Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where μ is known, but o2 is unknown. Consider the estimate for the variance. a) Show that ỡ² = 1 n i=1 c) Using these results, dev against the alternative H₁: 0² # of. (Xi -μ)² n C:= - Σ( X ₁ - 4) 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(ỡ² < ano²) = 2.5% and P(ỡ² > bno²) = 2.5%. test which can be used to test the hypothesis Ho: o² = of d) Assume we have n = 100, μ = 0 and we have observed ² = 4.81. Test the hypothesis Ho: o2 = 4 vs. H₁: 0² ‡ 4 at significance level 5%.
Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where μ is known, but o2 is unknown. Consider the estimate for the variance. a) Show that ỡ² = 1 n i=1 c) Using these results, dev against the alternative H₁: 0² # of. (Xi -μ)² n C:= - Σ( X ₁ - 4) 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(ỡ² < ano²) = 2.5% and P(ỡ² > bno²) = 2.5%. test which can be used to test the hypothesis Ho: o² = of d) Assume we have n = 100, μ = 0 and we have observed ² = 4.81. Test the hypothesis Ho: o2 = 4 vs. H₁: 0² ‡ 4 at significance level 5%.
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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could you please help me with part b and c , could you please write written explanations
Transcribed Image Text:Exercise 17. Let X₁,..., Xn ~ N(µ, o²) be i.i.d. where µ is known, but o² is unknown. Consider
the estimate
for the variance.
a) Show that
n
z² = ² Σ(X₁ −
-
n
i=1
C:=
:=
µ)²
n
2
Σ (Xi − μ)² ~ x² (n).
O
i=1
b) For a € (0, 1), denote the a-quantile of x²(n) by qan(a). Using this notation, find numbers
an, bn E R such that P(õ² < ano²) = 2.5% and P(õ² > bno²) = 2.5%.
c) Using these results, develop a test which can be used to test the hypothesis Ho: 0²
against the alternative H₁: 0² ‡ 0².
=
=
d) Assume we have n =
100, μ
0 and we have observed ²
Ho: 0²
= 4 vs. H₁: 0² ‡ 4 at significance level 5%.
=
0²
4.81. Test the hypothesis
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