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Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o² is unknown. Consider the estimate for the variance. a) Show that 7² == n n i=1 (Xi − µ)² - - TL C:= = 2 ( x ₁ = 1) ²³ ~ x² (n). i=1 b) For a € (0, 1), denote the a-quantile of x²(n) by qn(a). Using this notation, find numbers an, bn ER such that P(02 b₂0²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o2 = 0 against the alternative H₁:0² +0. d) Assume we have n = 100,= 0 and we have observed ² = 4.81. Test the hypothesis Ho: 024 vs. H₁: 024 at significance level 5%. =

Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o² is unknown. Consider the estimate for the variance. a) Show that 7² == n n i=1 (Xi − µ)² - - TL C:= = 2 ( x ₁ = 1) ²³ ~ x² (n). i=1 b) For a € (0, 1), denote the a-quantile of x²(n) by qn(a). Using this notation, find numbers an, bn ER such that P(02 b₂0²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o2 = 0 against the alternative H₁:0² +0. d) Assume we have n = 100,= 0 and we have observed ² = 4.81. Test the hypothesis Ho: 024 vs. H₁: 024 at significance level 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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Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o² is unknown. Consider the estimate for the variance. a) Show that 7² n == - Σ - n i=1 (Xi − µ)² 2 = [(x ² − ¹) ²³ ~ x² (n). i=1 C:= b) For a € (0, 1), denote the a-quantile of x²(n) by qn(a). Using this notation, find numbers an, bn ER such that P(02 <ano2) = 2.5% and P(02>b₂o²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o2 = 0 against the alternative H₁:0² +0. d) Assume we have n = 100,= 0 and we have observed ² = 4.81. Test the hypothesis Ho: 24 vs. H₁: 024 at significance level 5%.
Transcribed Image Text:Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o² is unknown. Consider the estimate for the variance. a) Show that 7² n == - Σ - n i=1 (Xi − µ)² 2 = [(x ² − ¹) ²³ ~ x² (n). i=1 C:= b) For a € (0, 1), denote the a-quantile of x²(n) by qn(a). Using this notation, find numbers an, bn ER such that P(02 <ano2) = 2.5% and P(02>b₂o²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o2 = 0 against the alternative H₁:0² +0. d) Assume we have n = 100,= 0 and we have observed ² = 4.81. Test the hypothesis Ho: 24 vs. H₁: 024 at significance level 5%.
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