1 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.

1. Find the probability that at least 4 of the 17 applicants are successful.
2. Find the expected number of successful applicants in the sample.
3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
4. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.