Sampling From A Non-Normal Population (Central Limit Theorem)

A random sample of \(40\) observations is selected from a population with mean \(20\) and standard deviation \(2.9\). Given that \(\bar{X}\) is the sample mean, state the approximate distribution of \(\bar{X}\). Hence, find

1. \(\mathrm{P}\left( \bar{X}\le 20.7 \right)\);
2. \(\mathrm{P}\left( \bar{X}>18.5 \right)\);
3. \(\mathrm{P}\left( 14<\bar{X}\le 19.5 \right)\);
4. the value of \(a\) if \(\mathrm{P}\left( \bar{X}>a \right)=0.05\).