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

Info icon This preview shows pages 131–135. Sign up to view the full content.

View Full Document Right Arrow Icon
Is r=0? Yes Yes No No i=1 n i n X
Image of page 131

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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,
Image of page 132
[ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) [ ] ( ) [ ] 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
Image of page 133

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon