5 A fair six-sided die has faces numbered \(1,2,3,4,5,6\). The score on one throw is denoted by \(X\).
- Write down the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 35 } { 12 }\).
Fayez has a six-sided die with faces numbered \(1,2,3,4,5,6\). He suspects that it is biased so that when it is thrown it is more likely to show a low number than a high number. In order to test his suspicion, he plans to throw the die 50 times. If the mean score is less than 3 he will conclude that the die is biased.
- Find the probability of a Type I error.
- With reference to this context, describe circumstances in which Fayez would make a Type II error.