Question

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.

Answer 1

If \(q(X,0)\sim N(0,1)\) is the Pivotal function for θ,
This implies the 95% confidence interval can be given by
\(P(Z_{0.025})\)
Now simplify this expression until an expression of this kind is obtained
\(P(Z_{0.025}\times A<0\)
Where A and B are constants.
For example, for the pivotal function formed in (b), Confidence Interval can be given by
\(P(\overline{x}+Z_{0.025}\times\frac{\sigma}{n}<0<\overline{x}+Z_{0.975}\times\frac{\sigma}{n})=0.95\)
The confidence interval is
\((\overline{x}-1.96\times\frac{\sigma}{n}, \overline{x}+1.96\times \frac{\sigma}{n})\)