The events \(\mathrm { A } , B\) are such that \(P ( A ) = 0.2 , P ( B ) = 0.3\). Determine the value of \(P ( A \cup B )\) when
\(A , B\) are mutually exclusive,
\(A , B\) are independent,
\(\quad A \subset B\).
Dewi, a candidate in an election, believes that \(45 \%\) of the electorate intend to vote for him. His agent, however, believes that the support for him is less than this. Given that \(p\) denotes the proportion of the electorate intending to vote for Dewi,
state hypotheses to be used to resolve this difference of opinion.
They decide to question a random sample of 60 electors. They define the critical region to be \(X \leq 20\), where \(X\) denotes the number in the sample intending to vote for Dewi.
Determine the significance level of this critical region.
If in fact \(p\) is actually 0.35 , calculate the probability of a Type II error.
Explain in context the meaning of a Type II error.
Explain briefly why this test is unsatisfactory. How could it be improved while keeping approximately the same significance level?