3 The mean of a sample of 80 independent observations of a continuous random variable \(Y\) is denoted by \(\bar { Y }\). It is given that \(\mathrm { P } ( \bar { Y } \leqslant 157.18 ) = 0.1\) and \(\mathrm { P } ( \bar { Y } \geqslant 164.76 ) = 0.7\).
- Calculate \(\mathrm { E } ( Y )\) and the standard deviation of \(Y\).
- State
(a) where in your calculations you have used the Central Limit Theorem,
(b) why it was necessary to use the Central Limit Theorem,
(c) why it was possible to use the Central Limit Theorem.