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2. This exercise is to provide more details to the arguments leading to V (r) = (N), nN where (1) n r = -1 and s² = (yi – rx;)², Σ - (2) i=1 see page 8 of Lecture 8. - Let zi = yi — Rx; and consider a simple random sample from a population 21,..., Zn Z1, ZN (3) (4) Let μz be the population mean, σ the population variance, z the sample με mean and s² the sample variance. (a) Show that μz = 0 and express s½ in terms of the quantities in (3). Express in terms by x, y and R. (b) Using the formulas in Lecture 3, find an expression for Ŵ(z) in terms of s, n and N, that is unbiased for V(z). (c) Expand - - με r − R = 1 (r − R) + (1 − ) (r − R). - - (5) με Show that (r - R) = (6) με με It can be argued that the second term in (5) is small compared to the first term. This is because ñ is a good estimator of μx and so is close to μx when n is large. Since R is a constant, it follows that - V (r) = V (r − R) ≈ V(2), (7) where the approximation is due to ignoring the second term in (5).

2. This exercise is to provide more details to the arguments leading to V (r) = (N), nN where (1) n r = -1 and s² = (yi – rx;)², Σ - (2) i=1 see page 8 of Lecture 8. - Let zi = yi — Rx; and consider a simple random sample from a population 21,..., Zn Z1, ZN (3) (4) Let μz be the population mean, σ the population variance, z the sample με mean and s² the sample variance. (a) Show that μz = 0 and express s½ in terms of the quantities in (3). Express in terms by x, y and R. (b) Using the formulas in Lecture 3, find an expression for Ŵ(z) in terms of s, n and N, that is unbiased for V(z). (c) Expand - - με r − R = 1 (r − R) + (1 − ) (r − R). - - (5) με Show that (r - R) = (6) με με It can be argued that the second term in (5) is small compared to the first term. This is because ñ is a good estimator of μx and so is close to μx when n is large. Since R is a constant, it follows that - V (r) = V (r − R) ≈ V(2), (7) where the approximation is due to ignoring the second term in (5).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2. This exercise is to provide more details to the arguments leading to V (r) = (N), nN where (1) n r = -1 and s² = (yi – rx;)², Σ - (2) i=1 see page 8 of Lecture 8. - Let zi = yi — Rx; and consider a simple random sample from a population 21,..., Zn Z1, ZN (3) (4) Let μz be the population mean, σ the population variance, z the sample με mean and s² the sample variance.
Transcribed Image Text:2. This exercise is to provide more details to the arguments leading to V (r) = (N), nN where (1) n r = -1 and s² = (yi – rx;)², Σ - (2) i=1 see page 8 of Lecture 8. - Let zi = yi — Rx; and consider a simple random sample from a population 21,..., Zn Z1, ZN (3) (4) Let μz be the population mean, σ the population variance, z the sample με mean and s² the sample variance.
(a) Show that μz = 0 and express s½ in terms of the quantities in (3). Express in terms by x, y and R. (b) Using the formulas in Lecture 3, find an expression for Ŵ(z) in terms of s, n and N, that is unbiased for V(z). (c) Expand - - με r − R = 1 (r − R) + (1 − ) (r − R). - - (5) με Show that (r - R) = (6) με με It can be argued that the second term in (5) is small compared to the first term. This is because ñ is a good estimator of μx and so is close to μx when n is large. Since R is a constant, it follows that - V (r) = V (r − R) ≈ V(2), (7) where the approximation is due to ignoring the second term in (5).
Transcribed Image Text:(a) Show that μz = 0 and express s½ in terms of the quantities in (3). Express in terms by x, y and R. (b) Using the formulas in Lecture 3, find an expression for Ŵ(z) in terms of s, n and N, that is unbiased for V(z). (c) Expand - - με r − R = 1 (r − R) + (1 − ) (r − R). - - (5) με Show that (r - R) = (6) με με It can be argued that the second term in (5) is small compared to the first term. This is because ñ is a good estimator of μx and so is close to μx when n is large. Since R is a constant, it follows that - V (r) = V (r − R) ≈ V(2), (7) where the approximation is due to ignoring the second term in (5).
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