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(Solved): Refer to QUESTION 1. (a) What are the distributions of: (i) (x_(i))/(\sigma \sqrt(i)); (ii) (x_(i)^( ...
Refer to QUESTION 1. (a) What are the distributions of: (i)
(x_(i))/(\sigma \sqrt(i)); (ii)
(x_(i)^(2))/(\sigma ^(2)i); and (iii)
\sum_(i=1)^n (x_(i)^(2))/(\sigma ^(2)i). (b) Prove or disprove that the expression of the asymptotic confidence interval (
nlongrightarrow\infty ) for
\sigma ^(2)is
hat(\sigma )^(2)(1 -1.96\sqrt((2)/(n)))where
del^(2)is the maximum likelihood estimate of
\sigma ^(2)found in Question 1 (d) (c) Given:
n=30 and Q=(30hat(\sigma )^(2))/(\sigma ^(2))?\chi _(30)^(2)where
hat(\sigma )^(2)is the maximum likelihood estimate of
\sigma ^(2)found in Question 1 (d) when
n=30. Find
aand
bwhich satisfy
0.025=P(Q<=a)=P(Q>=b)and hence write down a
95%confidence interval for
\sigma ^(2). (d) How can the
95%confidence interval in part (c) above be used to test the hypotheses
H_(0):
\sigma ^(2)=1versus
H_(1):\sigma ^(2)!=1? (e) Evaluate your
95%confidence interval in part (b) at
n=30and compare it with that in part (c). Which is better? Justify you answer (4)
