12 The variable \(X\) has the distribution \(\mathrm { B } \left( 50 , \frac { 1 } { 6 } \right)\). The probabilities \(\mathrm { P } ( X = r )\) for \(r = 0\) to 50 are given by the terms of the expansion of \(( a + b ) ^ { n }\) for specific values of \(a , b\) and \(n\).
- State the values of \(a\), \(b\) and \(n\).
A student has an ordinary 6 -sided dice. They suspect that it is biased so that it shows a 2 on fewer throws than it would if it were fair. In order to test the suspicion the dice is thrown 50 times and the number of 2 s is noted. The student then carries out a hypothesis test at the \(5 \%\) significance level.
- Write down suitable hypotheses for the test.
- Determine the rejection region for the test, showing the values of any relevant probabilities.