8 The proportion of left-handed adults in a country is known to be \(15 \%\). It is suggested that for mathematicians the proportion is greater than \(15 \%\). A random sample of 12 members of a university mathematics department is taken, and it is found to include five who are left-handed.

1. Stating your hypotheses, test whether the suggestion is justified, using a significance level as close to \(5 \%\) as possible.
2. In fact the significance test cannot be carried out at a significance level of exactly \(5 \%\). State the probability of making a Type I error in the test.
3. Find the probability of making a Type II error in the test for the case when the proportion of mathematicians who are left-handed is actually \(20 \%\).
4. Determine, as accurately as the tables of cumulative binomial probabilities allow, the actual proportion of mathematicians who are left-handed for which the probability of making a Type II error in the test is 0.01 .