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solutions3 - Stat 651 (Winter 2011) Kaizar Homework 3...

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Unformatted text preview: Stat 651 (Winter 2011) Kaizar Homework 3 Solutions 40 points total Exercises 1. Lohr, Chapter 2 problem 1 (8 points) (a) y U = 1 6 (98 + 102 + 154 + 133 + 190 + 175) = 142 (b) Plan 1: i. E[ y ] = 8 summationdisplay i =1 y i P( S = S i ) = 147 . 3 1 8 + 142 . 3 1 8 + 140 . 3 1 8 + 135 . 3 1 8 + 148 . 7 1 8 + 143 . 7 1 8 + 141 . 7 1 8 + 136 . 7 1 8 = 142 ii. V [ y ] = 8 summationdisplay i =1 ( y i- E[ y ]) 2 P( S = S i ) = 28 . 4 1 8 + 0 . 1 1 8 + 2 . 8 1 8 + 44 . 4 1 8 + 44 . 4 1 8 + 2 . 8 1 8 + 0 . 1 1 8 + 28 . 4 1 8 = 18 . 9 iii. Bias [ y ] = y- y U = 142- 142 = 0 iv. MSE[ y ] = variance + bias 2 = 18 . 9 + 0 = 18 . 9 Plan 2: i. E[ y ] = 3 summationdisplay i =1 y i P( S = S i ) = 135 . 3 1 4 + 143 . 7 1 2 + 147 . 3 1 4 = 142 . 5 1 ii. V[ y ] = 3 summationdisplay i =1 ( y i- E[ y ]) 2 P( S = S i ) = 51 . 4 1 4 + 1 . 4 1 2 + 23 . 4 1 4 = 19 . 4 iii. Bias [ y ] = y- y U = 142 . 5- 142 = 0 . 5 iv. MSE[ y ] = variance + bias 2 = 19 . 4 + 0 . 5 2 = 19 . 65 (c) I like sampling plan 1 better because it has smaller bias and smaller variance, and therefore smaller MSE. Sampling plan 1 will be closer to the true value more often. 2. Lohr, Chapter 2 problem 2 (8 points) (a) Find i : Let Z i represent whether or not unit i is included. unit 1: 1 = P(unit 1 included) = P( Z 1 = 1) = P( S = S 1 or S = S 3 ) = P( S = S 1 ) + P( S = S 3 ) because disjoint = 1 8 + 1 8 = 1 4 unit 2: 2 = P( S = S 2 ) + P( S = S 4 ) = 1 4 + 3 8 = 5 8 unit 3: 2 = P( S = S 1 ) + P( S = S 2 ) = 1 8 + 1 4 = 3 8 unit 4: 2 = P( S = S 3 ) + P( S = S 4 ) + P( S = S 5 ) = 1 8 + 3 8 + 1 8 = 5 8 2 unit 5: 2 = P( S = S 1 ) + P( S = S 5 ) = 1 8 + 1 8 = 1 4 unit 6: 2 = P( S = S 1 ) + P( S = S 3 ) + P( S = S 4 ) = 1 8 + 1 8 + 3 8 = 5 8 unit 7: 2 = P( S = S 2 ) + P( S = S 5 ) = 1 4 + 1 8 = 3 8 unit 8: 2 = P( S = S 2 ) + P( S = S 3 ) + P( S = S 4 ) + P( S = S 5 ) = 1 4 + 1 8 + 3 8 + 1 8 = 7 8 (b) The sampling distribution of t : Population: i 1 2 3 4 5 6 7 8 y...
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This note was uploaded on 07/26/2011 for the course STA 651 taught by Professor Kaizar during the Winter '11 term at Ohio State.

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solutions3 - Stat 651 (Winter 2011) Kaizar Homework 3...

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