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.

Random variables
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.

2021-05-24

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
$$(\overline{x}-1.96\times\frac{\sigma}{n}, \overline{x}+1.96\times \frac{\sigma}{n})$$