4. Let X₁,..., Xn N(μ,²). In this question, we will construct confidence interval for ². Let C~x²(r), and as usual for any a € [0, 1] we denote x² (r) to be P(C> x²(r)) = a. (a) Suppose that is known. By considering the distribution of μ Σ(*74), prove that i.i.d. prove that Σï-1(X₁ − µ¹)² Σï-1 (Xi − µ)² Xa/2(n) X₁-a/2 (n) is a 100(1-a) % confidence interval for o². (b) Suppose that is unknown. By considering the distribution of Σ(**). ;(*²=X) ²³. = 2 (n-1)S² 02 Σ#1(X - X) Σ(Χ. – X)2] X/2(n-1) X₁-a/2(n-1) is a 100(1-a) % confidence interval for o². (c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a) % confidence interval for a.
4. Let X₁,..., Xn N(μ,²). In this question, we will construct confidence interval for ². Let C~x²(r), and as usual for any a € [0, 1] we denote x² (r) to be P(C> x²(r)) = a. (a) Suppose that is known. By considering the distribution of μ Σ(*74), prove that i.i.d. prove that Σï-1(X₁ − µ¹)² Σï-1 (Xi − µ)² Xa/2(n) X₁-a/2 (n) is a 100(1-a) % confidence interval for o². (b) Suppose that is unknown. By considering the distribution of Σ(**). ;(*²=X) ²³. = 2 (n-1)S² 02 Σ#1(X - X) Σ(Χ. – X)2] X/2(n-1) X₁-a/2(n-1) is a 100(1-a) % confidence interval for o². (c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a) % confidence interval for a.
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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![4. Let X₁,..., Xn N(μ,2). In this question, we will construct confidence interval for ². Let
C~ x²(r), and as usual for any a € [0, 1] we denote x²(r) to be
P(C>x²(r)) = a.
(a) Suppose that μ is known. By considering the distribution of
prove that
i.i.d.
prove that
Xi
Σ(^-^)".
n
i=1
[Σï_₁(Xi −µ²)² Σï-₁ (Xi − µ)²]
X²/2(n)
Xỉa/z(n)
is a 100(1-a) % confidence interval for o².
(b) Suppose that is unknown. By considering the distribution of
2
Σ(**)
(X-X) (-1).5²
=
[Σ,(Χ. – X) Σ(Χ. – X)2]
X²/2(n-1) ¹X₁-a/2 (n − 1)
is a 100(1-a) % confidence interval for o².
(c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a)%
confidence interval for o.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3aa9238-d0bd-456b-acda-df0db22aaea2%2Feac3e39e-539e-4cef-8921-3644d73a62cf%2Fvsdazw_processed.png&w=3840&q=75)
Transcribed Image Text:4. Let X₁,..., Xn N(μ,2). In this question, we will construct confidence interval for ². Let
C~ x²(r), and as usual for any a € [0, 1] we denote x²(r) to be
P(C>x²(r)) = a.
(a) Suppose that μ is known. By considering the distribution of
prove that
i.i.d.
prove that
Xi
Σ(^-^)".
n
i=1
[Σï_₁(Xi −µ²)² Σï-₁ (Xi − µ)²]
X²/2(n)
Xỉa/z(n)
is a 100(1-a) % confidence interval for o².
(b) Suppose that is unknown. By considering the distribution of
2
Σ(**)
(X-X) (-1).5²
=
[Σ,(Χ. – X) Σ(Χ. – X)2]
X²/2(n-1) ¹X₁-a/2 (n − 1)
is a 100(1-a) % confidence interval for o².
(c) In the same setting as part (b), that is, suppose that is unknown. Construct a 100(1-a)%
confidence interval for o.
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