ ± i · ) where ¯ ± i · = 1 n n j =1 ± ij . Then clearly, we see that ¯ Y i · - ¯ Y ·· = ( μ + α i + ¯ ± i · ) - 1 r r X i =1 ( μ + α i + ¯ ± i · ) = ( μ + α i + ¯ ± i · ) - μ - 1 r r X i =1 α i ! + 1 r r X i =1 1 n n X j =1 ± ij = ( μ + α i + ¯ ± i · ) - μ - 1 r · 0 + 1 nr r X i =1 n X j =1 ± ij = μ + α i + ¯ ± i · - μ + ¯ ± ·· = α i + (¯ ± i · - ¯ ± ·· ) where we have used the “identifiability” assumption of the one-way ANOVA model that r i =1 α i = 0. ± Date : Submitted on February 4, 2010. 1
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STATS 203: HOMEWORK SOLUTION #2 2 (b). Using the result from part (a), we have that ¯ Y i · - ¯ Y ·· = α i + (¯ ± i · - ¯ ± ·· ). Hence, it follows directly from the quadratic expansion that r X i =1 ( ¯ Y i · - ¯ Y ·· ) 2 = r X i =1 [ α i + (¯ ± i · - ¯ ± ·· )] 2 = r X i =1 h α 2 i + 2 α i ± i · - ¯ ± ·· ) + (¯ ± i · - ¯ ± ·· ) 2 i = r X i =1 α 2 i + r X i =1 2 α i ± i · - ¯ ± ·· ) + r X i =1 ± i · - ¯ ± ·· ) 2 = r X i =1 α 2 i + 2 r X i =1 α i ± i · - ¯ ± ·· ) + r X i =1 ± i · - ¯ ± ·· ) 2 which proves the desired result. ± (c).
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ps2sol_2010 - STATS 203: HOMEWORK SOLUTION #2 THANKS TO...

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