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.

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&lt;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&lt;0&lt;x―+Z0.975×σn)=0.95 The confidence interval is (x―−1.96×σn,x―+1.96×σn)

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opatovaL

Answered question

2021-05-23

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 & Explanation

AGRFTr

Skilled2021-05-24Added 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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