Worked Example 6

The random variable \(X\) has mean \(25\) and standard deviation \(4.5\). A random sample of \(500\) observations is taken. Find the value of \(a\), correct to \(3\) decimal places, such that \(\mathrm{P}\left( \overline{X}>a \right)= 0.75\).

Explain whether it is necessary to apply the Central Limit Theorem in this case.