IT_Y4_new

Is r0 yes yes no no i1 n i n x the sample mean proved

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Is r=0? Yes Yes No No i=1 n i n X

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the sample mean. Proved that X, sample mean is an unbiased estimator of population mean θ . And also find its mean square error. (10 marks) Solution [ ] [ ] 1 1 i i n i n i X E X E n E X n n n θ θ = = = = = = 10. The quantity S 2 = 2 1 ( ) 1 i i n X X n = is called the sample variance and show that sample variance S is an unbiased estimator of population variance σ . (10 marks) [ ] [ ] [ ] [ ] 2 2 i n 1 i 2 2 i 2 n 1 i 2 i 2 n 1 i 2 i 2 X E - X E n X n X E S E 1) - (n 1) - (n ) X (X E S E 1 - n ) X (X S = = = = = = = We have,
[ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] ( ) [ ] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 i i 2 i 2 2 2 2 σ S E θ n σ n - θ σ n S E 1) - (n θ n σ X E ) X Var( X E and θ σ X E ) Var(X X E Y E Var(Y) Y E Y E - Y E Var(Y) = + + = + = + = + = + = + = = Therefore, Sample variance, S 2 is an unbiased estimator of population variance, 2 σ . 11. For any set at number x , ….., x prove algebraically that ( x – x ) = x - n x where x = x /n . (10 marks) ( ) x n - x x n x 2n - x x n x n x 2 - x x x x 2 - x ) x x x 2 (x ) x (x n 1 i 2 2 i 2 n 1 i 2 2 i 2 n 1 i 2 i n 1 i n 1 i 2 i n 1 i 2 i n 1 i 2 i 2 i n 1 i 2 i = = = = = = = = = + = + = + = + = 12. Write a method for determining when to stop generating new data to estimate a population mean. (10 marks) solution 1. Choose an acceptable value d for the standard deviation of the estimator. 2. Generate at least 100 data values. 3. Continue to generate additional data values, stopping when k data values is generated so that d k S , where S is the sample standard deviation based on those k values. 4. The estimate of θ is given by = = k 1 i i k X X . 1 n n 2 n i= 1 2 2 i= 1 i= 1 n i i i

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