ST3239: Survey Methodology - The Department of Statistics and ...

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Two special cases will be used later when n = 1, and n = 2. Theorem 2.1.3 For any i, j = 1, ..., n and s, t = 1, ..., N, (i) P (y i = u s ) = 1/N. (ii) P (y i = u s , y j = u t ) = 1 , i ≠ j, s ≠ t. N(N − 1) Proof. P (y k = u j ) = = P (y k = u s , y j = u t ) = = ∑ P (y 1 = u i1 , · · · , y k = u ik , · · · , y n = u in ) all (i 1 , · · · , i n ), but i k = j ( ) (N − n)! N − 1 (N − n)! (N − 1)! × (n − 1)! = × N! n − 1 N! (N − n)! = 1 N . ∑ P (y 1 = u i1 , · · · , y n = u in ) all (i 1 , · · · , i n ), but i k = s,i j = t ( ) (N − n)! N − 2 (N − n)! (N − 2)! × (n − 2)! = × N! n − 2 N! (N − n)! = 1 N(N − 1) . Example 1. A population contains {a, b, c, d}. We wish to draw a s.r.s of size 2. List all possible samples and find out the prob. of drawing {b, d}. Solution. Possible samples of size 2 are {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, The probability of drawing {b, d} is 1/6. 5
2.2 Estimation of population mean and total 2.2.1 Estimation of population mean Suppose that the population of size N has values {u 1 , u 2 , · · · , u N }, we can define 1) the population mean µ = u 1 + u 2 + · · · + u N N = 1 N N∑ u i , i=1 2) the population variance σ 2 = 1 N∑ (u i − µ) 2 . N i=1 We wish to estimate the quantities µ and σ 2 and to study the accuracy of their estimators. Suppose that a simple random sample of size n is drawn, resulting in {y 1 , y 2 , · · · , y n }. Then an obvious estimator for µ is the sample mean: n∑ ˆµ = ȳ = y i /n. i=1 Theorem 2.2.1 Proof. (i). By an ealier theorem, (i) E(y i ) = µ, V ar(y i ) = σ 2 . (ii) Cov(y i , y j ) = − σ2 , for i ≠ j. N − 1 N∑ N∑ 1 E(y i ) = u k P (y i = u k ) = u k k=1 k=1 N = µ. N∑ N∑ V ar(y i ) = (u k − µ) 2 P (y i = u k ) = (u k − µ) 2 1 k=1 k=1 N = σ2 . (ii). By defintion, Cov(y i , y j ) = E(y i y j ) − E(y i )E(y j ) = E(y i y j ) − µ 2 . Now, E(y i y j ) = = = = ∑ u s u t P (y i = u s , y j = u t ) = ∑ all s ≠ t all s ≠ t ⎡ ⎤ 1 ⎢ ∑ ⎥ ⎣ u s u t ⎦ = N(N − 1) 1 N(N − 1) 1 N(N − 1) u s u t − ∑ all s, t s=t [ ( (Nµ) 2 ∑ N − 1 N(N − 1) )] (u s − µ) 2 + Nµ 2 s=1 u s u t 1 N(N − 1) [ (Nµ) 2 − Nσ 2 − Nµ 2] = − σ2 N − 1 + µ2 . Thus, Cov(y i , y j ) = E(y i y j ) − µ 2 = − σ2 N−1 . 6 [( ∑ N ) ( N ) ] ∑ N∑ u s u t − u 2 s s=1 t=1 s=1
Page 1 and 2: ST3239: Survey Methodology by Wang
Page 3 and 4: 1.3 Why sample? If a sample is equa
Page 5: Proof. By the definition of s.r.s,
Page 9 and 10: Summary 1. How to draw a simple ran
Page 11 and 12: Central limit theorem: If n → ∞
Page 13 and 14: e.g. Suppose that a total of 1500 s
Page 15 and 16: 2.3.2 Estimation of population tota
Page 17 and 18: 2.4 Estimation of population propor
Page 19 and 20: Solution 2.5 Comparing estimates Su
Page 21 and 22: Proof. Note that ˆp 1 = X/n and ˆ

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