In the past, one quarter of job applicants at a certain firm had first-class degrees. A change is made in the job description and a director of the firm believes that, on average, the proportion of job applicants with first-class degrees has decreased.

In the month following the change, there were 35 job applicants, and \(r\) of these had first-class degrees. The firm carried out a hypothesis test at the 4% significance level to test the director’s belief.

(a) Use a binomial distribution to find the largest value of \(r\) that would provide sufficient evidence that the director’s belief is correct. [6]

In another month, the director carries out a similar test at the 4% significance level using the 35 job applicants from that month.

(b) Explain the meaning of a Type I error in this context, and state the probability of a Type I error. [2]

(c) Given that the proportion of job applicants with first-class degrees this year is actually 0.05, find the probability of a Type II error. [2]