Exercise 17. Let X₁,..., Xn~ N(μ, o²) be i.i.d. where u is known, but o2 is unknown. Consider the estimate for the variance. a) Show that 7² C := = 1 n Σ(Χ; - μ)2 i=1 72 2 Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x² (n) by qn (a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o24 vs. H₁: 024 at significance level 5%.
Exercise 17. Let X₁,..., Xn~ N(μ, o²) be i.i.d. where u is known, but o2 is unknown. Consider the estimate for the variance. a) Show that 7² C := = 1 n Σ(Χ; - μ)2 i=1 72 2 Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x² (n) by qn (a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o24 vs. H₁: 024 at significance level 5%.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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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
n
1
Σ(X₁-1
for the variance.
a) Show that
7²
C :=
i=1
-μ)²
72
2
·Σ( X ₁ = μ)² ~ x² (n).
O
i=1
b) For a € (0, 1), denote the a-quantile of x2 (n) by qn(a). Using this notation, find numbers
an, bn E R such that P(o² <ano²) = 2.5% and P(02>bno²) = 2.5%.
c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o
against the alternative H₁: 02 08.
d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis
Ho: o² = 4 vs. H₁: o² #4 at significance level 5%.
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