Checking the Independence.

Let \(X_i\sim N(\mu,\sigma^2) \) ; where \( [i=1,2,\ldots,n] \)
\(Z_i\sim N(0,1) \) ; where \( [i=1,2,\ldots,n] \)

Proof that \(\frac{(\bar X-\mu)}{\sigma}\) and \(\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}\) are independent, which implies \(\bar X\) and \(\sum_{i=1}^n(X_i-\bar X)^2\) are independent.

If i show that \(\bar X\) and \( (X_i-\bar X)\) are independent and draw conclusion that since \( \bar Z\) is only the function of \(\bar X\) and \((Z_i-\bar Z)^2\) is only the function of \( (X_i-\bar X)\), so \( \bar Z\) and \(\sum(Z_i-\bar Z)^2\) are independent. Is it suffices, ie, is the procedure to check the independence correct?
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