Since \(X_{1}, X_{2}, ... ,X_{n}\) are IID random Variables such that,

\(\overline{X}\sim N(0, \frac{1}{n})\)

This implies

\(X_{i}\sim N(0,1)\)

Now,

Pivotal function for Mean can be written as

\(Z=\frac{(\overline{x}-\mu)}{\frac{\sigma}{\sqrt{n}}}=\frac{(\overline{x}-0)}{\frac{1}{\sqrt{n}}}\)

\(\overline{X}\sim N(0, \frac{1}{n})\)

This implies

\(X_{i}\sim N(0,1)\)

Now,

Pivotal function for Mean can be written as

\(Z=\frac{(\overline{x}-\mu)}{\frac{\sigma}{\sqrt{n}}}=\frac{(\overline{x}-0)}{\frac{1}{\sqrt{n}}}\)