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Consider a r.s. {X₁: i = 1,2...,} of X ~ Binomial (1,p) with sample size n> 1 and unknown p € (0, 1). √n(Xn-p) d √Xn(1-x₂) (a) Show that prove it. (b) Use the result in (a) to derive the random interval for p with a probability 1 - a to include p. (c) Use the result in (b) to construct the 96% C.I. for p with n = 100 and Σi xi = 99. Then, compare the resulting interval with the possible range of p. What can you observe? (d) Consider a function g(t) = ln(- In t) with Delta method to get a "better" interval guess (96% C.I.) for ->> N(0,1). State clearly what large-n results (say WLLN, CLT, CMT, etc.) you use to

Consider a r.s. {X₁: i = 1,2...,} of X ~ Binomial (1,p) with sample size n> 1 and unknown p € (0, 1). √n(Xn-p) d √Xn(1-x₂) (a) Show that prove it. (b) Use the result in (a) to derive the random interval for p with a probability 1 - a to include p. (c) Use the result in (b) to construct the 96% C.I. for p with n = 100 and Σi xi = 99. Then, compare the resulting interval with the possible range of p. What can you observe? (d) Consider a function g(t) = ln(- In t) with Delta method to get a "better" interval guess (96% C.I.) for ->> N(0,1). State clearly what large-n results (say WLLN, CLT, CMT, etc.) you use to

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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Q4: Consider a r.s. {X₁: i = 1,2 ...,} of X~ Binomial (1,p) with sample size n > 1 and unknown p € (0, 1). √n(Xn-p) d √Xn (1-x₂) →N(0,1). State clearly what large-n results (say WLLN, CLT, CMT, etc.) you use to (a) Show that prove it. (b) Use the result in (a) to derive the random interval for p with a probability 1- a to include p. (c) Use the result in (b) to construct the 96% C.I. for p with n = 100 and Σi xi = 99. Then, compare the resulting interval with the possible range of p. What can you observe? (d) Consider a function g(t) = ln(- In t) with Delta method to get a "better" interval guess (96% C.I.) for p with n = 100 and Σ₁ x₁ = 99. ("Better" here means that the interval will not include any values out of the range of p.]
Transcribed Image Text:Q4: Consider a r.s. {X₁: i = 1,2 ...,} of X~ Binomial (1,p) with sample size n > 1 and unknown p € (0, 1). √n(Xn-p) d √Xn (1-x₂) →N(0,1). State clearly what large-n results (say WLLN, CLT, CMT, etc.) you use to (a) Show that prove it. (b) Use the result in (a) to derive the random interval for p with a probability 1- a to include p. (c) Use the result in (b) to construct the 96% C.I. for p with n = 100 and Σi xi = 99. Then, compare the resulting interval with the possible range of p. What can you observe? (d) Consider a function g(t) = ln(- In t) with Delta method to get a "better" interval guess (96% C.I.) for p with n = 100 and Σ₁ x₁ = 99. ("Better" here means that the interval will not include any values out of the range of p.]
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