The random variable \(R\) has the distribution \(\mathrm { B } ( 6 , p )\). A random observation of \(R\) is found to be 6. Carry out a \(5 \%\) significance test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.45\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p \neq 0.45\), showing all necessary details of your calculation.
The random variable \(S\) has the distribution \(\mathrm { B } ( n , p ) . \mathrm { H } _ { 0 }\) and \(\mathrm { H } _ { 1 }\) are as in part (i). A random observation of \(S\) is found to be 1 . Use tables to find the largest value of \(n\) for which \(\mathrm { H } _ { 0 }\) is not rejected. Show the values of any relevant probabilities.