Random variables X_{1},X_{2},...,X_{n} are independent and identically distributed. 0 is a parameter of their distribution. If q(X,0)\sim N(0,1) is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.

opatovaL 2021-05-23 Answered
Random variables X1,X2,...,Xn are independent and identically distributed. 0 is a parameter of their distribution.
If q(X,0)N(0,1) is a pivotal function for 0, explain how you would use this result to obtain a symmetrical 95% confidence interval for 0.
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AGRFTr
Answered 2021-05-24 Author has 95 answers

If q(X,0)N(0,1) is the Pivotal function for θ,
This implies the 95% confidence interval can be given by
P(Z0.025)
Now simplify this expression until an expression of this kind is obtained
P(Z0.025×A<0)
Where A and B are constants.
For example, for the pivotal function formed in (b), Confidence Interval can be given by
P(x+Z0.025×σn<0<x+Z0.975×σn)=0.95
The confidence interval is
(x1.96×σn,x+1.96×σn)

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