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.

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.

opatovaL 2021-05-23 Answered

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.

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Expert Answer

AGRFTr

Answered 2021-05-24 Author has 95 answers

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
$P (\overset{―}{x} + Z_{0.025} \times \frac{σ}{n} < 0 < \overset{―}{x} + Z_{0.975} \times \frac{σ}{n}) = 0.95$
The confidence interval is
$(\overset{―}{x} - 1.96 \times \frac{σ}{n}, \overset{―}{x} + 1.96 \times \frac{σ}{n})$

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